New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials

This paper reports a new numerical approach for numerically solving types of fractional variational problems. In our approach, we use the fractional integrals operational matrix, described in the sense of Riemann–Liouville, with the help of the Lagrange multiplier technique for converting the fractional variational problem into an easier problem that consisting of solving an algebraic equations system in the unknown coefficients. Several numerical examples are introduced, combined with their approximate solutions and comparisons with other numerical approaches, for confirming the accuracy and applicability of the proposed approach.     

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T_L,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.     

New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations

In this paper, using the exp-function method we obtain some new exact solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations. We show figures of some of the new solutions obtained here. We conclude that the exp-function method presents a wider applicability for handling nonlinear partial differential equations.     

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.     

A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only small dimension of Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for generalized pantograph equations.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.