Nonlocal solution of a nonlinear partial differential equation and its equivalent of nonlinear integral equation

In this paper, a set of nonlinear Volterra integral equations (NVIEs) of the second kind, with nonlocal condition is established from a nonlocal nonlinear partial differential equation (NPDE) of fractional order. The existence of a unique solution is guaranteed. Moreover, some properties concerning the stability of the solution are obtained. The modified Toeplitz matrix method (MTMM) is used, as a numerical method, to obtain a nonlinear algebraic system (NAS). In addition, many important theorems related to the existence of a unique solution of the NAS are considered and proved. Finally, numerical results are calculated, and the rate convergence of the error, in each case, is computed. © 2016 American Scientific Publishers All rights reserved.