Using New Operational Matrix for Solving Nonlinear Fractional Integral Equations

In this paper, a numerical method for solving nonlinear fractional integral equations (NFIE) is introduced. This method is based on the new basis functions (NFs) introduced in [M. Paripour and et al., Numerical solution of nonlinear Volterra Fredholm integral equations by using new basis functions, Communications in Numerical Analysis, (2013)]. Since the conventional operational matrices for fractional kernels are singular, the definition of these matrices is modified. In order to increase the accuracy of approximating integrals, the operational matrices are exactly calculated and parametrically presented. Then, the solution procedure is proposed and applied on NFIE. Furthermore, the error analysis is performed and rate of convergence is obtained. In addition, various numerical examples are provided for a wide range of fractional orders and nonlinearity of integral equations. Comparison of the results with the exact solutions and those reported in previous studies indicate the capability, salient accuracy, and superiority of the proposed method over similar ones.