NDLI: Computing Validated Solutions of Implicit Differential Equations

Content Provider	Springer Nature Link
Author	Hoefkens, Jens Berz, Martin Maki, Kyoko
Copyright Year	2003
Abstract	Ordinary differential equations (ODEs), including high-order implicit equations, describe important problems in mechanical and chemical engineering. However, the use of self-validated methods providing rigorous enclosures of the solution has mostly been limited to explicit and weakly nonlinear problems, and no general-purpose algorithm for the validated integration of general ODE initial value problems has been developed. Since most integration techniques for Differential Algebraic Equations (DAEs) are based on transformation to implicit ODEs, the integration of DAE initial value problems has traditionally been restricted to few hand-picked problems from the relatively small class of low-index systems. The recently developed Taylor model method combines high-order differential algebraic descriptions of functional dependencies with intervals for verification. It has proven its power in several applications, including verified integration of ODEs under avoidance of the wrapping effect. Recognizing antiderivation (integration) as a natural operation on Taylor models yields methods that treat DEs within a fully differential algebraic context as implicit equations made of conventional functions and antiderivation. This method has the potential to be applied to high-index DAE problems and allows the computation of guaranteed enclosures of final conditions from large initial regions for large classes of initial value problems. In the framework of this method, a Taylor model represents the highest derivative of the solution function occurring in the DE and all lower derivatives are treated as antiderivatives of this Taylor model. Consequently, one obtains a set of implicit equations involving only the highest derivative. Utilizing methods of verified inversion of functional dependencies described by Taylor models allows the computation of a guaranteed enclosure of the solution in the form of a Taylor model. The performance of the method is illustrated by detailed examples.
Starting Page	231
Ending Page	253
Page Count	23
File Format	PDF
ISSN	10197168
Journal	Advances in Computational Mathematics
Volume Number	19
Issue Number	1-3
e-ISSN	15729044
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2003-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theory of Computation Numeric Computing Mathematics Algebra Calculus of Variations and Optimal Control Optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Waveform relaxation methods for implicit differential equations

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

Jacobi interpolation approximations and their applications to singular differential equations

Numerical treatment of retarded differential–algebraic equations by collocation methods

Computing breaking points in implicit delay differential equations

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Solving differential equations with radial basis functions: multilevel methods and smoothing

Numerical solution of generalized Lyapunov equations

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Computing Validated Solutions of Implicit Differential Equations

Similar Documents

Waveform relaxation methods for implicit differential equations

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

Jacobi interpolation approximations and their applications to singular differential equations

Numerical treatment of retarded differential–algebraic equations by collocation methods

Computing breaking points in implicit delay differential equations

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Solving differential equations with radial basis functions: multilevel methods and smoothing

Numerical solution of generalized Lyapunov equations

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Computing Validated Solutions of Implicit Differential Equations