Convergence of Approximate Solution of Nonlinear Volterra-Fredholm Integral Equations

In this study, an effective technique upon compactly supported semi orthogonal cubic Bspline 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 SCIREA Journal of Physics http://www.scirea.org/journal/Physics December 26, 2016 Volume 1, Issue 2, December 2016