Study of Fractional-Order Boundary Value Problems Using Nondiscretization Numerical Method

Abstract

This paper is devoted to present a numerical scheme based on operational matrices to compute approximate solutions to fractional-order boundary value problems. For the mentioned operational matrices, we utilize fractional-order Bernoulli polynomials. Since fractional-order problems are usually difficult to treat for their corresponding analytical or exact solutions, therefore, we need sophisticated methods to find their best numerical solutions. The presented numerical scheme has the ability to reduce the proposed problem to the corresponding algebraic equations. The obtained algebraic equations are then solved by using the computational software MATLAB for the corresponding numerical results. The used method has the ability to save much more time and also is reliable to secure the proper amount of memory. Several examples are solved by using the considered method. Also, the solutions are compared with their exact solution graphically. In addition, the absolute errors for different scale values are presented graphically. In addition, we compare our results with the results of the shifted Legendre polynomials spectral method.