Integro-differential systems with a degenerate matrix multiplying the derivative. (English. Russian original) Zbl 1031.65148
Differ. Equ. 38, No. 5, 731-737 (2002); translation from Differ. Uravn. 38, No. 5, 692-697 (2002).
This paper is concerned with the existence of solutions to the system of integro-differential equations with a degenerate matrix multiplying the derivative
\[
A(t)x'(t)+B(t)x(t)+\int_{0}^{t}K(t,\tau,x(\tau)) d\tau=f(t), \quad t\in [0,1],
\]
\[ x(0)=a, \] where \(A(t)\) and \(B(t)\) are given \(n\times n\) matrices, \(K:\mathbb{R}^{n+2}\to \mathbb{R}^{n}\) and \(f(t)\) is a given \(n\)-dimensional vector function. The degeneracy is understood as the relation \(\operatorname {det} A(t)\equiv 0.\) A numerical solution method based on Euler’s implicit method and a quadrature formula using left rectangles are suggested.
\[ x(0)=a, \] where \(A(t)\) and \(B(t)\) are given \(n\times n\) matrices, \(K:\mathbb{R}^{n+2}\to \mathbb{R}^{n}\) and \(f(t)\) is a given \(n\)-dimensional vector function. The degeneracy is understood as the relation \(\operatorname {det} A(t)\equiv 0.\) A numerical solution method based on Euler’s implicit method and a quadrature formula using left rectangles are suggested.
Reviewer: Mouffak Benchohra (Sidi Bel Abbes)
MSC:
65R20 | Numerical methods for integral equations |
45J05 | Integro-ordinary differential equations |
45G15 | Systems of nonlinear integral equations |