20240119/2307.08078v2.json

{
    "title": "Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives^*",
    "abstract": "In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders , ).\nThe proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only  storage and  computational complexity, where  denotes the total number of time steps. Then we use a Legendre\nspectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established.\nWe demonstrate that the proposed scheme is unconditionally stable and convergent with an order of , where , , and  represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively.\nNumerical results are presented to validate the theoretical predictions.",
    "sections": [
        {
            "section_id": "1",
            "parent_section_id": null,
            "section_name": "1. Introduction",
            "text": "Fractional differential equations have wide applications in various fields of science, including physics, economics, engineering, chemistry, biology and others[11  ###reference_11###, 27  ###reference_27###, 28  ###reference_28###, 29  ###reference_29###, 42  ###reference_42###]. There are many kinds of definitions for the fractional derivatives,\nthe most used fractional derivatives are the Riemann-Liouville fractional derivative and the Caputo fractional derivative [31  ###reference_31###, 18  ###reference_18###]. However, both of these operators still present challenges in practical applications.\nTo be more precise, the Riemann-Liouville derivative of a constant is non-zero and the Laplace transform of this derivative contains terms that lack physical significance. The Caputo fractional derivative has successfully addressed both issues, however, its definition involves a singular kernel which poses challenges in analysis and computation. Caputo and Fabrizio [8  ###reference_8###] have proposed a novel definition of the fractional derivative with a smooth kernel, referred to as the Caputo and Fabrizio (CF) derivatives, which present distinct representations for the temporal and spatial variables. The representation in time variable\nis suitable to use the Laplace transform, and the spatial representation is more convenient to use the Fourier transform.\nAlthough there is ongoing debate regarding the mathematical properties of fractional derivatives with non-singular kernels [20  ###reference_20###, 34  ###reference_34###], numerous scholars remain interested in studying differential equations involving such derivatives due to their nice performance in various applications.\nConsidering the CF derivative offers two primary advantages: 1) The utilization of a regular kernel in non-local systems is motivated by its potential to accurately depict material heterogeneities and fluctuations of various scales, which cannot be adequately captured by classical local theories or fractional models with singular kernels, see, e.g., [8  ###reference_8###, 9  ###reference_9###];\n2) CF derivatives have numerical advantages. As we know, the truncation error of the numerical calculation for fractional operators with singular kernels is typically dependent on the order . For instance, in the case of the Caputo fractional derivative, employing classical L1 discretization results in an error of order , which becomes highly unfavorable when . In order to enhance actuarial accuracy, the utilization of higher-order methods will lead to an increase in computational complexity, particularly for problems with high dimensions. However, within the same approximation framework, the CF derivative has a higher truncation error, see Remark 2.2  ###reference_em2###.\nFurther properties and diverse applications of this fractional derivative can be found in various references, such as [9  ###reference_9###, 14  ###reference_14###, 6  ###reference_6###, 5  ###reference_5###, 30  ###reference_30###, 36  ###reference_36###, 16  ###reference_16###].\nLet , . In this paper, we are concerned with the numerical approximation of the multi-term time-fractional diffusion equation\nwith the initial conditions\nand the boundary condition\nwhere\n, and .  and  are given sufficiently smooth functions in their respective domains.\nIn addition,  is the Caputo-Fabrizio derivative\noperator of order  [8  ###reference_8###, 9  ###reference_9###] defined as\nIf , then - reduces to the single-term time-fractional diffusion equation.\nThe model of -, which describes the temporal flow of water within a leaky aquifer at various scales [5  ###reference_5###, 12  ###reference_12###], as well as the electro-magneto-hydrodynamic flow of non-Newtonian biofluids with heat transfer [1  ###reference_1###], etc. For the well-posedness of -, we refer to, e.g., [3  ###reference_3###, 4  ###reference_4###, 36  ###reference_36###].\nMany researchers have explored the numerical approximation of both single-term and multi-term time fractional diffusion equations.\nIn [24  ###reference_24###], Liu et al. proposed a finite difference method for solving time-fractional diffusion equations in both space and time domains.\nLin and Xu [23  ###reference_23###] utilized a finite difference scheme in time and Legendre spectral methods in space to numerically solve the time-fractional diffusion equations. Subsequently, Li and Xu [22  ###reference_22###]improved upon their previous work by proposing a space-time spectral method for these equations.\nFor the numerical treatment of multi-term time-fractional diffusion equations, [33  ###reference_33###] proposed a fully-discrete schemes for one- and two-dimensional multi-term time fractional sub-diffusion equations. These schemes combine the compact difference method for spatial discretization with L1 approximation for time discretization.\nThe Galerkin finite element method and the spectral method were introduced in [19  ###reference_19###] and [40  ###reference_40###, 13  ###reference_13###], respectively.\nZhao et al. [39  ###reference_39###] developed a fully-discrete scheme for a class of two-dimensional multi-term time-fractional diffusion equations with Caputo fractional derivatives, utilizing the finite element method in spatial direction and classical L1 approximation in temporal direction.\nAkman et al. [2  ###reference_2###] proposed a numerical approximation called the L1-2 formula for the Caputo-Fabrizio derivative using quadratic interpolation.\nIn [37  ###reference_37###], finite difference/spectral approximations for solving two-dimensional time CF fractional diffusion equation were proposed and analyzed. Later, a second order scheme [10  ###reference_10###] was devised for addressing this problem.\nA compact alternating direction implicit (ADI) difference scheme was proposed by [35  ###reference_35###] for solving the two-dimensional time-fractional diffusion equation.\nSimulating models with fractional derivatives presents a challenge due to their non-locality, which significantly impedes algorithm efficiency and necessitates greater memory storage compared to traditional local models.\nIn particular, for fractional models, the computational complexity of obtaining an approximate solution is  and the required memory storage is , which contrasts with local models that have a complexity of  and require a memory storage of , where  denotes the total number of time steps, see, e.g., [23  ###reference_23###, 37  ###reference_37###, 10  ###reference_10###].\nTo address this issue, several researchers have proposed efficient algorithms for computing the derivatives of Riemann-Liouville, Caputo, and Riesz fractional operators, see e.g., [17  ###reference_17###, 38  ###reference_38###, 21  ###reference_21###, 41  ###reference_41###] and the references therein.\nRecently, a fast compact finite difference method for quasi-linear time-fractional parabolic equations is presented and analyzed in [25  ###reference_25###]. Then, [26  ###reference_26###] proposed a fast second-order numerical scheme for approximating the Caputo-Fabrizio fractional derivative at node  with computational complexity of  and memory storage of .\nInspired by the above mentioned, we extend finite difference/spectral approximations for the multi-term Caputo-Fabrizio time-fractional diffusion equation (1.1  ###reference_###)-(1.3  ###reference_###).\nFirstly, we present a L1 formula for the Caputo-Fabrizio derivative. In this context, we introduce two discrete fractional differential operators, namely  and , which are essentially equivalent.\nHowever,  effectively utilizes the exponential kernel and incurs lower storage and computational costs compared to .\nThe idea of this approach is essentially identical to that of reference [26  ###reference_26###], albeit with a slightly different formulation in our case; specifically, the approximation is centered at point  and presented in a more concise manner. The error bounds associated with these two operators will be examined in detail.\nSecondly, we develop a semi-discrete scheme based on finite difference method for multi-term time-fractional derivatives, with complete proofs of its unconditional stability and convergence rate.\nA detailed error analysis is carried out for the semi-discrete problem, showing that the temporal accuracy is second order.\nFinally, we present the fully-discrete scheme based on the Legendre spectral collocation method for spatial discretization. We will investigate both the convergence order of this method and its implementation efficiency, while providing a rigorous proof of its spectral convergence in this paper.\nThe rest of this paper is organized as follows. In Section 2, a semi-discrete scheme is proposed for (1.1  ###reference_###)-(1.3  ###reference_###) based on fast L1 finite difference scheme. The stability and convergence analysis of the semi-discrete scheme is presented. In Section 3,\nwe construct a Legendre spectral collocation method for the spatial discretization of the semi-discrete scheme.\nError estimates are provided for the full discrete problem. Some numerical results are reported in Section 4. Finally, the conclusions are given in Section 5."
        },
        {
            "section_id": "2",
            "parent_section_id": null,
            "section_name": "2. Semi-discretization",
            "text": "Define , , where  is the time step."
        },
        {
            "section_id": "2.1",
            "parent_section_id": "2",
            "section_name": "2.1. Fast L1 formula for Caputo-Fabrizio derivative",
            "text": "We first give L1 approximation for fractional Caputo-Fabrizio derivative of function  defined by\nIn order to simplify the notations, we denote  for . The L1 formula is obtained by substituting the linear Lagrange interpolation of  into (2.1  ###reference_###). Precisely, the linear approximation of the function  on  is written as\nand the error in the approximation is\nThen we define the discrete fractional differential operator  by\nwhere  and\nThe right hand side of (2.4  ###reference_###) involves a sum of all previous solutions , which reflects the memory effect of the non-local fractional derivative. Thus it requires on average  storage and the total computational cost is  with  the total number of time steps. This makes both the computation and memory expensive, specially in the case of\nlong time integration. In order to overcome this difficulty, we propose a further approach to the fractional derivative.\nThe idea consists in first splitting the convolution integral in (2.1  ###reference_###) into a sum of history part and local part as follows:\nNote that a comparable treatment is employed in reference [17  ###reference_17###].\nThen the history part  can be rewritten as\nhence we have\nUsing the simple recurrence relation (2.5  ###reference_###), we define the discrete fractional differential operator  by\nIt is not difficult to see that  for . Comparing  in (2.4  ###reference_###) with  in (2.6  ###reference_###)-(2.7  ###reference_###), the former requires all the previous time step values of  while the latter only needs ,  and . This implies that approximating  by  considerably\nreduces the storage and computational costs as compared to . Roughly speaking, replacing  by  allows to reduce the storage cost\nfrom  to , and the computational cost from  to .\nThe fast algorithm of Caputo derivative in [17  ###reference_17###] should be noted to retain an additional truncation error , whereas the fast algorithm of CF derivative does not introduce this error. Furthermore, it is worth mentioning that other algorithms, such as parallel computational methods [15  ###reference_15###], result in an augmented spatial complexity.\nThe following lemma provides an error bound for approximation .\nSuppose that . For any , let\nThen\nProof. We consider proving the following estimate by mathematical induction:\nFirst we have\nwhere . Therefore, (2.8  ###reference_###) holds for . Now suppose that (2.8  ###reference_###) holds for , we need to prove that it holds also for . Similar to the proof of , we can easily get that\nBy combining (2.5  ###reference_###) and (2.7  ###reference_###), we obtain\nThe estimate (2.8  ###reference_###) is proved. Hence\nwhich prove the conclusion of the lemma.\nThe second rate of convergence of L1 formula has been proven in [2  ###reference_2###] by different methods, here we obtained identical results herein. Note that the rate of convergence of L1 formula for classical Caputo fractional derivative with order  is , this result seems reasonable since Caputo-Fabrizio derivative has smooth kernel."
        },
        {
            "section_id": "2.2",
            "parent_section_id": "2",
            "section_name": "2.2. Discretization in time",
            "text": "We denote  and . Then from (2.6  ###reference_###)-(2.7  ###reference_###) and Lemma 2.1  ###reference_em1###, the time fractional derivative (1.5  ###reference_###) at  can be approximated by\nwhere\nThen Eq.(1.1  ###reference_###) can be rewritten as\nwhere\nwith  for . Notice that\nwe denote\nthe above equations are recast into\nLet  be the approximation for , and . Then the semi-discrete problem of Eq. (1.1  ###reference_###) can be written as\nwhere\nMoreover, by utilizing relation\nwe can easily derive an alternative formulation of (2.15  ###reference_###)-(2.18  ###reference_###) as follows\nwhere\nSince , (2.19  ###reference_###)-(2.22  ###reference_###) can be also obtained by using  in Eq.(1.1  ###reference_###).\nIt is noteworthy that equations (2.15  ###reference_###)-(2.18  ###reference_###) offer computational advantages over equations (2.19  ###reference_###)-(2.22  ###reference_###). This is primarily attributed to the straightforward recurrence relation presented in equations (2.6  ###reference_###) and (2.7  ###reference_###). However, (2.19  ###reference_###)-(2.22  ###reference_###) is more appropriate for our analysis than (2.15  ###reference_###)-(2.18  ###reference_###), hence it play a crucial role in the subsequent sections.\nLet  be defined by (2.12  ###reference_###), then there exists a constant  such that\nProof. Without loss of generality, we assume that . By the definition of  and the inequalities of (2.10  ###reference_###), we have\nOn the other hand, since\nfor , we get\nThis implies that\nTherefore, there exists a constant  such that\nwhich prove the conclusion of the lemma.\nLet the coefficients  be defined by (2.9  ###reference_###), then for every ,\nProof.  can be easily obtained by the definition of  and the monotone property of the function . Finally, note that\nUsing the above equalities and the fact  completes the proof of the lemma.\n(2.26  ###reference_###) gives a easy way to compute all the coefficients .\nLet the coefficients  be defined by (2.23  ###reference_###), then\nProof. By Lemma 2.2  ###reference_em2###, and the definition of , we can readily arrive at these conclusions."
        },
        {
            "section_id": "2.3",
            "parent_section_id": "2",
            "section_name": "2.3. Stability and convergence analysis of the semi-discrete scheme",
            "text": "To discuss the stability and convergence of the semi-discrete scheme, we introduce functional spaces equipped with standard norms and inner products that will be utilized subsequently. Let  is the space of measurable functions whose square is Lebesgue integrable in . Then\nThe inner products of  and  are defined, respectively, by\nand the corresponding norms by\nThe norm  of the space   is defined by\nIn this paper, instead of using the above standard -norm, we prefer to define  by\nIt is widely acknowledged that the standard -norm and the norm defined by (2.27  ###reference_###) are equivalent; therefore, we will adopt the latter in subsequent discussions.\nThe variational (weak) formulation of the Eqs.(2.15  ###reference_###) and (2.16  ###reference_###)/(2.20  ###reference_###), subject to the boundary condition (2.18  ###reference_###), can be expressed as finding  such that for\nwhere .\nFor the semi-discretized problem (2.28  ###reference_###)-(2.29  ###reference_###), we can establish a stability result as follows.\nThe semi-discretized problem (2.28  ###reference_###)-(2.29  ###reference_###) is unconditionally stable in the sense that for all , it holds\nProof. By mathematical induction. First of all, when , we have\nNotice that , taking  and using the Cauchy-Schwarz inequality, we obtain immediately\nNow, suppose\nTaking  in (2.29  ###reference_###) gives\nHence, by using (2.30  ###reference_###) and Lemma 2.3  ###reference_em3###, we have\nThus, the proof is completed.\nIn the proof of the following theorem, we will demonstrate that  is bounded. As shown in equation (2.25  ###reference_###), . Therefore, it follows that  is also bounded.\nWe now conduct an error analysis for the solution of the semi-discretized problem.\nAssuming  and .\nLet  be the exact solution of (1.1  ###reference_###)-(1.3  ###reference_###),  be the solution of semi-discretized problem (2.28  ###reference_###)-(2.29  ###reference_###) with initial condition , then the following error estimates hold:\nwhere the constant  is defined in (2.24  ###reference_###) and  is the length of .\nProof. We shall establish the following estimate through a process of mathematical induction:\nLet , . By combining (2.13  ###reference_###) and (2.28  ###reference_###), the error  satisfies\nTaking  yields . This, in conjunction with (2.24  ###reference_###), yields\nTherefore, (2.32  ###reference_###) holds for . Assuming that (2.32  ###reference_###) holds for all , it is necessary to demonstrate its validity for . By combining (2.13  ###reference_###), (2.14  ###reference_###) and (2.29  ###reference_###), for , we have\nLet  in the above equation, then\nUsing the induction assumption and Lemma 2.3  ###reference_em3###, we derive\nNext we show that  is bounded. Considering that function  is decreasing on , we have\nby combining equation (2.26  ###reference_###).\nTherefore,\nConsequently we obtain, for all  such that ,"
        },
        {
            "section_id": "3",
            "parent_section_id": null,
            "section_name": "3. Full discretization",
            "text": ""
        },
        {
            "section_id": "3.1",
            "parent_section_id": "3",
            "section_name": "3.1. A shifted Legendre collocation method in space",
            "text": "We shall begin by providing a comprehensive overview of fundamental definitions and properties pertaining to Legendre Gauss-type quadratures.\nLet  denote the space of algebraic polynomials of degree less than or\nequal to  with respect to variable , and  be the Legendre polynomial\nof degree  on the interval . Then the discrete space, denoted by .\nLet  be the -orthogonal projection operator from  into , associated to the norm  defined in (2.27  ###reference_###), that is, for all , define , such that, ,\nFrom [7  ###reference_7###], the following estimate of projection holds:\nDefine the Legendre-Gauss-Lobatto nodes and weights as  and , , ,\nwhere  are the zeroes of , and\nMoreover, the following quadrature holds\nThe discrete inner product and norm defined as follow, for any continuous functions ,\nFrom [32  ###reference_32###], the discrete norm  is equivalent to the standard -norm in . If we denote \nand  as the nodes and\nweights of shifted Legendre-Gauss-Lobatto quadratures on , then one can easily show that\nThus, we define the discrete inner product and norm on  as follows\nIt is not difficult to obtain that\nand\nWe introduce the operator of interpolation at the  shifted Legendre-Gauss-Lobatto nodes, denoted by , i.e., , , such that\nThe interpolation error estimate (see [7  ###reference_7###]) is\nNow consider the spectral discretization to the problem - as follows: find , such that\nwhere\nFor  given, the well-posedness of the problem (3.7  ###reference_###) is guaranteed by the well-known Lax-Milgram Lemma."
        },
        {
            "section_id": "3.2",
            "parent_section_id": "3",
            "section_name": "3.2. Convergence analysis of the full discretization scheme",
            "text": "To simplify matters, we present the semi-discretized problem (2.28  ###reference_###)-(2.29  ###reference_###) in a compact form: find , such that\nwhere\nWe denote by  the norm associated to the bilinear form :\nIt follows from (3.3  ###reference_###) that for all  the\ndiscrete norm  is equivalent to the norm  defined in (2.27  ###reference_###).\nAssuming  and .\nLet  is the solution of the problem (3.7  ###reference_###) with the initial condition  taken to be ,  the solution of the semi-discretized problem (2.28  ###reference_###)-(2.29  ###reference_###). Suppose that  with , for ,\nthen there exists a constant  such that\nProof. For any , denote . It is direct to check that\nBy virtue of (3.4  ###reference_###) gives\nhence\nFor the last term, by definition, we have\nand\nIt is known that the following result holds (see e.g. [32  ###reference_32###, 7  ###reference_7###]): , , ,\nThus for , ,  we have\nApplying the above results to (3.9  ###reference_###) and (3.10  ###reference_###), we obtain\nand\nLet , using (3.8  ###reference_###) and the norm equivalence, for , we have\nand\nBy triangular inequality\nfor , we obtain\nand\nThe above estimate specially holds for , which implies\nand\nSimilar to the proof of Theorem 2.2  ###reference_hm2###, we can immediately get the following conclusions:\nNotice that\nand the boundedness of  and , then there exists a constant  such that"
        },
        {
            "section_id": "4",
            "parent_section_id": null,
            "section_name": "4. Numerical validation",
            "text": ""
        },
        {
            "section_id": "4.1",
            "parent_section_id": "4",
            "section_name": "4.1. Implementation",
            "text": "We provide a comprehensive account of the implementation of problem (3.7  ###reference_###) using the shifted Legendre collocation method.\nConsidering problem (3.7  ###reference_###), we express the function  in terms of the Lagrangian interpolants based on the shifted Legendre-Gauss-Lobatto points , i.e.,\nwhere , unknowns of the discrete solution.  is the Lagrangian polynomials defined in , which satisfies\nwhere  is the Kronecker symbols. Taking (4.5  ###reference_###) into (3.7  ###reference_###), and notice that the homogeneous Dirichlet boundary condition (1.3  ###reference_###), then choosing each test function  to be  , we have\nDefine the matrices\nThen, we obtain the matrix representation of the above equation in the following form:\nThe linear system (4.2  ###reference_###) can be solved in particular by the LU factorization or other related computational techniques.\nFinally, we discuss about the calculation of . When , the initial condition  taken to be\nIt is not difficult to see that\nwhich implies that  satisfies interpolation condition (3.5  ###reference_###). When , suppose that\nthen\nFurthermore,\nIn a word, we can easily obtain  at each iteration of time-step."
        },
        {
            "section_id": "4.2",
            "parent_section_id": "4",
            "section_name": "4.2. Numerical results",
            "text": "We present a series of numerical results to validate our theoretical propositions.\nFirstly, to investigate the computational performance of two discrete fractional differential operators  and , we test three examples from [2  ###reference_2###]. Denote .\nConsider the function  , the Caputo-Fabrizio fractional derivative of\norder  with  of  is written as\nConsider the function , the Caputo-Fabrizio fractional derivative of\norder  with  of  is written as\nConsider the function , the Caputo-Fabrizio fractional derivative of\norder  with  of  is written as\nThe proofs based on the method of integration by parts can be found in [8  ###reference_8###] and [2  ###reference_2###].\nWe choose  in Example 4.1  ###reference_xamp1### and  in Example 4.2  ###reference_xamp2### and Example 4.3  ###reference_xamp3###, and set , . Define the errors\nfor  and  operators, respectively, where  is the last time step. Tables 1  ###reference_###\u20133  ###reference_### give the\nnumerical results of approximation error and CPU time with three examples. Here CPU time represents the total computation time, that is, the whole time for computing the approximations of Caputo-Fabrizio fractional derivatives at every time step. The convergence rates in Tables are given by\nTables 1  ###reference_###\u20133  ###reference_### demonstrate that the errors of both  and  approximations are virtually identical, as a result of their equivalence ().\nMoreover, both approximations have achieved second-order convergence of error, as stated in Lemma 2.1  ###reference_em1###.\nHowever, we observe that the CPU time of  approximation increases linearly with respect to , while the  approximation increases almost quadratically.\nThis suggests that the  operator holds promise as it requires less storage and incurs lower computational costs than the  operator during computation.\nSecondly, we provide preliminary computational findings to demonstrate the efficacy of the finite difference/shifted Legendre collocation method (abbreviated as FCM).\nConsider the following three-term time-fractional diffusion equations:\nwhere , , and\nwith , . The exact solution of the Eq.(4.3  ###reference_###) is , which is sufficiently smooth. In our experiments, we set the parameters .\nThe following error norms have been used as the error indicator:\nWe test Example 4.4  ###reference_xamp4### with four cases: Case 1. , then (4.3  ###reference_###) reduces to the single-term time-fractional diffusion equation; Case 2. , , ; Case 3. , , ; Case 4. , , . In time discretization, we use  operator. Table 4  ###reference_### shows the errors and temporal accuracy of FCM with polynomial degree  at  for different cases of . Here convergence rates are given by\nIt can be observed that the FCM exhibits a second-order temporal convergence rate, , which is consistent with our theoretical analysis.\nNext, we check the spatial accuracy with respect to the polynomial degree . In order to avoid the contamination of temporal error, we need\nfix the time step  sufficiently small. Here we take , and terminate computing at  for saving time.\nFig. 4.1  ###reference_### shows the errors with respect to polynomial degree  at  in semi-log scale. Evidently, the spatial discretization exhibits exponential convergence as demonstrated by the nearly linear curves depicted in this figure. The aforementioned is known as spectral accuracy as expected since the exact solution is a sufficiently smooth function with respect to the space variable.\n\n###figure_1### \n###figure_2### To further verify the numerical validity, we finally test a two-dimensional problem.\nConsider the following three-term time-fractional diffusion equations:\nwhere , , and\nThe exact solution of the Eq. (4.4  ###reference_###) is , which is sufficiently smooth. In our experiments, we set the parameters .\nIn this case, we denote  and  as the nodes and weights of shifted Legendre-Gauss-Lobatto quadratures on . Then we express the function  in terms of the two-dimensional Lagrangian interpolants based on the shifted Legendre-Gauss-Lobatto points ,\nwhere , unknowns of the discrete solution.  and  are the Lagrangian polynomials defined in  and , i.e.,\nwhere  and  are the Kronecker symbols. A linear system such as (4.2  ###reference_###) can be readily derived.\nHere we take . Fig. 4.2  ###reference_### shows the errors with respect to polynomial degree  in semi-log scale.\nThanks to the fast scheme (2.7  ###reference_###), a small time step does not significantly escalate the computational burden in the time direction,\nthereby the proposed method is effective even for handling high-dimensional problems.\n\n###figure_3### \n###figure_4###"
        },
        {
            "section_id": "5",
            "parent_section_id": null,
            "section_name": "5. Concluding remarks",
            "text": "In this work, we have developed a fully discrete scheme for the multi-term time-fractional diffusion\nequations with Caputo-Fabrizio derivatives. The proposed approach utilizes the finite difference method to approximate multi-term fractional derivatives in time and employs the Legendre spectral collocation method for spatial discretization.\nSpecifically, we use the exponential property of Caputo-Fabrizio derivative to give a recursive difference calculation scheme, which offers benefits in terms of computational complexity and storage capacity.\nThe proposed scheme has been proved to be unconditionally stable and convergent with order . Numerical results show good agreement with the theoretical analysis. Due to its high resolution feature in spectral approximation, the proposed method can be extended to handle multi-term time-fractional diffusion equations in higher spatial dimensions."
        }
    ],
    "appendix": [],
    "tables": {
        "1": {
            "table_html": "<figure class=\"ltx_table\" id=\"S4.T1\">\n<figcaption class=\"ltx_caption ltx_centering\" style=\"font-size:90%;\"><span class=\"ltx_tag ltx_tag_table\">Table 1. </span>Comparisons of  with  for Example <a class=\"ltx_ref\" href=\"#S4.Thmexamp1\" title=\"Example 4.1. \u2023 4.2. Numerical results \u2023 4. Numerical validation \u2023 Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives^*\"><span class=\"ltx_text ltx_ref_tag\">4.1</span></a>.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S4.T1.58\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S4.T1.13.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.5.1.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.6.2.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.7.3.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.8.4.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.9.5.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.10.6.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.11.7.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.12.8.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T1.13.9.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S4.T1.22.18\">\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.14.10.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.15.11.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.16.12.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.17.13.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.18.14.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.19.15.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.20.16.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.21.17.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T1.22.18.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.31.27\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.23.19.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.24.20.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.25.21.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.26.22.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.27.23.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.28.24.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.29.25.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.30.26.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.31.27.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.40.36\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.32.28.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.33.29.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.34.30.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.35.31.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.36.32.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.37.33.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.38.34.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.39.35.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.40.36.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.49.45\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.41.37.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.42.38.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.43.39.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.44.40.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.45.41.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.46.42.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.47.43.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.48.44.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T1.49.45.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.58.54\">\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.50.46.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.51.47.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.52.48.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.53.49.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.54.50.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.55.51.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.56.52.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.57.53.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T1.58.54.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 1. Comparisons of  with  for Example 4.1."
        },
        "2": {
            "table_html": "<figure class=\"ltx_table\" id=\"S4.T2\">\n<figcaption class=\"ltx_caption ltx_centering\" style=\"font-size:90%;\"><span class=\"ltx_tag ltx_tag_table\">Table 2. </span>Comparisons of  with  for Example <a class=\"ltx_ref\" href=\"#S4.Thmexamp2\" title=\"Example 4.2. \u2023 4.2. Numerical results \u2023 4. Numerical validation \u2023 Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives^*\"><span class=\"ltx_text ltx_ref_tag\">4.2</span></a>.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S4.T2.58\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S4.T2.13.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.5.1.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.6.2.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.7.3.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.8.4.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.9.5.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.10.6.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.11.7.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.12.8.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T2.13.9.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S4.T2.22.18\">\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.14.10.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.15.11.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.16.12.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.17.13.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.18.14.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.19.15.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.20.16.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.21.17.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T2.22.18.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T2.31.27\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.23.19.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.24.20.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.25.21.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.26.22.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.27.23.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.28.24.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.29.25.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.30.26.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.31.27.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T2.40.36\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.32.28.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.33.29.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.34.30.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.35.31.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.36.32.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.37.33.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.38.34.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.39.35.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.40.36.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T2.49.45\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.41.37.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.42.38.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.43.39.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.44.40.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.45.41.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.46.42.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.47.43.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.48.44.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T2.49.45.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T2.58.54\">\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.50.46.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.51.47.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.52.48.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.53.49.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.54.50.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.55.51.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.56.52.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.57.53.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T2.58.54.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 2. Comparisons of  with  for Example 4.2."
        },
        "3": {
            "table_html": "<figure class=\"ltx_table\" id=\"S4.T3\">\n<figcaption class=\"ltx_caption ltx_centering\" style=\"font-size:90%;\"><span class=\"ltx_tag ltx_tag_table\">Table 3. </span>Comparisons of  with  for Example <a class=\"ltx_ref\" href=\"#S4.Thmexamp3\" title=\"Example 4.3. \u2023 4.2. Numerical results \u2023 4. Numerical validation \u2023 Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives^*\"><span class=\"ltx_text ltx_ref_tag\">4.3</span></a>.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S4.T3.58\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S4.T3.13.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.5.1.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.6.2.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.7.3.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.8.4.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.9.5.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.10.6.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.11.7.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.12.8.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T3.13.9.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S4.T3.22.18\">\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.14.10.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.15.11.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.16.12.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.17.13.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.18.14.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.19.15.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.20.16.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.21.17.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T3.22.18.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T3.31.27\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.23.19.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.24.20.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.25.21.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.26.22.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.27.23.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.28.24.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.29.25.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.30.26.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.31.27.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T3.40.36\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.32.28.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.33.29.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.34.30.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.35.31.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.36.32.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.37.33.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.38.34.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.39.35.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.40.36.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T3.49.45\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.41.37.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.42.38.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.43.39.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.44.40.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.45.41.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.46.42.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.47.43.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.48.44.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T3.49.45.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T3.58.54\">\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.50.46.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.51.47.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.52.48.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.53.49.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.54.50.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.55.51.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.56.52.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.57.53.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T3.58.54.9\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 3. Comparisons of  with  for Example 4.3."
        },
        "4": {
            "table_html": "<figure class=\"ltx_table\" id=\"S4.T4\">\n<figcaption class=\"ltx_caption ltx_centering\" style=\"font-size:90%;\"><span class=\"ltx_tag ltx_tag_table\">Table 4. </span>Numerical convergence of FCM in the temporal direction for Example <a class=\"ltx_ref\" href=\"#S4.Thmexamp4\" title=\"Example 4.4. \u2023 4.2. Numerical results \u2023 4. Numerical validation \u2023 Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives^*\"><span class=\"ltx_text ltx_ref_tag\">4.4</span></a>.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S4.T4.160\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S4.T4.8.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.1.1.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.2.2.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.3.3.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.4.4.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.5.5.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.6.6.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.7.7.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t\" id=\"S4.T4.8.8.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S4.T4.15.15\">\n<td class=\"ltx_td ltx_border_t\" id=\"S4.T4.15.15.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.9.9.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.10.10.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.11.11.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.12.12.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.13.13.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.14.14.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.15.15.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.23.23\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.16.16.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.17.17.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.18.18.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.19.19.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.20.20.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.21.21.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.22.22.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.23.23.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.31.31\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.24.24.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.25.25.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.26.26.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.27.27.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.28.28.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.29.29.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.30.30.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.31.31.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.39.39\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.32.32.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.33.33.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.34.34.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.35.35.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.36.36.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.37.37.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.38.38.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.39.39.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.46.46\">\n<td class=\"ltx_td\" id=\"S4.T4.46.46.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.40.40.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.41.41.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.42.42.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.43.43.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.44.44.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.45.45.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.46.46.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.53.53\">\n<td class=\"ltx_td ltx_border_t\" id=\"S4.T4.53.53.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.47.47.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.48.48.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.49.49.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.50.50.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.51.51.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.52.52.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.53.53.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.61.61\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.54.54.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.55.55.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.56.56.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.57.57.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.58.58.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.59.59.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.60.60.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.61.61.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.69.69\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.62.62.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.63.63.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.64.64.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.65.65.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.66.66.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.67.67.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.68.68.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.69.69.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.77.77\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.70.70.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.71.71.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.72.72.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.73.73.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.74.74.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.75.75.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.76.76.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.77.77.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.84.84\">\n<td class=\"ltx_td\" id=\"S4.T4.84.84.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.78.78.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.79.79.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.80.80.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.81.81.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.82.82.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.83.83.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.84.84.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.91.91\">\n<td class=\"ltx_td ltx_border_t\" id=\"S4.T4.91.91.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.85.85.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.86.86.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.87.87.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.88.88.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.89.89.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.90.90.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.91.91.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.99.99\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.92.92.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.93.93.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.94.94.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.95.95.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.96.96.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.97.97.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.98.98.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.99.99.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.107.107\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.100.100.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.101.101.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.102.102.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.103.103.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.104.104.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.105.105.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.106.106.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.107.107.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.115.115\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.108.108.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.109.109.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.110.110.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.111.111.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.112.112.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.113.113.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.114.114.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.115.115.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.122.122\">\n<td class=\"ltx_td\" id=\"S4.T4.122.122.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.116.116.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.117.117.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.118.118.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.119.119.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.120.120.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.121.121.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.122.122.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.129.129\">\n<td class=\"ltx_td ltx_border_t\" id=\"S4.T4.129.129.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.123.123.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.124.124.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.125.125.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.126.126.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.127.127.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.128.128.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_t\" id=\"S4.T4.129.129.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.137.137\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.130.130.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.131.131.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.132.132.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.133.133.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.134.134.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.135.135.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.136.136.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.137.137.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.145.145\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.138.138.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.139.139.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.140.140.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.141.141.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.142.142.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.143.143.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.144.144.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.145.145.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.153.153\">\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.146.146.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.147.147.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.148.148.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.149.149.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.150.150.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.151.151.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.152.152.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left\" id=\"S4.T4.153.153.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T4.160.160\">\n<td class=\"ltx_td ltx_border_b\" id=\"S4.T4.160.160.8\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.154.154.1\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.155.155.2\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.156.156.3\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.157.157.4\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.158.158.5\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.159.159.6\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n<td class=\"ltx_td ltx_align_left ltx_border_b\" id=\"S4.T4.160.160.7\" style=\"padding-top:1.25pt;padding-bottom:1.25pt;\"></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 4. Numerical convergence of FCM in the temporal direction for Example 4.4."
        }
    },
    "image_paths": {
        "1(a)": {
            "figure_path": "2307.08078v2_figure_1(a).png",
            "caption": "Figure 4.1. Numerical convergence of FCM in the spatial direction for Example 4.4.",
            "url": "http://arxiv.org/html/2307.08078v2/extracted/5334211/Fig1-1.jpg"
        },
        "1(b)": {
            "figure_path": "2307.08078v2_figure_1(b).png",
            "caption": "Figure 4.1. Numerical convergence of FCM in the spatial direction for Example 4.4.",
            "url": "http://arxiv.org/html/2307.08078v2/extracted/5334211/Fig1-2.jpg"
        },
        "2(a)": {
            "figure_path": "2307.08078v2_figure_2(a).png",
            "caption": "Figure 4.2. Numerical convergence of FCM in the spatial direction for Example 4.5.",
            "url": "http://arxiv.org/html/2307.08078v2/extracted/5334211/Fig2-1.jpg"
        },
        "2(b)": {
            "figure_path": "2307.08078v2_figure_2(b).png",
            "caption": "Figure 4.2. Numerical convergence of FCM in the spatial direction for Example 4.5.",
            "url": "http://arxiv.org/html/2307.08078v2/extracted/5334211/Fig2-2.jpg"
        }
    },
    "validation": true,
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