The Adomian Decomposition Method for a Type of Fractional Differential Equations

Peng Guo

doi:10.4236/jamp.2019.710166

Abstract

Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.

Highlights

Fractional calculus can be dated back to the end of 17th century
Fractional differential equations are widely used in many fields
We can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations

Summary

Introduction

Fractional calculus can be dated back to the end of 17th century. In 1695, Leibniz and L’Hospital have discussed 1/2 order derivative, which is regarded as the birth of fractional differential equation. We give an analytical solution of the time fractional differential equation of the following form. Many earlier researchers use it instead of Caputo derivative, but for the Riemann-Liouville derivative we need to specify the values of certain fractional derivatives of the unknown solution at the initial conditions. It has a clearly physical meaning and can be measured Another reason we choose Caputo derivative is that under homogeneous conditions the equations with Riemann-Liouville operator are equivalent to the equations with Caputo operator, if we choose Caputo derivative it allows us to specify inhomogeneous initial conditions, too, if we needed.

Fractional Integral and Fractional Derivative

Adomian Decomposition Method

Numerical Examples

Conclusion