In this paper, we apply the extended triangular Operational matrices of fractional order to solve the fractional voltrra model for population growth of a species in a closed system. The fractional derivative is considered in the Caputo sense. This technique is based on generalized Operational matrix of triangular functions. The introduced method reduces the proposed problem for solving a system of algebraic equations. Illustrative examples are included to demonstrate the validity and the applicability of the proposed method...

In this article we implement an Operational matrix of fractional integration for Legendre polynomials. We proposed an algorithm to obtain an approximation solution for fractional differential equations, described in Riemann-Liouville sense, based on shifted Legendre polynomials. This method was applied to solve linear multiorder fractional differential equation with initial conditions, and the exact solutions obtained for some illustrated examples. Numerical results reveal that this method gives ideal approximation for linear multi-order fractional differential 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 de nition of these matrices is modi ed. 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.

In this article, we have discussed a new application of modification of hat functions on nonlinear multi-order fractional differential equations. The Operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. The method provides the solution in the form of a rapidly convergent series. Furthermore, error analysis of the proposed method is provided under several mild conditions. Three numerical examples are given to show the efficiency and accuracy of the method. Illustrative examples are included to demonstrate the validity, efficiency, and applicability of the method.