Fractional Differential Equations

Fractional-order Legendre functions for solving fractional-order differential equations

A new analytical modelling for fractional telegraph equation via Laplace transform

By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.

https://iopscience.iop.org/article/10.1088/0031-9112/10/5/018/pdf

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

In this paper, we use the fractional complex transform and the (G'/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann—Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.

https://iopscience.iop.org/article/10.1088/1674-1056/22/11/110202/pdf

Exact solutions of nonlinear fractional differential equations by (G?/G)-expansion method

A generalized fractional sub-equation method for fractional differential equations with variable coefficients☆

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of series solution.

Homotopy analysis method for higher-order fractional integro-differential equations

Homotopy perturbation method for two dimensional time-fractional wave equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear time-fractional Tricomi-type equation (TFTTE), which is obtained from the standard one-dimensional linear Tricomi-type equation by replacing the first-order time derivative with a fractional derivative (of order ?, with 1?<?????2). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h k?+?1?+?? 2), where h, ? and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective.

Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(hk+1+τ2−α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.

Bo Tang

Papers

A generalized fractional sub-equation method for fractional differential equations with variable coefficients☆

Homotopy analysis method for higher-order fractional integro-differential equations

Homotopy perturbation method for two dimensional time-fractional wave equation

Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation