Iterative solution of systems of linear differential equations

Parallel processing has made iterative methods an attractive alternative for solving large systems of initial value problems. Iterative methods for initial value problems have a history of more than a century, and in the works of Picard (1893) and Lindelöf (1894) they were given a firm theoretical basis. In particular, the superlinear convergence on finite intervals is included in Lindelöf (1894).

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Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case

Axioms ◽

10.3390/axioms8020056 ◽

2019 ◽

Vol 8 (2) ◽

pp. 56 ◽

Cited By ~ 2

Author(s):

Galina Kurina

Keyword(s):

Differential Equations ◽

Asymptotic Solution ◽

Initial Value Problems ◽

Singular Matrix ◽

Singularly Perturbed Systems ◽

English Transl ◽

Initial Value ◽

Boundary Functions ◽

Linear Differential ◽

Orthogonal Projectors

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) ′ (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

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Locally Linearized Runge-Kutta method of Dormand and Prince for large systems of initial value problems

Journal of Computational Physics ◽

10.1016/j.jcp.2020.109946 ◽

2021 ◽

Vol 426 ◽

pp. 109946

Author(s):

F.S. Naranjo-Noda ◽

J.C. Jimenez

Keyword(s):

Initial Value Problems ◽

Kutta Method ◽

Initial Value ◽

Runge Kutta Method ◽

Large Systems ◽

Runge Kutta

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A Method for Solving Initial Value Problems for Linear Differential Equations in Hilbert Space Based on The Cayley Transform

Numerical Functional Analysis and Optimization ◽

10.1080/01630569308816534 ◽

1993 ◽

Vol 14 (5-6) ◽

pp. 459-473 ◽

Cited By ~ 12

Author(s):

Damir Z. Arov ◽

Ivan P. Gavrilyuk

Keyword(s):

Hilbert Space ◽

Differential Equations ◽

Initial Value Problems ◽

Linear Differential Equations ◽

Cayley Transform ◽

Initial Value ◽

Linear Differential

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On the iterative methods for solving fractional initial value problems: new perspective

Journal of Fractional Calculus and Nonlinear Systems ◽

10.48185/jfcns.v2i1.297 ◽

2021 ◽

Vol 2 (1) ◽

pp. 76-81

Author(s):

Qasem M. Al-Mdallal ◽

Mohamed Ali Hajji ◽

Thabet Abdeljawad

Keyword(s):

Approximate Solution ◽

Numerical Solution ◽

Iterative Methods ◽

Initial Value Problems ◽

Short Communication ◽

Initial Value ◽

New Perspective

In this short communication, we introduce a new perspective for a numerical solution of fractional initial value problems (FIVPs). Basically, we split the considered FIVP into FIVPs on subdomains which can be solved iteratively to obtain the approximate solution for the whole domain.

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Limit theorems for solutions of stochastic differential equation problems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171280000087 ◽

1980 ◽

Vol 3 (1) ◽

pp. 113-149 ◽

Cited By ~ 2

Author(s):

J. Vom Scheidt ◽

W. Purkert

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Stochastic Differential Equation ◽

Initial Value Problems ◽

Limit Theorems ◽

Eigenvalue Problems ◽

Random Processes ◽

Inhomogeneous Term ◽

Initial Value ◽

Linear Differential

In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.Limit theorems are proved for the eigenvalues and the eigenfunctions of eigenvalue problems and for the solutions of boundary value problems and initial value problems.

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The iterative solution of initial value problems for ordinary differential equations

USSR Computational Mathematics and Mathematical Physics ◽

10.1016/0041-5553(82)90036-2 ◽

1982 ◽

Vol 22 (2) ◽

pp. 58-72

Author(s):

H. Bauch

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Value Problems ◽

Iterative Solution ◽

Initial Value

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Waveform iterative methods for parallel solution of initial value problems

Proceedings Scalable Parallel Libraries Conference ◽

10.1109/splc.1994.377001 ◽

2002 ◽

Cited By ~ 2

Author(s):

A. Lumsdaine ◽

J.M. Squyres ◽

M.W. Reichelt

Keyword(s):

Iterative Methods ◽

Initial Value Problems ◽

Initial Value ◽

Parallel Solution

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Numerical solution of linear differential equations by Walsh polynomials approach

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2020.57.2.1459 ◽

2020 ◽

Vol 57 (2) ◽

pp. 217-254

Author(s):

◽

Rodolfo Toledo

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Harmonic Analysis ◽

Initial Value Problems ◽

Linear Differential Equations ◽

Previous Article ◽

Initial Value ◽

Constant Coefficients ◽

Linear Differential

AbstractIn 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

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