Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

Abstract

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

Optimized Trigonometric Spline Function with Conjugate GradientMethod for Solving FDEs

“…Homotopy analysis [11], matrix approach method for solving FDE discussed in [12], fractional explicit Adams method used in [13]Muhammad I. Bhatti and Md.Habibur Rahman used B-polynomial bases to solve FDE [14], discrete Prabhakar fractional operator studied in [15], a spectral Tau method investigated by Hari Mohan Srivastava and et al [16].…”