| { | |
| "id": "DAECM01", | |
| "url": "https://www.bmtdynamics.org/cgi-bin/display.pl?name=DAECM01", | |
| "title": "Computing Validated Solutions of Implicit Differential Equations", | |
| "pdfs": [ | |
| { | |
| "url": "https://www.bmtdynamics.org/pub/papers/DAECM01/DAECM01.pdf", | |
| "filename": "DAECM01.pdf", | |
| "size_bytes": null | |
| } | |
| ], | |
| "page_text": "Publications\nReprint Server\nComputing Validated Solutions of Implicit Differential Equations\nAbstract\nOrdinary differential equations (ODEs), including high-order implicit equations, describe\nimportant problems in mechanical and chemical engineering. However, the use of selfvalidated\nmethods providing rigorous enclosures of the solution has mostly been limited to\nexplicit and weakly nonlinear problems, and no general-purpose algorithm for the validated\nintegration of general ODE initial value problems has been developed. Since most integration\ntechniques for Differential Algebraic Equations (DAEs) are based on transformation to implicit\nODEs, the integration of DAE initial value problems has traditionally been restricted to\nfew hand-picked problems from the relatively small class of low-index systems. The recently\ndeveloped Taylor model method combines high-order differential algebraic descriptions of\nfunctional dependencies with intervals for verification. It has proven its power in several applications,\nincluding verified integration of ODEs under avoidance of the wrapping effect.\nRecognizing antiderivation (integration) as a natural operation on Taylor models yields methods\nthat treat DEs within a fully differential algebraic context as implicit equations made of\nconventional functions and antiderivation. This method has the potential to be applied to highindex\nDAE problems and allows the computation of guaranteed enclosures of final conditions\nfrom large initial regions for large classes of initial value problems. In the framework of this\nmethod, a Taylor model represents the highest derivative of the solution function occurring in\nthe DE and all lower derivatives are treated as antiderivatives of this Taylor model. Consequently,\none obtains a set of implicit equations involving only the highest derivative. Utilizing\nmethods of verified inversion of functional dependencies described by Taylor models allows\nthe computation of a guaranteed enclosure of the solution in the form of a Taylor model. The\nperformance of the method is illustrated by detailed examples.\nJ. Hoefkens, M. Berz, K. Makino,\nAdvances in Computational Mathematics\n19\n(2003) 231-253\nDownload\nClick on the icon to download the corresponding file.\nDownload Adobe PDF version (179086 Bytes).\nGo Back\nto the reprint server.\nGo Back\nto the home page.\nThis page is maintained by\nKyoko Makino\n. Please contact her if there are any problems with it." | |
| } |