# Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel

D.N. Sidorov, A.N. Tynda, I.R. MuftahovIntegral equations are in the core of many mathematical models in physics, economics and ecology. Volterra integral equations of the first kind with jump discontinuous kernels play important role in such models and they are considered in this article. Regularization method and sufficient conditions are derived for existence and uniqueness of the solution of such integral equations. An efficient numerical method based on the mid-rectangular quadrature rule is proposed for these equations with jump discontinuous kernels. The accuracy of proposed numerical method is $O(N^{-1})$. The model examples demonstrate efficiency of proposed method: errors, two mesh differences and orders of convergent.Full text

- Keywords
- Volterra integral equations of the 1st kind; evolving systems; Glushkov integral model; numerical method.
- References
- 1. Sidorov D.N. Metody analiza integral'nykh dinamicheskikh modelei: teoriya i prilozheniya [Methods of Analysis of Integral Dynamic Models: Theory and Applications]. Irkutsk, Izd. IGU, 2013. 293 p.

2. Boikov I.V., Tynda A.N. Approximate Solution of Nonlinear Integral Equations of Developing Systems Theory. Differential Equations, 2003, vol. 39, no. 9, pp. 1277-1288. DOI: 10.1023/B:DIEQ.0000012695.06431.c4

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 & 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. Micke A. The Treatment of Integral Equations with Discontinuous Kernels Using Product Type Quadrature Formula. Computing, 1989, vol. 42, pp. 207-223. DOI: 10.1007/BF02239749

6. Sidorov D.N. Volterra Equations of the First Kind with Discontinuous Kernels in the Theory of Evolving Systems Control. Stud. Inform. Univ., 2011, vol. 9, pp. 135-146.

7. Sidorov D.N. On Parametric Families of Solutions of Volterra Integral Equations of the First Kind with Piecewise Smooth Kernel. Differential Equations, 2013, vol. 49, no. 2, pp. 210-216. DOI: 10.1134/S0012266113020079

8. Sidorov D.N. Solvability of Systems of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernels. Russian Mathematics, 2013, vol. 57, issue 1, pp. 62-72.

9. Tynda A.N. Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations. Comp. Meth. Appl. Math., 2006, vol. 6, no. 4, pp. 436-442.