A novel semi-analytical approach for solving nonlinear Volterra integro-differential equations

In this work, we present an efficient semi-analytical method based on the Taylor series for solving nonlinear Volterra integro-differential equations, namely the differential transform method (DTM). The DTM provides a recursive relation for the coefficients of the Taylor series that is derived from the given equations. We provide a new recursive relation for the nonlinear Volterra integro-differential equations with complex nonlinear kernels. Since the DTM is based on the Taylor series, it is difficult to obtain accurate approximate solutions in a large domain. To overcome this difficulty, the standard DTM is applied in each subdomain, called the multistage differential transform method (MsDTM). We also present an convergence analysis for the proposed method. To demonstrate the efficiency of the proposed method, several numerical examples are performed and support the results in our analysis.