Computational DIQM Scheme for Solving Nonlinear Volterra Integro-Differential Equations

Document Type : Original Article

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Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt.

Abstract

The authors in this work develop the differential quadrature method (DQM) to solve the nonlinear Volterra integro-differential equation by introducing an integration matrix operator that is fully combined with the differentiation matrix in (DQM) method. The obtained method (DIQM) transforms the discretized nonlinear Volterra integro-differential equation into a nonlinear algebraic system of equations which is solved iteratively by Newton's method.The advantage of the differential quadrature method appears when it is used to solve boundary-value, initial-value, linear or nonlinear differential equations that DQM requires less grid points to obtain acceptable accuracy unlike finite difference method (FDM), finite element method (FEM) and finite volume method (FVM) which may need more number of grid points to obtain the solution. The efficiency of the (DIQM) method is examined by solving three examples where the error norms and convergence rates achieve the expected exponential behavior and it is easy to implement for solving other kinds of integro-differential equations.

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