TITLE:
Numerical Solution for Initial and Boundary Value Problems of Fractional Order
AUTHORS:
A. M. Kawala
KEYWORDS:
Fractional Differential Equations, Shifted Legendre Polynomials, Operational Matrix, Caputo Derivative
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.12,
December
19,
2018
ABSTRACT: Fractional calculus has been used in many fields, such as engineering, population, medicine, fluid mechanics and different
fields of chemistry and physics. These fields were found to be best described
using fractional differential equations (FDEs) to model their processes and
equations. One of the well-known methods for solving fractional differential equations
is the Shifted Legendre operational matrix (LOM) method. In this article, I
proposed a numerical method based on Shifted Legendre polynomials for solving a
class of fractional differential equations. A fractional order operational
matrix of Legendre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational
matrix, the initial and boundary equations are transformed into the products of
several matrixes and by scattering the coefficients and the products of
matrixes. I got a system of linear equations. Results obtained by using the
proposed method (LOM) presented here show that the numerical method is very
effective and appropriate for solving initial and boundary value problems of
fractional ordinary differential equations. Moreover, some numerical examples
are provided and the comparison is presented between
the obtained results and those analytical results achieved that have proved the
method’s validity.