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A new numerical method for solving two-dimensional Volterra–Fredholm integral equations

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Abstract

In this paper, a numerical method for solving nonlinear two-dimensional Volterra–Fredholm integral equations is presented. The approximate solution is expressed as expansion of two-dimensional delta basis functions (2D-DFs). Afterward, using the properties of 2D-DFs and their operational matrix of integration together with collocation method the numerical solution of these equations is reduced to the solution of a nonlinear system of algebraic equations. Moreover, it is proved in a theorem that the method is convergence and error is \(O(h^2)\). Furthermore, error analysis of the proposed method is provided under several mild conditions. Finally, the effectiveness of the method is illustrated in some numerical experiments.

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Acknowledgments

The authors is very grateful to referees and editors for their constructive comments and suggestions, which helped to improve the paper.

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Authors and Affiliations

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

    Farshid Mirzaee & Elham Hadadiyan

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  1. Farshid Mirzaee
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  2. Elham Hadadiyan
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Correspondence to Farshid Mirzaee.

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Mirzaee, F., Hadadiyan, E. A new numerical method for solving two-dimensional Volterra–Fredholm integral equations. J. Appl. Math. Comput. 52, 489–513 (2016). https://doi.org/10.1007/s12190-015-0951-1

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  • DOI: https://doi.org/10.1007/s12190-015-0951-1

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