Fractional Differential Equations

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.

http://www.kau.edu.sa/Files/130/Researches/62005_33049.pdf

A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

A new jacobi operational matrix: an application for solving fractional differential equations

Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

In this article we propose a new fractional derivative without singular
 kernel. We consider the potential application for modeling the steady
 heat-conduction problem. The analytical solution of the fractional-order heat
 flow is also obtained by means of the Laplace transform.

/pdf/a-new-fractional-derivative-without-singular-kernel-9et0fa2lhw.pdf

A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL Application to the Modelling of the Steady Heat Flow

We will consider He's variational iteration method for solving fractional
Riccati differential equation. This method is based on the use of Lagrange multipliers for
identification of optimal value of a parameter in a functional. This technique provides a
sequence of functions which converges to the exact solution of the problem. The present
method performs extremely well in terms of efficiency and simplicity.

/pdf/he-s-variational-iteration-method-for-solving-fractional-1gqxi5vlad.pdf

He's Variational Iteration Method for Solving Fractional Riccati Differential Equation

The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn–Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann–Liouville derivative. Some relevant examples are investigated.

Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

The fractional subequation method is applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order. The accuracy and efficiency of the scheme are discussed for these illustrative examples.

/pdf/fractional-subequation-method-for-cahn-hilliard-and-klein-33igx4c81g.pdf

Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations

Solving a multi-order fractional differential equation using homotopy analysis method

In this paper the fractional sub-equation method is used to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation.The fractional derivative is described in the Jumarie’s modified Riemann-Liouville sense. Two illustrative examples are given, showing the accuracy and convenience of the method.

Haleh Tajadodi

Papers

He's Variational Iteration Method for Solving Fractional Riccati Differential Equation

Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations

Solving a multi-order fractional differential equation using homotopy analysis method

Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation