Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

Abstract

In this article, an effective approach has been proposed to obtain the approximate solutions of linear and nonlinear two-dimensional Volterra integro-differential equations. First, the two-dimensional wavelets are introduced, and using it the operational matrices of integration, differentiation, and product have been constructed. Then, by utilizing the properties and matrices of wavelets along with the collocation point, the matrix form of Volterra integro-differential equations has been derived. This approach reduces the two-dimensional linear and nonlinear Volterra integro-differential equations into the system of linear and nonlinear algebraic equations respectively. The convergence analysis and error analysis have been extensively studied by the help of two-dimensional wavelets approximation. Some illustrative examples are examined to clarify the accuracy and effectiveness of the proposed scheme. The graphical representations obtained by two proposed wavelets have been plotted to justify the applicability and validity of the method.