Convergence of Approximate Solution of Nonlinear Volterra-Fredholm Integral Equations

Volume 1, Issue 2, December 2016     |     PP. 125-143      |     PDF (540 K)    |     Pub. Date: December 26, 2016
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Monireh Nosrati Sahlan, Department of Mathematics and Computer Science, Technical Faculty, University of Bonab, Box 55517-61167, Bonab, Iran
Hamid Reza Marasi, Department of Mathematics and Computer Science, Technical Faculty, University of Bonab, Box 55517-61167, Bonab, Iran

In this study, an effective technique upon compactly supported semi orthogonal cubic B-spline wavelets for solving nonlinear Volterra-Fredholm integral equations is proposed. Properties of B-spline wavelets and function approximation by them are first presented and the exponential convergence rate of the approximation, Ο(2-4j ), is proved. For solving the nonlinear Volterra-Fredholm integral equation, the unknown function of problem is approximated by cubic B-spline wavelets. Then Properties of these functions are used to reduce nonlinear mixed integral equation to some algebraic system. For solving the mentioned system, Galerkin and collocation methods are applied. In the both methods, Cubic B-spline wavelets are used as testing and weighting functions. Convergence and error analysis of the method is described through some proved theorems. Because of having vanishing moments, compact support and semi orthogonality properties of these wavelets, operational matrices of the Galerkin and collocation methods are very sparse. In fact the entries with significant magnitude are in the diagonal of operational matrices, and other entries are very small and hence can be set to zero without significantly affecting the solution. Because of having low memory requirement, high speed and accuracy of the method, the presented procedure is more practical with respect to many of other methods for solving this class of integral equations. The method is computationally attractive and applications are demonstrated through illustrative examples. As is shown in the reported tables of examples, compare the error of three methods, we can find that the presented method get better approximate solution.

Fredholm-Volterra-Hammerstein integral equations, collocation method, Galerkin method, Cubic B-spline wavelets, error analysis

Cite this paper
Monireh Nosrati Sahlan, Hamid Reza Marasi, Convergence of Approximate Solution of Nonlinear Volterra-Fredholm Integral Equations , SCIREA Journal of Physics. Volume 1, Issue 2, December 2016 | PP. 125-143.


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