Solution of Fractional Harmonic Oscillator in a Fractional B-poly Basis

Abstract

An algorithm for approximating solutions to fractional-order differential equations in fractional polynomial basis is presented. A finite generalized fractional-order basis set is obtained from the modified Bernstein Polynomials, where α is the fractional-order of the modified Bernstein type polynomials (B-polys). The algorithm determines the desired solution in terms of continuous finite number of generalized fractional polynomials in a closed interval and makes use of Galerkin method to calculate the unknown expansion coefficients for constructing the approximate solution to the fractional differential equations. The Caputo's definition for a fractional derivative is used to evaluate derivatives of the polynomials. Each term in a differential equation is converted into matrix form and the final matrix problem is inverted to construct a solution of the fractional differential equations. However, the accuracy and the efficiency of the algorithm rely on the size of the set of B-polys. Furthermore, a recursive definition for generating fractional B-polys and the analytic formulism for calculating fractional derivatives are presented. The current algorithm is applied to solve the fractional harmonic oscillator problem and a number of linear and non-linear fractional differential equations. An excellent agreement is obtained between desired and exact solutions. Furthermore, the current algorithm has great potential to be implemented in other disciplines, when there are no exact solutions available to the fractional differential equations.