A new hybrid method for solving nonlinear fractional differential equations

In this paper, numerical solution of initial and boundary value problems for nonlinear fractional differential equations is considered by pseudospectral method. In order to avoid solving systems of nonlinear equations resulting from the method, the residual function of the problem is constructed, as well as a suggested unconstrained optimization model solved by PSOGSA algorithm. Furthermore, the research inspects and discusses the spectral accuracy of Chebyshev polynomials in the approximation theory. The following scheme is tested for a number of prominent examples, and the obtained results demonstrate the accuracy and efficiency of the proposed method

A Direct Analytic Method for Solution of Linear Fredholm Integral and Integro-Differential Equations of the Second Kind

In this paper, a direct analytic method is given for the solution of the linear Fredholm integral and integro-differential equations of the second kind, which is based on the span of the known function, under the action of the operator defined by the kernel. The necessary conditions for using this method are so weak that extends its applicability. The solved examples show the strength of this method

Numerical Solution of Higher-Order Linear and Nonlinear Ordinary Differential Equations with Orthogonal Rational Legendre Functions

In this paper, we describe a method for the solution of linear and nonlinear ordinary differential equations ODE’s of arbitrary order with initial or boundary conditions. In this direction we first investigate some properties of orthogonal rational Legendre functions, and then we give the least square method based on these basis functions for the solution of such equations. In this method the solution of an ODE is reduced to a minimization problem, which is then numerically solved via Maple 16. Finally results of this method which are obtained in the form of continuous functions, will be compared with the numerical results in other references

A Good Approximate Solution for Li´enard Equation in a Large Interval Using Block Pulse Functions

In this paper, the Block pulse functions (BPFs) and their operational matrices of integration and differentiation are used to solve Li´enard equation in a large interval. This method converts the equation to a system of nonlinear algebraic equations whose solution is the coeffi- cients of Block pulse expansion of the solution of the Li´enard equation. Moreover, this method is examined by comparing the results with the results obtained by the Adomian decomposition method (ADM) and the Variational iteration method (VIM

The Two-Term Abel’s Integral Equation

In this article we investigate the two-term Abel’s integral equations. We will do this in two different ways and show that such equation is reducible to an integro−differential equation of Volterra type

Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets

Chebyshev Wavelets Method for Solution of Nonlinear Fractional Integrodifferential Equations in a Large Interval

An efficient Chebyshev wavelets method for solving a class of nonlinear fractional integrodifferential equations in a large interval is developed, and a new technique for computing nonlinear terms in such equations is proposed. Existence of a unique solution for such equations is proved. Convergence and error analysis of the proposed method are investigated. Moreover in order to show efficiency of the proposed method, the new approach is compared with some numerical methods