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Example 2. Euler's equations for compressible gas flow in one dimension are\n\n\[ \begin{cases} {\rho }_{t} + {\left( \rho v\right) }_{x} & = 0\;\text{ (conservation of mass) } \\ {\left( \rho v\right) }_{t} + {\left( \rho {v}^{2} + p\right) }_{x} & = 0\;\text{ (conservation of momentum) } \\ {\left( \rho E\right) }_{t... | in \( \mathbb{R} \times \left( {0,\infty }\right) \) . Here \( \rho \) is the mass density, \( v \) the velocity, and \( E \) the energy density per unit mass. We assume\n\n\[ E = e + \frac{{v}^{2}}{2} \]\n\nwhere \( e \) is the internal energy per unit mass and the term \( \frac{{v}^{2}}{2} \) corresponds to the kinet... | Yes |
For the \( p \) -system (6), we have\n\n\[ \n{\mathbf{u}}_{t} + \mathbf{B}\left( \mathbf{u}\right) {\mathbf{u}}_{x} = 0 \n\]\n\nfor\n\n\[ \n\mathbf{B}\left( z\right) = D\mathbf{F}\left( z\right) = \left( \begin{matrix} 0 & - 1 \\ - {p}^{\prime }\left( {z}_{1}\right) & 0 \end{matrix}\right) .\n\] | The eigenvalues are \( {\lambda }_{1} = - \sigma ,{\lambda }_{2} = \sigma \), for \( \sigma \mathrel{\text{:=}} {p}^{\prime }{\left( {z}_{1}\right) }^{1/2} \) . These are real and distinct provided we hereafter suppose the strict hyperbolicity condition\n\n(41)\n\n\[ \n{p}^{\prime } > 0\text{.} \n\]\n\nFor the nonlinea... | Yes |
Euler's equations (9) comprise a strictly hyperbolic system provided we assume \( p > 0 \) and\n\n(42)\n\n\[ \frac{\partial p}{\partial \rho } > 0,\frac{\partial p}{\partial e} > 0 \] | Let us rather change variables and regard the density \( \rho \), velocity \( v \) and internal energy \( e \) as the unknowns. We can then rewrite Euler’s equations (9) in terms of these quantities, and, so doing, obtain after some calculations the system\n\n(43)\n\n\[ \left\{ \begin{array}{l} {\rho }_{t} + v{\rho }_{... | Yes |
In the case of a scalar conservation law (i.e. \( m = 1 \) ), for any convex \( \Phi \) we can find a corresponding flux function \( \Psi \) | \[ \Psi \left( z\right) = {\int }_{{z}_{0}}^{z}{\Phi }^{\prime }\left( w\right) {F}^{\prime }\left( w\right) {dw}\;\left( {z \in \mathbb{R}}\right) . \] | "Yes" |
For the \( p \) -system we have \( m = 2 \) . To verify (25),(26) we must find \( \Phi ,\Psi \), with \( \Phi \) convex and\n\n\[ \left( {{\Phi }_{{z}_{1}},{\Phi }_{{z}_{2}}}\right) \left( \begin{matrix} 0 & - 1 \\ - {p}^{\prime }\left( {z}_{1}\right) & 0 \end{matrix}\right) = \left( \begin{matrix} {\Psi }_{{z}_{1}} \\... | A solution is\n\n\[ \Phi \left( z\right) = \frac{{z}_{2}^{2}}{2} + {\int }_{0}^{{z}_{1}}p\left( w\right) {dw},\;\Psi \left( z\right) = - p\left( {z}_{1}\right) {z}_{2}\;\left( {z \in {\mathbb{R}}^{2}}\right) . \]\n\nNote \( \Phi \) is convex, since \( {p}^{\prime } > 0 \) . | Yes |
Clairaut's equation from differential geometry is the PDE\n\n\[ \nx \cdot {Du} + f\left( {Du}\right) = u \]\n\nwhere \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is given. | A complete integral is\n\n\[ \nu\left( {x;a}\right) = a \cdot x + f\left( a\right) \;\left( {x \in U}\right) \]\n\nfor \( a \in {\mathbb{R}}^{n} \). | Yes |
The eikonal equation from geometric optics is the PDE\n\n\[ \left| {Du}\right| = 1\text{.} \] | A complete integral is\n\n\[ u\left( {x;a, b}\right) = a \cdot x + b\;\left( {x \in U}\right) \] \n\nfor \( x \in U, a \in \partial B\left( {0,1}\right), b \in \mathbb{R} \) . | Yes |
The Hamilton-Jacobi equation from mechanics is in its simplest form the partial differential equation\n\n\[ \n{u}_{t} + H\left( {Du}\right) = 0 \]\n\nwhere \( H : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) . Here \( u \) depends on \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{R}}^{n} \) and \( t \in \mat... | A complete integral is\n\n\[ \nu\left( {x, t;a, b}\right) = a \cdot x - {tH}\left( a\right) + b\;\left( {x \in {\mathbb{R}}^{n}, t \geq 0}\right) \]\n\nwhere \( a \in {\mathbb{R}}^{n}, b \in \mathbb{R} \) . | Yes |
Consider the PDE\n\n\[ \n{u}^{2}\left( {1 + {\left| Du\right| }^{2}}\right) = 1 \n\] | A complete integral is\n\n\[ \nu\left( {x, a}\right) = \pm {\left( 1 - {\left| x - a\right| }^{2}\right) }^{1/2}\;\left( {\left| {x - a}\right| < 1}\right) .\n\nWe compute\n\n\[ \n{D}_{a}u = \frac{\mp \left( {x - a}\right) }{{\left( 1 - {\left| x - a\right| }^{2}\right) }^{1/2}} = 0 \n\]\n\nprovided \( a = \phi \left( ... | No |
Let \( H\left( p\right) = {\left| p\right| }^{2}, h \equiv 0 \) in Example 3 above. Then\n\n\[ \n{u}^{\prime }\left( {x, t;a}\right) = x \cdot a - t{\left| a\right| }^{2}.\n\] | We calculate the envelope by setting \( {D}_{a}{u}^{\prime } = x - {2ta} = 0 \) . Hence \( a = \frac{x}{2t} \) , and so\n\n\[ \n{v}^{\prime }\left( {x, t}\right) = x \cdot \frac{x}{2t} - t{\left| \frac{x}{2t}\right| }^{2} = \frac{{\left| x\right| }^{2}}{4t}\;\left( {x \in {\mathbb{R}}^{n}, t > 0}\right)\n\]\n\nsolves t... | Yes |
We demonstrate the utility of equations (17) by explicitly solving the problem\n\n(18)\n\n\[ \n\\begin{cases} {x}_{1}{u}_{{x}_{2}} - {x}_{2}{u}_{{x}_{1}} & = u & & \\text{ in }U \\\\ u & = g & & \\text{ on }\\Gamma , \\end{cases} \n\]\n\nwhere \( U \) is the quadrant \( \\left\\{ {{x}_{1} > 0,{x}_{2} > 0}\\right\\} \) ... | The PDE in (18) is of the form (12), for \( \\mathbf{b} = \\left( {-{x}_{2},{x}_{1}}\\right) \) and \( c = - 1 \) . Thus the equations (17) read\n\n(19)\n\n\[ \n\\left\\{ \\begin{array}{l} {\\dot{x}}^{1} = - {x}^{2},{\\dot{x}}^{2} = {x}^{1} \\\\ \\dot{z} = z. \\end{array}\\right. \n\]\n\nAccordingly we have\n\n\[ \n\\l... | Yes |
\[ \begin{cases} {u}_{{x}_{1}} + {u}_{{x}_{2}} & = {u}^{2} & & \text{ in }U \\ u & = g & & \text{ on }\Gamma . \end{cases} \] | Now \( U \) is the half-space \( \left\{ {{x}_{2} > 0}\right\} \) and \( \Gamma = \left\{ {{x}_{2} = 0}\right\} = \partial U \) . Here \( \mathbf{b} = \left( {1,1}\right) \) and \( c = - {z}^{2} \) . Then (21) becomes \[ \begin{cases} {\dot{x}}^{1} & = 1,\;{\dot{x}}^{2} = 1 \\ \dot{z} & = {z}^{2}. \end{cases} \] Conseq... | Yes |
Consider the fully nonlinear problem\n\n\[ \n\\begin{cases} {u}_{{x}_{1}}{u}_{{x}_{2}} & = u\\;\\text{ in }U \\\\ u & = {x}_{2}^{2}\\;\\text{ on }\\Gamma \n\\end{cases} \n\]\n\nwhere \( U = \\left\\{ {{x}_{1} > 0}\\right\\} ,\\Gamma = \\left\\{ {{x}_{1} = 0}\\right\\} = \\partial U \) . | Here \( F\\left( {p, z, x}\\right) = {p}_{1}{p}_{2} - z \), and hence the characteristic ODE (11) become\n\n\[ \n\\begin{cases} {\\dot{p}}^{1} & = {p}^{1},{\\dot{p}}^{2} = {p}^{2} \\\\ \\dot{z} & = 2{p}^{1}{p}^{2} \\\\ {\\dot{x}}^{1} & = {p}^{2},{\\dot{x}}^{2} = {p}^{1} \n\\end{cases} \n\]\n\nWe integrate these equatio... | Yes |
Can we solve the linear boundary-value problem\n\n(53)\n\n\\[ \n\\begin{cases} \\mathbf{b} \\cdot {Du} & = 0 & & \\text{ in }U \\\\ u & = g & & \\text{ on }\\Gamma ? \\end{cases} \n\\] | Invoking Theorem 2, we see that there exists a unique solution \( u \) defined near \( \\Gamma \) and indeed that \( u\\left( {\\mathbf{x}\\left( s\\right) }\\right) \\equiv u\\left( {\\mathbf{x}\\left( 0\\right) }\\right) = g\\left( {x}^{0}\\right) \) for each solution of the ODE (52), with the initial condition \( \\... | Yes |
As an instance of a quasilinear first-order PDE, we turn now to the scalar conservation law\n\n\[ G\left( {{Du},{u}_{t}, u, x, t}\right) = {u}_{t} + \operatorname{div}\mathbf{F}\left( u\right) \]\n\n(56)\n\n\[ = {u}_{t} + {\mathbf{F}}^{\prime }\left( u\right) \cdot {Du} = 0 \]\n\nin \( U = {\mathbb{R}}^{n} \times \left... | Since the direction \( t = {x}_{n + 1} \) plays a special role, we appropriately modify our notation. Writing now \( q = \left( {p,{p}_{n + 1}}\right) \) and \( y = \left( {x, t}\right) \), we have\n\n\[ G\left( {q, z, y}\right) = {p}_{n + 1} + {\mathbf{F}}^{\prime }\left( z\right) \cdot p \]\n\nand consequently\n\n\[ ... | Yes |
Example 6 (Characteristics for the Hamilton-Jacobi equation). We look now at the general Hamilton-Jacobi PDE\n\n(62)\n\n\[ G\left( {{Du},{u}_{t}, u, x, t}\right) = {u}_{t} + H\left( {{Du}, x}\right) = 0, \]\n\nwhere \( {Du} = {D}_{x}u = \left( {{u}_{{x}_{1}},\ldots ,{u}_{{x}_{n}}}\right) \) . | Then writing \( q = \left( {p,{p}_{n + 1}}\right), y = \left( {x, t}\right) \) , we have\n\n\[ G\left( {q, z, y}\right) = {p}_{n + 1} + H\left( {p, x}\right) \]\n\nand so\n\n\[ {D}_{q}G = \left( {{D}_{p}H\left( {p, x}\right) ,1}\right) ,{D}_{y}G = \left( {{D}_{x}H\left( {p, x}\right) ,0}\right) ,{D}_{z}G = 0. \]\n\nThu... | Yes |
Let us consider the initial-value problem for Burgers’ equation:\n\n\[ \n\\begin{cases} {u}_{t} + {\\left( \\frac{{u}^{2}}{2}\\right) }_{x} & = 0\\;\\text{ in }\\mathbb{R} \\times \\left( {0,\\infty }\\right) \\\\ u & = g\\;\\text{ on }\\mathbb{R} \\times \\{ t = 0\\} , \\end{cases} \n\]\n\nwith the initial data\n\n\[ ... | According to the characteristic equations (cf. §3.2.5) any smooth solution \( u \) of (13),(14) takes the constant value \( {z}^{0} = g\\left( {x}^{0}\\right) \) along the projected characteristic\n\n\[ \n\\mathbf{y}\\left( s\\right) = \\left( {g\\left( {x}^{0}\\right) s + {x}^{0}, s}\\right) \\;\\left( {s \\geq 0}\\ri... | Yes |
Example 2 (Rarefaction waves and nonphysical shocks). Again consider the initial-value problem (13), for which now we take\n\n(15)\n\n\\[ \n g\\left( x\\right) = \\left\\{ \\begin{array}{ll} 0 & \\text{ if }x < 0 \\\\ 1 & \\text{ if }x > 0 \\end{array}\\right. \n\\]\n\nThe method of characteristics this time does not l... |  A \ | Yes |
Example 3. We again return to Burgers' equation (13), now for the initial function\n\n\[ g\\left( x\\right) = \\left\\{ \\begin{array}{ll} 0 & \\text{ if }x < 0 \\\\ 1 & \\text{ if }0 \\leq x \\leq 1 \\\\ 0 & \\text{ if }x > 1 \\end{array}\\right. \] | For \( 0 \\leq t \\leq 2 \), we may combine the analysis in Examples 1 and 2 above to find\n\n\[ u\\left( {x, t}\\right) \\mathrel{\\text{:=}} \\left\\{ \\begin{array}{ll} 0 & \\text{ if }x < 0 \\\\ \\frac{x}{t} & \\text{ if }0 < x < t \\\\ 1 & \\text{ if }t < x < 1 + \\frac{t}{2} \\\\ 0 & \\text{ if }x > 1 + \\frac{t}... | Yes |
Let \( U \subset {\mathbb{R}}^{n} \) be a bounded, open set with smooth boundary. We consider the initial/boundary-value problem for the heat equation\n\n\[ \begin{cases} {u}_{t} - {\Delta u} = 0 & \text{ in }U \times \left( {0,\infty }\right) \\ u = 0 & \text{ on }\partial U \times \lbrack 0,\infty ) \\ u = g & \text{... | Will this work? To find out, we compute\n\n\[ {u}_{t}\left( {x, t}\right) = {v}^{\prime }\left( t\right) w\left( x\right) ,{\Delta u}\left( {x, t}\right) = v\left( t\right) {\Delta w}\left( x\right) . \]\n\nHence\n\n\[ 0 = {u}_{t}\left( {x, t}\right) - {\Delta u}\left( {x, t}\right) = {v}^{\prime }\left( t\right) w\lef... | Yes |
Let us next apply the separation of variables technique to discover a solution of the porous medium equation\n\n\[ \n{u}_{t} - \Delta \left( {u}^{\gamma }\right) = 0\;\text{ in }{\mathbb{R}}^{n} \times \left( {0,\infty }\right) ,\n\]\n\nwhere \( u \geq 0 \) and \( \gamma > 1 \) is a constant. | As in the previous example, we seek a solution of the form\n\n\[ \nu\left( {x, t}\right) = v\left( t\right) w\left( x\right) \;\left( {x \in {\mathbb{R}}^{n}, t \geq 0}\right) .\n\]\n\nInserting into (11), we discover that\n\n\[ \n\frac{{v}^{\prime }\left( t\right) }{v{\left( t\right) }^{\gamma }} = \mu = \frac{\Delta ... | Yes |
Let us turn once again to the Hamilton-Jacobi equation\n\n\[ \n{u}_{t} + H\left( {Du}\right) = 0\;\text{ in }{\mathbb{R}}^{n} \times \left( {0,\infty }\right) \n\]\n\nand look for a solution \( u \) having the form\n\n\[ \nu\left( {x, t}\right) = w\left( x\right) + v\left( t\right) \;\left( {x \in {\mathbb{R}}^{n}, t \... | Then\n\n\[ \n0 = {u}_{t}\left( {x, t}\right) + H\left( {{Du}\left( {x, t}\right) }\right) = {v}^{\prime }\left( t\right) + H\left( {{Dw}\left( x\right) }\right) \n\]\n\nif and only if\n\n\[ \nH\left( {{Dw}\left( x\right) }\right) = \mu = - {v}^{\prime }\left( t\right) \;\left( {x \in {\mathbb{R}}^{n}, t > 0}\right) \n\... | Yes |
Consider again the initial-value problem for the heat equation\n\n\[ \n\\begin{cases} {u}_{t} - {\\Delta u} = 0 & \\text{ in }{\\mathbb{R}}^{n} \\times \\left( {0,\\infty }\\right) \\\\ u = g & \\text{ on }{\\mathbb{R}}^{n} \\times \\{ t = 0\\} . \\end{cases} \n\] | We establish a new method for solving (16) by computing \( \\widehat{u} \), the Fourier transform of \( u \) in the spatial variables \( x \) only. Thus\n\n\[ \n\\begin{cases} {\\widehat{u}}_{t} + {\\left| y\\right| }^{2}\\widehat{u} & = 0\\;\\text{ for }t > 0 \\\\ \\widehat{u} & = \\widehat{g}\\;\\text{ for }t = 0, \\... | Yes |
Let us next look at the initial-value problem for Schrödinger's equation\n\n\[ \n\\begin{cases} i{u}_{t} + {\\Delta u} & = 0\\;\\text{ in }{\\mathbb{R}}^{n} \\times \\left( {0,\\infty }\\right) \\\\ u & = g\\;\\text{ on }{\\mathbb{R}}^{n} \\times \\{ t = 0\\} . \\end{cases} \n\]\n\nHere \( u \) and \( g \) are complex-... | If we formally replace \( t \) by \( {it} \) on the right-hand side of (18), we obtain the formula\n\n\[ \nu\left( {x, t}\\right) = \\frac{1}{{\\left( 4\\pi it\\right) }^{n/2}}{\\int }_{{\\mathbb{R}}^{n}}{e}^{\\frac{i{\\left| x - y\\right| }^{2}}{4t}}g\\left( y\\right) {dy}\\;\\left( {x \\in {\\mathbb{R}}^{n}, t > 0}\\... | Yes |
We next analyze the initial-value problem for the wave equation\n\n\[ \left\{ \begin{matrix} {u}_{tt} - {\Delta u} = 0 & \text{ in }{\mathbb{R}}^{n} \times \left( {0,\infty }\right) \\ u = g,{u}_{t} = h & \text{ on }{\mathbb{R}}^{n} \times \{ t = 0\} \end{matrix}\right. \] | where for simplicity we suppose the initial velocity to be zero. Take as before \( \widehat{u} \) to be the Fourier transform of \( u \) in the variable \( x \in {\mathbb{R}}^{n} \) . Then\n\n\[ \left\{ \begin{matrix} {\widehat{u}}_{tt} + {\left| y\right| }^{2}\widehat{u} = 0 & \text{ for }t > 0 \\ \widehat{u} = \wideh... | Yes |
We claim now that if \( n = {2k} + 1 \) is odd and\n\n(37)\n\n\[ \gamma \mathrel{\text{:=}} \frac{{\left( -1\right) }^{k}}{2{\left( 2\pi \right) }^{2k}}\frac{{\partial }^{2k}}{\partial {s}^{2k}}\widetilde{g} = \frac{{\left( -1\right) }^{k}}{2{\left( 2\pi \right) }^{2k}}\mathcal{R}\left( {{\Delta }^{k}g}\right) ,\n\]\n\... | To confirm this, note first that \( u = g \) on \( {\mathbb{R}}^{n} \times \{ t = 0\} \) in view of Theorem 5(ii). Consequently we need only check the second initial condition, that\n\n\[ {u}_{t}\left( {x,0}\right) = - {\int }_{{S}^{n - 1}}{\gamma }_{s}\left( {x \cdot \omega ,\omega }\right) {dS} = 0. \]\n\nIn view of ... | Yes |
Example 7 (Huygens' principle for hyperbolic systems). A linear system of first-order PDE\n\n\\[ \n{\\mathbf{u}}_{t} + \\mathop{\\sum }\\limits_{{j = 1}}^{n}{B}_{j}{\\mathbf{u}}_{{x}_{j}} = 0 \n\\]\n\nfor the unknown \\( \\mathbf{u} : {\\mathbb{R}}^{n} \\times \\lbrack 0,\\infty ) \\rightarrow {\\mathbb{R}}^{m},\\mathb... | So let us suppose that \\( \\mathbf{u} \\) is a smooth solution of (39) and take the Radon transform in the variables \\( x \\). We deduce from Theorem 3(ii) that for each fixed \\( \\omega \\in {S}^{n - 1} \\)\n\n\\[ \n\\begin{cases} {\\widetilde{\\mathbf{u}}}_{t} + \\mathbf{B}\\left( \\omega \\right) {\\widetilde{\\m... | Yes |
What PDE does \( {v}^{\# } \) satisfy? | We compute\n\n\[ \Delta {v}^{\# }\left( {x, s}\right) = {\int }_{0}^{\infty }{e}^{-{st}}{\Delta v}\left( {x, t}\right) {dt} = {\int }_{0}^{\infty }{e}^{-{st}}{v}_{t}\left( {x, t}\right) {dt} \]\n\n\[ = s{\int }_{0}^{\infty }{e}^{-{st}}v\left( {x, t}\right) {dt} + {\left. {e}^{-{st}}v\right| }_{t = 0}^{t = \infty } = s{... | Yes |
We illustrate this idea by studying formally the effects of small diffusion upon the transport of dye within a moving fluid in \( {\mathbb{R}}^{2} \). Suppose we are given a smooth vector field \( \mathbf{b} : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2},\mathbf{b} = \left( {{b}^{1},{b}^{2}}\right) \) , representing t... | Consider now for \( \varepsilon > 0 \) the singular perturbation:\n\n\[ - {\varepsilon \Delta }{u}^{\varepsilon } + \operatorname{div}\left( {{u}^{\varepsilon }\mathbf{b}}\right) = {\delta }_{0}\;\text{ in }{\mathbb{R}}^{2}. \]\n\nThe new term \ | Yes |
What happens to \( {u}^{\varepsilon } \) as \( \varepsilon \rightarrow 0 \) ? | LEMMA (Asymptotics). Suppose that \( k, l : \mathbb{R} \rightarrow \mathbb{R} \) are continuous functions, that \( l \) grows at most linearly and that \( k \) grows at least quadratically. Assume also there exists a unique point \( {y}_{0} \in \mathbb{R} \) such that\n\n\[ k\left( {y}_{0}\right) = \mathop{\min }\limit... | No |
Let us once more turn our attention to the wave equation\n\n\[ \n{u}_{tt} - {\Delta u} = 0\;\text{ in }{\mathbb{R}}^{n} \times \left( {0,\infty }\right) ,\n\]\n\nand we now regard the solution \( u \) as taking complex values. We fix \( \varepsilon > 0 \) and seek a solution \( u = {u}^{\varepsilon } \) of (27) having ... | Substituting (28) into (27), we find after some computations that\n\n\[ \n0 = {u}_{tt}^{\varepsilon } - \Delta {u}^{\varepsilon } = {e}^{i{p}^{\varepsilon }/\varepsilon }\left( {\frac{i{p}_{tt}^{\varepsilon }}{\varepsilon }{a}^{\varepsilon } - {\left( \frac{{p}_{t}^{\varepsilon }}{\varepsilon }\right) }^{2}{a}^{\vareps... | Yes |
Let \( U \) denote an open, bounded subset of \( {\mathbb{R}}^{n} \), with smooth boundary \( \partial U \), and consider this boundary-value problem for a divergence structure PDE:\n\n\[ \begin{cases} - \mathop{\sum }\limits_{{i, j = 1}}^{n}{\left( {a}^{ij}\left( \frac{x}{\varepsilon }\right) {u}_{{x}_{i}}^{\varepsilo... | In the following heuristic discussion let us assume\n\n\[ {u}^{\varepsilon } \rightarrow u\;\text{ as }\varepsilon \rightarrow 0 \]\n\nin some suitable sense and try to determine an equation which \( u \) satisfies. The trick is to suppose \( {u}^{\varepsilon } \) admits the following two-scale expansion:\n\n\[ {u}^{\v... | Yes |
Let \( n = 1, U = \left( {0,2}\right) \), and\n\n\[ u\left( x\right) = \left\{ \begin{array}{ll} x & \text{ if }0 < x \leq 1 \\ 1 & \text{ if }1 \leq x < 2. \end{array}\right. \]\n\nDefine\n\n\[ v\left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ if }0 < x \leq 1 \\ 0 & \text{ if }1 < x < 2. \end{array}\right. \]\n... | But we easily calculate\n\n\[ {\int }_{0}^{2}u{\phi }^{\prime }{dx} = {\int }_{0}^{1}x{\phi }^{\prime }{dx} + {\int }_{1}^{2}{\phi }^{\prime }{dx} \]\n\n\[ = - {\int }_{0}^{1}{\phi dx} + \phi \left( 1\right) - \phi \left( 1\right) = - {\int }_{0}^{2}{v\phi dx} \]\n\nas required. | Yes |
Example 2. Let \( n = 1, U = \left( {0,2}\right) \), and\n\n\[ u\left( x\right) = \left\{ \begin{array}{ll} x & \text{ if }0 < x \leq 1 \\ 2 & \text{ if }1 < x < 2 \end{array}\right. \]\n\nWe assert \( {u}^{\prime } \) does not exist in the weak sense. To check this, we must show there does not exist any function \( v ... | Suppose, to the contrary,(5) were valid for some \( v \) and all \( \phi \) . Then\n\n(6)\n\n\[ - {\int }_{0}^{2}{v\phi dx} = {\int }_{0}^{2}u{\phi }^{\prime }{dx} = {\int }_{0}^{1}x{\phi }^{\prime }{dx} + 2{\int }_{1}^{2}{\phi }^{\prime }{dx} \]\n\n\[ = - {\int }_{0}^{1}{\phi dx} - \phi \left( 1\right) \]\n\nChoose a ... | Yes |
For which values of \( \alpha > 0, n, p \) does \( u \) belong to \( {W}^{1, p}\left( U\right) \) ? | To answer, note first that \( u \) is smooth away from 0, with\n\n\[ {u}_{{x}_{i}}\left( x\right) = \frac{-\alpha {x}_{i}}{{\left| x\right| }^{\alpha + 2}}\;\left( {x \neq 0}\right) \]\n\nand so\n\n\[ \left| {{Du}\left( x\right) }\right| = \frac{\left| \alpha \right| }{{\left| x\right| }^{\alpha + 1}}\;\left( {x \neq 0... | Yes |
Example 1 (Dirichlet's principle). Take\n\n\[ L\left( {p, z, x}\right) = \frac{1}{2}{\left| p\right| }^{2}. \]\n\nThen \( {L}_{{p}_{i}} = {p}_{i}\left( {i = 1,\ldots, n}\right) ,{L}_{z} = 0 \) ; and so the Euler-Lagrange equation associated with the functional\n\n\[ I\left\lbrack w\right\rbrack \mathrel{\text{:=}} \fra... | This fact is Dirichlet's principle, previously introduced in §2.2.5. | No |
Example 2 (Generalized Dirichlet's principle). Write\n\n\[ \nL\left( {p, z, x}\right) = \frac{1}{2}\mathop{\sum }\limits_{{i, j = 1}}^{n}{a}^{ij}\left( x\right) {p}_{i}{p}_{j} - {zf}\left( x\right) , \n\] \n\nwhere \( {a}^{ij} = {a}^{ji}\left( {i, j = 1,\ldots, n}\right) \) . Then \( {L}_{{p}_{i}} = \mathop{\sum }\limi... | We will see later (in \( §{8.1.3} \) and \( §{8.2} \) ) that the uniform ellipticity condition on the \( {a}^{ij}\left( {i, j = 1,\ldots, n}\right) \) is a natural further assumption, required to prove the existence of a minimizer. Consequently from the nonlinear viewpoint of the calculus of variations, the divergence ... | No |
Example 4 (Minimal surfaces). Let\n\n\\[ \nL\\left( {p, z, x}\\right) = {\\left( 1 + {\\left| p\\right| }^{2}\\right) }^{1/2} \n\\]\n\nso that\n\n\\[ \nI\\left\\lbrack w\\right\\rbrack = {\\int }_{U}{\\left( 1 + {\\left| Dw\\right| }^{2}\\right) }^{1/2}{dx} \n\\]\n\nis the area of the graph of the function \\( w : U \\... | The expression \\( \\operatorname{div}\\left( \\frac{Du}{{\\left( 1 + {\\left| Du\\right| }^{2}\\right) }^{1/2}}\\right) \\) on the left side of (10) is \\( n \\) times the mean curvature of the graph of \\( u \\) . Thus a minimal surface has zero mean curvature. | Yes |
If \( L = L\left( {p, z}\right) \) does not depend upon the independent variables \( x \), then the integral (1) is invariant under translations in space. | To be specific, select \( k \in \{ 1,\ldots, n\} \) and define\n\n\[ \mathbf{x}\left( {x,\tau }\right) \mathrel{\text{:=}} x + \tau {e}_{k},\;w\left( {x,\tau }\right) \mathrel{\text{:=}} u\left( {x + \tau {e}_{k}}\right) . \]\n\nThen\n\n\[ \mathbf{v} = {e}_{k},\;m = {u}_{{x}_{k}} \]\n\nConsequently if \( u \) is a crit... | No |
Example 2 (Scaling invariance). The functional\n\n\[ \nI\left\lbrack w\right\rbrack = {\int }_{U}{\left| Dw\right| }^{p}{dx} \]\n\nsmooth minimizers \( u \) of which solve the \( p \) -Laplacian equation\n\n\[ \n\operatorname{div}\left( {{\left| Du\right| }^{p - 2}{Du}}\right) = 0 \]\n\nis invariant under the scaling t... | To be consistent with previous notation, we put \( \lambda = {e}^{\tau } \) and define\n\n\[ \n\mathbf{x}\left( {x,\tau }\right) \mathrel{\text{:=}} {e}^{\tau }x,\;w\left( {x,\tau }\right) \mathrel{\text{:=}} {e}^{\tau \frac{n - p}{p}}u\left( {{e}^{\tau }x}\right) . \]\n\nThen\n\n\[ \n\mathbf{v} = x,\;m = {Du} \cdot x ... | Yes |
Consider the integral expression\n\n\[ I\left\lbrack w\right\rbrack = {\int }_{0}^{T}{\int }_{{\mathbb{R}}^{n}}\frac{1}{2}{w}_{t}^{2} - \left( {\frac{1}{2}{\left| Dw\right| }^{2} + F\left( w\right) }\right) {dxdt} \]\ndefined for functions \( w = w\left( {x, t}\right) \) with, say, compact support. As usual, we write \... | The integrand of (11) does not depend on the time variable \( t \) and is consequently invariant under shifts in this variable. Noether's Theorem implies that this invariance forces a conservation law, in this case conservation of energy. More precisely, we define\n\n\[ \mathbf{x}\left( {x, t,\tau }\right) \mathrel{\te... | Yes |
Example 1 (Reaction-diffusion equations). Let us investigate the solvability of the initial/boundary-value problem for the reaction-diffusion system\n\n\[ \n\\begin{cases} {\\mathbf{u}}_{t} - \\Delta \\mathbf{u} & = \\mathbf{f}\\left( \\mathbf{u}\\right) & & \\text{ in }{U}_{T} \\\\ \\mathbf{u} & = \\mathbf{0} & & \\te... | THEOREM 2 (Existence). There exists a unique weak solution of (2).\n\nProof. 1. We will apply Banach's Theorem in the space\n\n\[ \nX = C\\left( {\\left\\lbrack {0, T}\\right\\rbrack ;{L}^{2}\\left( {U;{\\mathbb{R}}^{m}}\\right) }\\right) \n\]\n\nwith the norm\n\n\[ \n\\parallel \\mathbf{v}\\parallel \\mathrel{\\text{:... | Yes |
The \( p \) -system is this collection of two conservation laws:\n\n\[\n\begin{cases} {u}_{t}^{1} - {u}_{x}^{2} & = 0\;\text{ (compatibility condition) } \\ {u}_{t}^{2} - p{\left( {u}^{1}\right) }_{x} & = 0\;\text{ (Newton’s law) } \end{cases}\n\]\n\nin \( \mathbb{R} \times \left( {0,\infty }\right) \), where \( p : \m... | Here\n\n\[\n\mathbf{F}\left( z\right) = \left( {-{z}_{2}, - p\left( {z}_{1}\right) }\right)\n\]\n\nfor \( z = \left( {{z}_{1},{z}_{2}}\right) \) . The \( p \) -system arises as a rewritten form of the scalar quasilinear wave equation\n\n\[\n{u}_{tt} - {\left( p\left( {u}_{x}\right) \right) }_{x} = 0\;\text{ in }\mathbb... | Yes |
Euler's equations for compressible gas flow in one dimension are\n\n\[ \n\begin{cases} {\rho }_{t} + {\left( \rho v\right) }_{x} & = 0\;\text{ (conservation of mass) } \\ {\left( \rho v\right) }_{t} + {\left( \rho {v}^{2} + p\right) }_{x} & = 0\;\text{ (conservation of momentum) } \\ {\left( \rho E\right) }_{t} + {\lef... | \[ \left\{ \begin{array}{l} {F}^{1}\left( z\right) = {z}_{2} \\ {F}^{2}\left( z\right) = \frac{{\left( {z}_{2}\right) }^{2}}{{z}_{1}} + p\left( {{z}_{1},\frac{{z}_{3}}{{z}_{1}} - \frac{1}{2}{\left( \frac{{z}_{2}}{{z}_{1}}\right) }^{2}}\right) \\ {F}^{3}\left( z\right) = \frac{{z}_{2}{z}_{3}}{{z}_{1}} + p\left( {{z}_{1}... | Yes |
Example 3. The one-dimensional shallow water equations are\n\n\\[ \n\\begin{cases} {h}_{t} + {\\left( vh\\right) }_{x} & = 0\\;\\text{ (conservation of mass) } \\\\ {\\left( vh\\right) }_{t} + {\\left( {v}^{2}h + \\frac{{h}^{2}}{2}\\right) }_{x} & = 0\\;\\text{ (conservation of momentum) } \\end{cases} \n\\] \n\nin \\(... | Here\n\n\\[ \n\\mathbf{F}\\left( z\\right) = \\left( {{z}_{2},\\frac{{z}_{2}^{2}}{{z}_{1}} + \\frac{{z}_{1}^{2}}{2}}\\right) \n\\] \n\nfor \\( z = \\left( {{z}_{1},{z}_{2}}\\right) ,{z}_{1} > 0 \\) . | Yes |
For the \( p \) -system (6), we have\n\n\[ \n{\mathbf{u}}_{t} + \mathbf{B}\left( \mathbf{u}\right) {\mathbf{u}}_{x} = 0 \n\]\n\nfor\n\n\[ \n\mathbf{B}\left( z\right) = D\mathbf{F}\left( z\right) = \left( \begin{matrix} 0 & - 1 \\ - {p}^{\prime }\left( {z}_{1}\right) & 0 \end{matrix}\right) .\n\] | The eigenvalues are \( {\lambda }_{1} = - \sigma ,{\lambda }_{2} = \sigma \), for \( \sigma \mathrel{\text{:=}} {p}^{\prime }{\left( {z}_{1}\right) }^{1/2} \) . These are real and distinct provided we hereafter suppose the strict hyperbolicity condition\n\n(41)\n\n\[ \n{p}^{\prime } > 0\text{.} \n\]\n\nFor the nonlinea... | Yes |
Euler's equations (9) comprise a strictly hyperbolic system provided we assume \( p > 0 \) and\n\n(42)\n\n\[ \frac{\partial p}{\partial \rho } > 0,\frac{\partial p}{\partial e} > 0 \] | Let us rather change variables and regard the density \( \rho \), velocity \( v \) and internal energy \( e \) as the unknowns. We can then rewrite Euler’s equations (9) in terms of these quantities and, in so doing, obtain after some calculations the system\n\n(43)\n\n\[ \left\{ \begin{array}{l} {\rho }_{t} + v{\rho }... | Yes |
In the case of a scalar conservation law (i.e. \( m = 1 \) ), for any convex \( \Phi \) we can find a corresponding flux function \( \Psi \) | \[ \Psi \left( z\right) \mathrel{\text{:=}} {\int }_{{z}_{0}}^{z}{\Phi }^{\prime }\left( w\right) {F}^{\prime }\left( w\right) {dw}\;\left( {z \in \mathbb{R}}\right) . \] | Yes |
For the \( p \) -system we have \( m = 2 \) . To verify (25),(26) we must find \( \Phi ,\Psi \), with \( \Phi \) convex and\n\n\[ \n\\left( {{\\Phi }_{{z}_{1}},{\Phi }_{{z}_{2}}}\\right) \\left( \\begin{matrix} 0 & - 1 \\ - {p}^{\\prime }\\left( {z}_{1}\\right) & 0 \\end{matrix}\\right) = \\left( \\begin{matrix} {\\Psi... | A solution is\n\n\[ \n\\Phi \\left( z\\right) = \\frac{{z}_{2}^{2}}{2} + P\\left( {z}_{1}\\right) ,\\;\\Psi \\left( z\\right) = - p\\left( {z}_{1}\\right) {z}_{2}\\;\\left( {z \\in {\\mathbb{R}}^{2}}\\right) ,\n\]\n\nwhere \( {P}^{\\prime } = p \) . Note \( \\Phi \) is convex, since \( {p}^{\\prime } > 0 \) . | Yes |
Example 2.2 (Coupled Vibrations) Consider the system of two masses \( {m}_{1} \) and \( {m}_{2} \) in Figure 1 connected to each other by a spring with spring constant \( {k}_{2} \) and to the walls by springs with spring constants \( {k}_{1} \) and \( {k}_{3} \) respectively. Let \( u\left( t\right) \) be the displace... | \[ {m}_{1}{u}^{\prime \prime } = - c{u}^{\prime } - \left( {{k}_{1} + {k}_{2}}\right) u + {k}_{2}v \] \[ {m}_{2}{v}^{\prime \prime } = - c{v}^{\prime } - \left( {{k}_{2} + {k}_{3}}\right) v + {k}_{2}u. \] Here we have a system of two second-order equations, and we define \( {x}_{1} \mathrel{\text{:=}} u \) , \( {x}_{2}... | Yes |
Example 2.5 Let \( \\mathbb{A} \) be the set of all \( n \\times 1 \) continuously differentiable vector functions on an interval \( I \) and let \( \\mathbb{B} \) be the set of all \( n \\times 1 \) continuous vector functions on an interval \( I \) and note that \( \\mathbb{A} \) and \( \\mathbb{B} \) are linear spac... | \[ \n= \\alpha {x}^{\\prime }\\left( t\\right) + \\beta {y}^{\\prime }\\left( t\\right) - {\\alpha A}\\left( t\\right) x\\left( t\\right) - {\\beta A}\\left( t\\right) y\\left( t\\right)\n\]\n\n\[ \n= \\alpha \\left\\lbrack {{x}^{\\prime }\\left( t\\right) - A\\left( t\\right) x\\left( t\\right) }\\right\\rbrack + \\be... | Yes |
Theorem 2.7 Assume we have exactly \( n \) constant \( n \times 1 \) vectors\n\n\[{\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n}\]\n\nand \( C \) is the column matrix \( C = \left\lbrack {{\psi }_{1}{\psi }_{2}\cdots {\psi }_{n}}\right\rbrack \) . Then \( {\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n} \) are linearly depende... | Proof Let \( {\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n} \) and \( C \) be as in the statement of this theorem. Then\n\n\[\det C = 0\]\n\nif and only if there is a nontrivial vector\n\n\[\left\lbrack \begin{matrix} {c}_{1} \\ {c}_{2} \\ \vdots \\ {c}_{n} \end{matrix}\right\rbrack\]\n\nsuch that\n\n\[C\left\lbrack \begi... | Yes |
Example 2.8 Since\n\n\[ \det \left\lbrack \begin{matrix} 1 & 2 & - 4 \\ 2 & 1 & 1 \\ - 3 & - 1 & - 3 \end{matrix}\right\rbrack = 0 \]\n\nthe vectors\n\n\[ {\psi }_{1} = \left\lbrack \begin{matrix} 1 \\ 2 \\ - 3 \end{matrix}\right\rbrack ,\;{\psi }_{2} = \left\lbrack \begin{matrix} 2 \\ 1 \\ - 1 \end{matrix}\right\rbrac... | \( \bigtriangleup \) | No |
Example 2.10 Show that the three vector functions \( {\phi }_{1},{\phi }_{2},{\phi }_{3} \) defined by\n\n\[ \n{\phi }_{1}\left( t\right) = \left\lbrack \begin{array}{l} t \\ t \end{array}\right\rbrack ,\;{\phi }_{2}\left( t\right) = \left\lbrack \begin{matrix} {t}^{2} \\ t \end{matrix}\right\rbrack ,\;{\phi }_{3}\left... | To see this, assume \( {c}_{1},{c}_{2},{c}_{3} \) are constants such that\n\n\[ \n{c}_{1}{\phi }_{1}\left( t\right) + {c}_{2}{\phi }_{2}\left( t\right) + {c}_{3}{\phi }_{3}\left( t\right) = 0 \n\]\n\nfor all \( t \in I \) . Then\n\n\[ \n{c}_{1}\left\lbrack \begin{array}{l} t \\ t \end{array}\right\rbrack + {c}_{2}\left... | Yes |
Theorem 2.11 The linear vector differential equation (2.3) has n linearly independent solutions on \( I \), and if \( {\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{n} \) are \( n \) linearly independent solutions on \( I \), then\n\n\[ x = {c}_{1}{\phi }_{1} + {c}_{2}{\phi }_{2} + \cdots + {c}_{n}{\phi }_{n} \]\n\n(2.5)\n\n... | Proof Let \( {\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n} \) be \( n \) linearly independent constant \( n \times 1 \) vectors and let \( {t}_{0} \in I \) . Then let \( {\phi }_{i} \) be the solution of the IVP\n\n\[ {x}^{\prime } = A\left( t\right) x,\;x\left( {t}_{0}\right) = {\psi }_{i}, \]\n\nfor \( 1 \leq i \leq n ... | Yes |
Example 2.13 Find eigenpairs for\n\n\[ A = \left\lbrack \begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}\right\rbrack \] | The characteristic equation of \( A \) is\n\n\[ \det \left( {A - {\lambda I}}\right) = \left| \begin{matrix} - \lambda & 1 \\ - 2 & - 3 - \lambda \end{matrix}\right| = 0 \]\n\nSimplifying, we have\n\n\[ {\lambda }^{2} + {3\lambda } + 2 = \left( {\lambda + 2}\right) \left( {\lambda + 1}\right) = 0. \]\n\nHence the eigen... | Yes |
Theorem 2.14 If \( {\lambda }_{0},{x}_{0} \) is an eigenpair for the constant \( n \times n \) matrix \( A \) , then \[ x\left( t\right) = {e}^{{\lambda }_{0}t}{x}_{0},\;t \in \mathbb{R}, \] defines a solution \( x \) of \[ {x}^{\prime } = {Ax} \] (2.7) on \( \mathbb{R} \) . | Proof Let \[ x\left( t\right) = {e}^{{\lambda }_{0}t}{x}_{0} \] then \[ {x}^{\prime }\left( t\right) = {\lambda }_{0}{e}^{{\lambda }_{0}t}{x}_{0} \] \[ = {e}^{{\lambda }_{0}t}{\lambda }_{0}{x}_{0} \] \[ = {e}^{{\lambda }_{0}t}A{x}_{0} \] \[ = A{e}^{{\lambda }_{0}t}{x}_{0} \] \[ = {Ax}\left( t\right) \] for \( t \in \ma... | Yes |
Solve the differential equation\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}\right\rbrack x \n\] | From Example 2.13 we get that the eigenpairs for\n\n\[ \nA \mathrel{\text{:=}} \left\lbrack \begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}\right\rbrack \n\]\n\nare\n\[ \n- 2,\left\lbrack \begin{matrix} 1 \\ - 2 \end{matrix}\right\rbrack \text{ and } - 1,\left\lbrack \begin{matrix} 1 \\ - 1 \end{matrix}\right\rbrack \n\... | No |
Theorem 2.16 If \( x = u + {iv} \) is a complex vector-valued solution of (2.3), where \( u, v \) are real vector-valued functions, then \( u, v \) are real vector-valued solutions of (2.3). | Proof Assume \( x \) is as in the statement of the theorem. Then\n\n\[ \n{x}^{\prime }\left( t\right) = {u}^{\prime }\left( t\right) + i{v}^{\prime }\left( t\right) = A\left( t\right) \left\lbrack {u\left( t\right) + {iv}\left( t\right) }\right\rbrack ,\;\text{ for }\;t \in I, \n\] \n\nor \n\n\[ \n{u}^{\prime }\left( t... | Yes |
Example 2.17 Solve the differential equation\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} 3 & 1 \\ - {13} & - 3 \end{matrix}\right\rbrack x \n\] | The characteristic equation of the coefficient matrix is\n\n\[ \n{\lambda }^{2} + 4 = 0 \n\]\n\nand the eigenvalues are\n\n\[ \n{\lambda }_{1} = {2i},\;{\lambda }_{2} = - {2i} \n\]\n\nTo find an eigenvector corresponding to \( {\lambda }_{1} = {2i} \), we solve\n\n\[ \n\left( {A - {2iI}}\right) x = 0 \n\]\n\nor\n\[ \n\... | Yes |
Theorem 2.20 Assume \( A \) is a continuous \( n \times n \) matrix function on an interval \( I \) and assume that \( \Phi \) defined by\n\n\[ \Phi \left( t\right) = \left\lbrack {{\phi }_{1}\left( t\right) ,{\phi }_{2}\left( t\right) ,\cdots ,{\phi }_{n}\left( t\right) }\right\rbrack ,\;t \in I,\n\]\n\nis the \( n \t... | Proof Assume \( {\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{n} \) are solutions of (2.3) on \( I \) and define the \( n \times n \) matrix function \( \Phi \) by\n\n\[ \Phi \left( t\right) = \left\lbrack {{\phi }_{1}\left( t\right) ,{\phi }_{2}\left( t\right) ,\cdots ,{\phi }_{n}\left( t\right) }\right\rbrack ,\;t \in I.\... | No |
Theorem 2.21 (Existence-Uniqueness Theorem) Assume A is a continuous matrix function on an interval I. Then the IVP\n\n\[ \n{X}^{\prime } = A\left( t\right) X,\;X\left( {t}_{0}\right) = {X}_{0}, \]\n\nwhere \( {t}_{0} \in I \) and \( {X}_{0} \) is an \( n \times n \) constant matrix, has a unique solution \( X \) that ... | Proof This theorem follows from Theorem 2.3 and the fact that \( X \) is a solution of the matrix equation (2.9) iff each of its columns is a solution of the vector equation (2.3). | No |
Corollary 2.24 Assume \( {\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{n} \) are \( n \) solutions of the vector equation (2.3) on \( I \) and \( \Phi \) is the matrix function with columns \( {\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{n} \) . Then either\n\n(a) \( \det \Phi \left( t\right) = 0,\; \) for all \( t \in I \) ,\n... | Proof The first statement of this theorem follows immediately from Liouville's formula in Theorem 2.23. The proof of the statements concerning linear independence and linear dependence is left as an exercise (see Exercise 2.24). | No |
Example 2.25 Show that the vector functions \( {\phi }_{1},{\phi }_{2} \) defined by\n\n\[ \n{\phi }_{1}\left( t\right) = \left\lbrack \begin{matrix} \cos \left( {2t}\right) \\ - 3\cos \left( {2t}\right) - 2\sin \left( {2t}\right) \end{matrix}\right\rbrack ,\;{\phi }_{2}\left( t\right) = \left\lbrack \begin{matrix} \si... | In Example 2.17 we saw that \( {\phi }_{1},{\phi }_{2} \) are solutions of the vector differential equation (2.8) on \( \mathbb{R} \) . Let \( \Phi \) be the matrix function with columns \( {\phi }_{1} \) and \( {\phi }_{2} \), respectively. Then\n\n\[ \n\det \Phi \left( 0\right) = \left| \begin{matrix} 1 & 0 \\ - 3 & ... | Yes |
Let's show that the four real solutions computed in Example 2.18 involving the oscillations of two masses are linearly independent on \( \mathbb{R} \) . | If we evaluate each solution at \( t = 0 \), then we obtain the following determinant:\n\n\[ \left| \begin{matrix} 1 & 0 & 1 & 0 \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} & - \frac{1}{2} & \frac{\sqrt{11}}{2} \\ 1 & 0 & - 1 & 0 \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{2} & - \frac{\sqrt{11}}{2} \end{matrix}\right| ... | Yes |
Example 2.27 Show that the vector functions \( {\phi }_{1},{\phi }_{2} \) defined by\n\n\[ \n{\phi }_{1}\left( t\right) = \left\lbrack \begin{matrix} {t}^{2} \\ 1 \end{matrix}\right\rbrack ,\;{\phi }_{2}\left( t\right) = \left\lbrack \begin{matrix} t \cdot \left| t\right| \\ 1 \end{matrix}\right\rbrack \n\]\n\nfor \( t... | Assume \( {c}_{1},{c}_{2} \) are constants such that\n\n\[ \n{c}_{1}{\phi }_{1}\left( t\right) + {c}_{2}{\phi }_{2}\left( t\right) = 0 \n\]\n\nfor \( t \in \mathbb{R} \) . Then\n\[ \n{c}_{1}\left\lbrack \begin{matrix} {t}^{2} \\ 1 \end{matrix}\right\rbrack + {c}_{2}\left\lbrack \begin{matrix} t \cdot \left| t\right| \\... | Yes |
An \( n \times n \) matrix function \( \Phi \) is a fundamental matrix for the vector differential equation (2.3) iff the columns of \( \Phi \) are \( n \) linearly independent solutions of (2.3) on I. If \( \Phi \) is a fundamental matrix for the vector differential equation (2.3), then a general solution \( x \) of (... | Proof Assume \( \Phi \) is an \( n \times n \) matrix function whose columns are linearly independent solutions of (2.3) on \( I \) . Since the columns of \( \Phi \) are solutions of (2.3), we have by Theorem 2.20 that \( \Phi \) is a solution of the matrix equation (2.9). Since the columns of \( \Phi \) are linearly i... | No |
Find a fundamental matrix \( \Phi \) for\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} - 2 & 3 \\ 2 & 3 \end{matrix}\right\rbrack x \]\n\n(2.10)\n\nVerify that \( \Phi \) is a fundamental matrix and then write down a general solution of this vector differential equation in terms of this fundamental matrix. | The characteristic equation is\n\n\[ \n{\lambda }^{2} - \lambda - {12} = 0 \]\n\nand so the eigenvalues are \( {\lambda }_{1} = - 3,{\lambda }_{2} = 4 \) . Corresponding eigenvectors\n\nare\n\[ \n\left\lbrack \begin{matrix} 3 \\ - 1 \end{matrix}\right\rbrack \text{ and }\left\lbrack \begin{array}{l} 1 \\ 2 \end{array}\... | Yes |
Theorem 2.31 If \( \Phi \) is a fundamental matrix for (2.3), then \( \Psi = {\Phi C} \) where \( C \) is an arbitrary \( n \times n \) nonsingular constant matrix is a general fundamental matrix of (2.3). | Proof Assume \( \Phi \) is a fundamental matrix for (2.3) and set\n\n\[ \Psi = {\Phi C} \]\n\nwhere \( C \) is an \( n \times n \) constant matrix. Then \( \Psi \) is continuously differentiable on \( I \) and\n\n\[ {\Psi }^{\prime }\left( t\right) = {\Phi }^{\prime }\left( t\right) C \]\n\n\[ = A\left( t\right) \Phi \... | Yes |
Verify the Cayley-Hamilton Theorem (Theorem 2.33) directly for the matrix\n\n\[ A = \left\lbrack \begin{array}{ll} 2 & 3 \\ 4 & 1 \end{array}\right\rbrack \] | The characteristic equation for \( A \) is\n\n\[ \left| \begin{matrix} 2 - \lambda & 3 \\ 4 & 1 - \lambda \end{matrix}\right| = {\lambda }^{2} - {3\lambda } - {10} = 0.\n\nNow\n\n\[ {A}^{2} - {3A} - {10I} = \left\lbrack \begin{matrix} {16} & 9 \\ {12} & {13} \end{matrix}\right\rbrack - \left\lbrack \begin{matrix} 6 & 9... | Yes |
Theorem 2.35 (Putzer Algorithm for Finding \( {e}^{At} \) ) Let \( {\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{n} \) be the (not necessarily distinct) eigenvalues of the matrix \( A \) . Then\n\n\[ \n{e}^{At} = \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}{p}_{k + 1}\left( t\right) {M}_{k} \]\n\nwhere \( {M}_{0} \mat... | Proof Let\n\n\[ \n\Phi \left( t\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}{p}_{k + 1}\left( t\right) {M}_{k},\;\text{ for }t \in \mathbb{R}, \]\n\nwhere \( {p}_{k},1 \leq k \leq n \) and \( {M}_{k},0 \leq k \leq n \) are as in the statement of this theorem. Then by the uniqueness theorem (Theo... | Yes |
Use the Putzer algorithm (Theorem 2.35) to find \( {e}^{At} \) when\n\n\[ A \mathrel{\text{:=}} \left\lbrack \begin{matrix} 1 & 1 \\ - 1 & 3 \end{matrix}\right\rbrack \] | The characteristic equation for \( A \) is\n\n\[ {\lambda }^{2} - {4\lambda } + 4 = 0 \]\n\nso \( {\lambda }_{1} = {\lambda }_{2} = 2 \) are the eigenvalues. By the Putzer algorithm (Theorem 2.35),\n\n\[ {e}^{At} = \mathop{\sum }\limits_{{k = 0}}^{1}{p}_{k + 1}\left( t\right) {M}_{k} = {p}_{1}\left( t\right) {M}_{0} + ... | Yes |
Example 2.37 (Complex Eigenvalues) Use the Putzer algorithm (Theorem 2.35) to find \( {e}^{At} \) when\n\n\[ A \mathrel{\text{:=}} \left\lbrack \begin{array}{ll} 1 & - 1 \\ 5 & - 1 \end{array}\right\rbrack \]\n\nThe characteristic equation for \( A \) is\n\n\[ {\lambda }^{2} + 4 = 0 \]\n\nso \( {\lambda }_{1} = {2i},{\... | \[ {e}^{At} = \mathop{\sum }\limits_{{k = 0}}^{1}{p}_{k + 1}\left( t\right) {M}_{k} = {p}_{1}\left( t\right) {M}_{0} + {p}_{2}\left( t\right) {M}_{1}. \]\n\nNow\n\n\[ {M}_{0} = I = \left\lbrack \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right\rbrack \]\n\nand\n\n\[ {M}_{1} = A - {\lambda }_{1}I = \left\lbrack \begin{... | Yes |
Theorem 2.39 Assume \( A \) and \( B \) are \( n \times n \) constant matrices. Then\n\n(i) \( \frac{d}{dt}{e}^{At} = A{e}^{At},\; \) for \( t \in \mathbb{R} \) ,\n\n(ii) \( \det \left\lbrack {e}^{At}\right\rbrack \neq 0 \), for \( t \in \mathbb{R} \) and \( {e}^{At} \) is a fundamental matrix for \( \left( {2.7}\right... | Proof The result (i) follows immediately from the definition of \( {e}^{At} \) .\n\nSince \( {e}^{At} \) is the identity matrix at \( t = 0 \) and \( \det \left( I\right) = 1 \neq 0 \), we get from Corollary 2.24 that \( \det \left( {e}^{At}\right) \neq 0 \), for all \( t \in \mathbb{R} \) and so (ii) holds.\n\nWe now ... | No |
Theorem 2.40 (Variation of Constants Formula) Assume that \( A \) is an \( n \times n \) continuous matrix function on an interval \( I, b \) is a continuous \( n \times 1 \) vector function on \( I \), and \( \Phi \) is a fundamental matrix for (2.3). Then the solution of the IVP\n\n\[ \n{x}^{\prime } = A\left( t\righ... | Proof The uniqueness of the solution of the given IVP follows from Theorem 2.3. Let \( \Phi \) be a fundamental matrix for (2.3) and set\n\n\[ \nx\left( t\right) = \Phi \left( t\right) {\Phi }^{-1}\left( {t}_{0}\right) {x}_{0} + \Phi \left( t\right) {\int }_{{t}_{0}}^{t}{\Phi }^{-1}\left( s\right) b\left( s\right) {ds}... | Yes |
Assume \( A \) is an \( n \times n \) constant matrix and \( b \) is a continuous \( n \times 1 \) vector function on an interval \( I \) . Then the solution \( x \) of the IVP\n\n\[ \n{x}^{\prime } = {Ax} + b\left( t\right) ,\;x\left( {t}_{0}\right) = {x}_{0}, \n\]\n\nwhere \( {t}_{0} \in I,{x}_{0} \in {\mathbb{R}}^{n... | Proof Letting \( \Phi \left( t\right) = {e}^{At} \) in the general variation of constants formula in Theorem 2.40, we get, using the fact that \( {\left\{ {e}^{At}\right\} }^{-1} = {e}^{-{At}} \) ,\n\n\[ \nx\left( t\right) = \Phi \left( t\right) {\Phi }^{-1}\left( {t}_{0}\right) {x}_{0} + \Phi \left( t\right) {\int }_{... | Yes |
Theorem 2.42 Assume \( A\left( t\right) \) is a continuous \( n \times n \) matrix function on an interval I. If\n\n\[ A\left( t\right) A\left( s\right) = A\left( s\right) A\left( t\right) \]\n\nfor all \( t, s \in I \), then\n\n\[ \Phi \left( t\right) \mathrel{\text{:=}} {e}^{{\int }_{{t}_{0}}^{t}A\left( s\right) {ds}... | Proof Let\n\n\[ \Phi \left( t\right) \mathrel{\text{:=}} {e}^{{\int }_{{t}_{0}}^{t}A\left( s\right) {ds}} \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{k!}{\left\lbrack {\int }_{{t}_{0}}^{t}A\left( s\right) ds\right\rbrack }^{k}. \]\n\nWe leave it to the reader to show that the infinite series o... | No |
Use the variation of constants formula to solve the IVP\n\n\\[ \n{x}^{\\prime } = \\left\\lbrack \\begin{matrix} 1 & 1 \\\\ - 1 & 3 \\end{matrix}\\right\\rbrack x + \\left\\lbrack \\begin{matrix} {e}^{2t} \\\\ 2{e}^{2t} \\end{matrix}\\right\\rbrack ,\\;x\\left( 0\\right) = \\left\\lbrack \\begin{array}{l} 2 \\\\ 1 \\en... | Since\n\n\\[ \nA \\mathrel{\\text{:=}} \\left\\lbrack \\begin{matrix} 1 & 1 \\\\ - 1 & 3 \\end{matrix}\\right\\rbrack \n\\]\n\nwe have from Example 2.36 that\n\n\\[ \n{e}^{At} = {e}^{2t}\\left\\lbrack \\begin{matrix} 1 - t & t \\\\ - t & 1 + t \\end{matrix}\\right\\rbrack .\n\\]\n\nFrom the variation of constants formu... | Yes |
Use Theorem 2.40 and Theorem 2.42 to help you solve the IVP\n\n\\[ \n{x}^{\\prime } = \\left\\lbrack \\begin{matrix} \\frac{1}{t} & 0 \\\\ 0 & \\frac{2}{t} \\end{matrix}\\right\\rbrack x + \\left\\lbrack \\begin{matrix} {t}^{2} \\\\ t \\end{matrix}\\right\\rbrack ,\\;x\\left( 1\\right) = \\left\\lbrack \\begin{matrix} ... | By Theorem 2.42,\n\n\\[ \n\\Phi \\left( t\\right) = {e}^{{\\int }_{1}^{t}A\\left( s\\right) {ds}} = {e}^{{\\int }_{1}^{t}\\left\\lbrack \\begin{matrix} \\frac{1}{s} & 0 \\\\ 0 & \\frac{2}{s} \\end{matrix}\\right\\rbrack {ds}}\n\\]\n\n\\[ \n= {e}^{\\left\\lbrack \\begin{matrix} \\ln t & 0 \\\\ 0 & 2\\ln t \\end{matrix}\... | Yes |
Three important examples of norms on \( {\mathbb{R}}^{n} \) are\n\n(i) the Euclidean norm \( \left( {{l}_{2}\text{norm}}\right) \) defined by\n\n\[ \parallel x{\parallel }_{2} \mathrel{\text{:=}} \sqrt{{x}_{1}^{2} + {x}_{2}^{2} + \cdots + {x}_{n}^{2}} \]\n\n(ii) the maximum norm \( \left( {{l}_{\infty }\text{norm}}\rig... | We leave it to the reader to check that these examples are actually norms. | No |
Theorem 2.49 (Stability Theorem) Assume A is an \( n \times n \) constant matrix.\n\n(iii) If all the eigenvalues of \( A \) have negative real parts, then the trivial solution of \( {x}^{\prime } = {Ax} \) is globally asymptotically stable on \( \lbrack 0,\infty ) \) . | We now prove part \( \left( {iii}\right) \) of this theorem. By part \( \left( {ii}\right) \) the trivial solution is stable on \( \lbrack 0,\infty ) \), so it remains to show that every solution approaches the zero vector as \( t \rightarrow \infty \) . Let \( {\lambda }_{1},\cdots ,{\lambda }_{n} \) be the eigenvalue... | Yes |
Example 2.51 Determine the stability of the trivial solution of\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}\right\rbrack x \n\]\n\non \( \lbrack 0,\infty ) \) . | The characteristic equation is\n\n\[ \n{\lambda }^{2} + 1 = 0 \n\]\n\nand hence the eigenvalues are \( {\lambda }_{1} = i,{\lambda }_{2} = - i \) . Since both eigenvalues have zero real parts and both eigenvalues are simple, the trivial solution is stable on \( \lbrack 0,\infty ) \) . | Yes |
Example 2.52 Determine the stability of the trivial solution of\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} - 2 & 1 & 0 \\ - 1 & - 2 & 0 \\ 0 & 0 & 1 \end{matrix}\right\rbrack x \n\]\n\non \( \lbrack 0,\infty ) \) . | The characteristic equation is\n\n\[ \n\left( {{\lambda }^{2} + {4\lambda } + 5}\right) \left( {\lambda - 1}\right) = 0 \n\]\n\nand hence the eigenvalues are \( {\lambda }_{1} = - 2 + i,{\lambda }_{2} = - 2 - i \), and \( {\lambda }_{3} = 1 \) . Since the eigenvalue \( {\lambda }_{3} \) has a positive real part, the tr... | Yes |
Theorem 2.54 The matrix norm induced by the vector norm \( \parallel \cdot \parallel \) is given by\n\n\[ \parallel A\parallel = \mathop{\max }\limits_{{x \neq 0}}\frac{\parallel {Ax}\parallel }{\parallel x\parallel } \]\n\nIn particular,\n\n\[ \parallel {Ax}\parallel \leq \parallel A\parallel \cdot \parallel x\paralle... | Proof This result follows from the following statement. If \( x \neq 0 \), then\n\n\[ \frac{\parallel {Ax}\parallel }{\parallel x\parallel } = \parallel A\left( \frac{x}{\parallel x\parallel }\right) \parallel = \parallel {Ay}\parallel \]\n\nwhere \( y = \frac{x}{\parallel x\parallel } \) is a unit vector. | Yes |
Theorem 2.55 The matrix norm induced by the traffic norm ( \( {l}_{1} \) norm) is given by\n\n\[ \parallel A{\parallel }_{1} = \mathop{\max }\limits_{{1 \leq j \leq n}}\mathop{\sum }\limits_{{i = 1}}^{n}\left| {a}_{ij}\right| \] | Proof Let \( \parallel \cdot {\parallel }_{1} \) be the traffic norm on \( {\mathbb{R}}^{n} \), let \( A \in {M}_{n}, x \in {\mathbb{R}}^{n} \), and consider\n\n\[ \parallel {Ax}{\parallel }_{1} = {\begin{Vmatrix}\left( \begin{matrix} \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{1j}{x}_{j} \\ \cdots \\ \mathop{\sum }\limits... | Yes |
Corollary 2.60 If \( \mu \left( A\right) < 0 \), then the trivial solution of the vector equation \( {x}^{\prime } = {Ax} \) is globally asymptotically stable. | Proof This follows from part (v) of Theorem 2.59 and Theorem 2.49. | Yes |
if \( {\mu }_{1} \) corresponds to the traffic vector norm, then, for \( A \in {M}_{n} \n\n\[ \n{\mu }_{1}\left( A\right) = \mathop{\max }\limits_{{1 \leq j \leq n}}\left\{ {{a}_{jj} + \mathop{\sum }\limits_{{i = 1, i \neq j}}^{n}\left| {a}_{ij}\right| }\right\} \n\] | Proof We will prove part (i) here. The proof of part (ii) is Exercise 2.59 and the proof of part (iii) is nontrivial, but is left to the reader. Using Theorem 2.55\n\n\[ \n{\mu }_{1}\left( A\right) = \mathop{\lim }\limits_{{h \rightarrow 0 + }}\frac{\parallel I + {hA}{\parallel }_{1} - 1}{h} \n\]\n\n\[ \n= \mathop{\lim... | No |
The trivial solution of the vector equation\n\n\[ \n{x}^{\prime } = \left( \begin{matrix} - {3.3} & {.3} & 3 & - {.4} \\ 1 & - 2 & 1 & {1.3} \\ - {1.2} & {.4} & - 5 & {.2} \\ - 1 & {.8} & {.5} & - 2 \end{matrix}\right) x \n\]\n\n(2.15)\n\nis globally asymptotically stable because | \[ \n{\mu }_{1}\left( A\right) = - {.1} < 0 \n\]\n\nwhere \( A \) is the coefficient matrix in (2.15). Note that \( {\mu }_{\infty }\left( A\right) = {1.3} > 0 \) . \( \;\bigtriangleup \) | Yes |
Theorem 2.63 (Jordan Canonical Form) If \( A \) is an \( n \times n \) constant matrix, then there is a nonsingular \( n \times n \) constant matrix \( P \) so that \( A = {PJ}{P}^{-1} \) , where \( J \) is a block diagonal matrix of the form\n\n\[ J = \left\lbrack \begin{array}{llll} {J}_{1} & 0 & \cdots & 0 \\ 0 & {J... | Proof We will only prove this theorem for \( 2 \times 2 \) matrices \( A \) . For a proof of the general result, see Horn and Johnson [24]. There are two cases:\n\nCase 1: A has two linearly independent eigenvectors \( {x}^{1} \) and \( {x}^{2} \) .\n\nIn this case we have eigenpairs \( {\lambda }_{1},{x}^{1} \) and \(... | No |
Theorem 2.64 (Log of a Matrix) If \( C \) is an \( n \times n \) nonsingular matrix, then there is a matrix \( B \) such that\n\n\[ \n{e}^{B} = C\text{.}\n\] | Proof We will just prove this theorem for \( 2 \times 2 \) matrices. Let \( {\mu }_{1},{\mu }_{2} \) be the eigenvalues of \( \mathrm{C} \) . Since \( C \) is nonsingular \( {\mu }_{1},{\mu }_{2} \neq 0 \) . First we prove the result for two special cases.\n\nCase 1. Assume\n\n\[ \nC = \left\lbrack \begin{array}{ll} {\... | Yes |
Find a log of the matrix\n\n\[ C = \left\lbrack \begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right\rbrack \] | The characteristic equation for \( C \) is\n\n\[ {\lambda }^{2} - {6\lambda } + 5 = 0 \]\n\nand so the eigenvalues are \( {\lambda }_{1} = 1,{\lambda }_{2} = 5 \) . The Jordan canonical form (see Theorem 2.63) of \( C \) is\n\n\[ J = \left\lbrack \begin{array}{ll} 1 & 0 \\ 0 & 5 \end{array}\right\rbrack \]\n\nEigenpair... | Yes |
Consider the scalar differential equation\n\n\[ \n{x}^{\prime } = \left( {{\sin }^{2}t}\right) x \n\] | A general solution of this differential equation is\n\n\[ \n\phi \left( t\right) = c{e}^{\frac{1}{2}t - \frac{1}{4}\sin \left( {2t}\right) }. \n\]\n\nNote that even though the coefficient function in our differential equation is periodic with minimum period \( \pi \), the only period \( \pi \) solution of our different... | Yes |
Theorem 2.67 (Floquet’s Theorem) If \( \Phi \) is a fundamental matrix for the Floquet system \( {x}^{\prime } = A\left( t\right) x \), where the matrix function \( A \) is continuous on \( \mathbb{R} \) and has minimum positive period \( \omega \), then the matrix function \( \Psi \) defined by \( \Psi \left( t\right)... | Proof Assume \( \Phi \) is a fundamental matrix for the Floquet system \( {x}^{\prime } = \) \( A\left( t\right) x \) . Define the matrix function \( \Psi \) by\n\n\[ \Psi \left( t\right) = \Phi \left( {t + \omega }\right) \]\n\nfor \( t \in \mathbb{R} \) . Then\n\n\[ {\Psi }^{\prime }\left( t\right) = {\Phi }^{\prime ... | Yes |
Find the Floquet multipliers for the scalar differential equation\n\n\[ \n{x}^{\prime } = \left( {{\sin }^{2}t}\right) x \n\] | In Example 2.66 we saw that a nontrivial solution of this differential equation is\n\n\[ \n\phi \left( t\right) = {e}^{\frac{1}{2}t - \frac{1}{4}\sin \left( {2t}\right) }. \n\]\n\nHence\n\n\[ \nc \mathrel{\text{:=}} {\phi }^{-1}\left( 0\right) \phi \left( \pi \right) = {e}^{\frac{\pi }{2}} \n\]\n\nand so \( \mu = {e}^{... | Yes |
Find the Floquet multipliers for the Floquet system\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} 1 & 1 \\ 0 & \frac{\left( \cos t + \sin t\right) }{\left( 2 + \sin t - \cos t\right) } \end{matrix}\right\rbrack x. \n\] | Solving this equation first for \( {x}_{2} \) and then \( {x}_{1} \), we get that\n\n\[ \n{x}_{2}\left( t\right) = \beta \left( {2 + \sin t - \cos t}\right) \n\]\n\n\[ \n{x}_{1}\left( t\right) = \alpha {e}^{t} - \beta \left( {2 + \sin t}\right) \n\]\n\nfor \( t \in \mathbb{R} \) . It follows that a fundamental matrix f... | Yes |
Theorem 2.71 Let \( \Phi \left( t\right) = P\left( t\right) {e}^{Bt} \) be as in Floquet’s theorem (Theorem 2.67). Then \( x \) is a solution of the Floquet system \( {x}^{\prime } = A\left( t\right) x \) iff the vector function \( y \) defined by \( y\left( t\right) = {P}^{-1}\left( t\right) x\left( t\right), t \in \m... | Proof Assume \( x \) is a solution of the Floquet system \( {x}^{\prime } = A\left( t\right) x \) . Then\n\n\[ x\left( t\right) = \Phi \left( t\right) {x}_{0} \]\n\nfor some \( n \times 1 \) constant vector \( {x}_{0} \) . Let \( y\left( t\right) = {P}^{-1}\left( t\right) x\left( t\right) \) . Then\n\n\[ y\left( t\righ... | Yes |
Theorem 2.72 Let \( {\mu }_{1},{\mu }_{2},\cdots ,{\mu }_{n} \) be the Floquet multipliers of the Floquet system \( {x}^{\prime } = A\left( t\right) x \) . Then the trivial solution is\n\n(i) globally asymptotically stable on \( \lbrack 0,\infty ) \) iff \( \left| {\mu }_{i}\right| < 1,1 \leq i \leq n \) ;\n\n(ii) stab... | Proof We will just prove this theorem for the two-dimensional case. Let \( \Phi \left( t\right) = P\left( t\right) {e}^{Bt} \) and \( C \) be as in Floquet’s theorem. Recall that in the proof of Floquet’s theorem, \( B \) was picked so that\n\n\[ {e}^{B\omega } = C\text{.}\]\n\nBy the Jordan canonical form theorem (The... | Yes |
The number \( {\mu }_{0} \) is a Floquet multiplier of the Floquet system \( {x}^{\prime } = A\left( t\right) x \) iff there is a nontrivial solution \( x \) such that\n\n\[ x\left( {t + \omega }\right) = {\mu }_{0}x\left( t\right) \]\n\nfor all \( t \in \mathbb{R} \) . Consequently, the Floquet system has a nontrivial... | Proof First assume \( {\mu }_{0} \) is a Floquet multiplier of the Floquet system \( {x}^{\prime } = \) \( A\left( t\right) x \) . Then \( {\mu }_{0} \) is an eigenvalue of\n\n\[ C \mathrel{\text{:=}} {\Phi }^{-1}\left( 0\right) \Phi \left( \omega \right) \]\n\nwhere \( \Phi \) is a fundamental matrix of \( {x}^{\prime... | Yes |
Theorem 2.74 Assume \( {\mu }_{1},{\mu }_{2},\cdots ,{\mu }_{n} \) are the Floquet multipliers of the Floquet system \( {x}^{\prime } = A\left( t\right) x \) . Then\n\n\[ \n{\mu }_{1}{\mu }_{2}\cdots {\mu }_{n} = {e}^{{\int }_{0}^{\omega }\operatorname{tr}\left\lbrack {A\left( t\right) }\right\rbrack {dt}}.\n\] | Proof Let \( \Phi \) be the solution of the matrix IVP\n\n\[ \n{X}^{\prime } = A\left( t\right) X,\;X\left( 0\right) = I.\n\]\n\nThen \( \Phi \) is a fundamental matrix for \( {x}^{\prime } = A\left( t\right) x \) and\n\n\[ \nC \mathrel{\text{:=}} {\Phi }^{-1}\left( 0\right) \Phi \left( \omega \right) = \Phi \left( \om... | Yes |
Consider the scalar differential equation (Hill's equation)\n\n\[ \n{y}^{\prime \prime } + q\left( t\right) y = 0, \n\]\n\nwhere we assume that \( q \) is a continuous periodic function on \( \mathbb{R} \) with minimum positive period \( \omega \) . | Writing Hill’s equation as a system in the standard way, we get the Floquet system\n\n\[ \n{x}^{\prime } = \left\lbrack \begin{matrix} 0 & 1 \\ - q\left( t\right) & 0 \end{matrix}\right\rbrack x. \n\]\n\nBy the Floquet multipliers of Hill's equation we mean the Floquet multipliers of the preceding Floquet system. It fo... | Yes |
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