Solvability And Numerical Solutions Of Systems Of Nonlinear Volterra Integral Equations Of The First Kind With Piecewise Continuous Kernels

The existence theorem for systems of nonlinear Volterra integral equations kernels of the first kind with piecewise continuous is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with piecewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is O(1/N).

Keywords

Volterra integral equations; discontinuous kernel; ill-posed problem; evolving dynamical systems; quadrature; Dekker - Brent method.

References

1. Sidorov D.N., Tynda A.N., Muftahov I.R. Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 3, pp. 107-115. (in Russian) DOI: 10.14529/mmp140311
2. Sidorov D.N. Integral Dynamical Models: Singularities, Signals and Control. World Scientific Series on Nonlinear Science, ser. A, vol. 87, Singapore, World Sc. Publ., 2015. 260 p.
3. Sidorov D.N. Solution to the Volterra Integral Equations of the First Kind with Discontinuous Kernels. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 18 (277), pp. 44-52. (in Russian)
4. Markova E.V., Sidorov D.N. On One Integral Volterra Model of Developing Dynamical Systems. Automation and Remote Control, 2014, vol. 75, no 3, pp. 413-421. DOI: 10.1134/S0005117914030011
5. Sidorov D.N. Solution to Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels. Russian Mathematics, 2013, vol. 57, pp. 62-72. DOI: 10.3103/S1066369X13010064
6. Brent R.P. An Algorithm with Guaranteed Convergence for Finding a Zero of a Function. The Computer Journal, 1971, vol. 14, no 4, pp. 422-425. DOI: 10.1093/comjnl/14.4.422
7. Trenogin V.A. Funktsional'nyy analiz [Functional Analysis]. Moscow, Fizmatlit, 2002. (in Russian)