School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Document Type


Publication Date



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.


©Attribution 3.0 Unported (CC BY 3.0)

Creative Commons License

Creative Commons Attribution 3.0 License
This work is licensed under a Creative Commons Attribution 3.0 License.

Publication Title

Physics and Technical Sciences



Included in

Mathematics Commons



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.