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Numerical solution of two dimensional stochastic Volterra–Fredholm integral equations via operational matrix method based on hat functions

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Abstract

Numerical solution of two-dimensional (2D) stochastic integral equations due to randomness has its own difficulties. For instance, most of them do not have analytical solution or obtaining their analytical solution is very hard. So, one of the essential requirement is presenting an efficient method to approximate the solutions of these equations with proper precision. In this paper, we introduce a numerical technique based on two dimensional modification of hat functions (2D MHFs) to estimate the numerical solution of 2D linear stochastic Volterra–Fredholm integral equations. In this approach, first the mentioned equation is transformed into a linear system of algebraic equations and then this system is solved via a direct or numerical method. Furthermore, it is proved that the order of convergence of the proposed method is \(O(h^3)\). Finally, accuracy of this scheme is measured by solving two test examples via described technique.

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Acknowledgements

The authors would like to express our very great appreciation to editor and anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.

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  1. Faculty of Mathematical Sciences and Statistics, Malayer University, P. O. Box 65719-95863, Malayer, Iran

    Farshid Mirzaee & Nasrin Samadyar

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  1. Farshid Mirzaee
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  2. Nasrin Samadyar
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Correspondence to Farshid Mirzaee.

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Mirzaee, F., Samadyar, N. Numerical solution of two dimensional stochastic Volterra–Fredholm integral equations via operational matrix method based on hat functions. SeMA 77, 227–241 (2020). https://doi.org/10.1007/s40324-020-00213-2

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