Numerical Solutions of Linear Fractional Order BVP by Galerkin Residual Method with Differentiable Polynomials

The main objective of this paper is to solve numerically the differential equations of fractional order with homogeneous boundary conditions by the Galerkin weighted residual method. In this method, linear combinations of some types of functions are used to find the approximate solutions which must satisfy the homogeneous boundary conditions. Such type of functions should be differentiated and integrated easily, so the piecewise polynomials, namely, Bernstein, Bernoulli and Modified Legendre polynomials are used as basis functions in this paper. The fractional derivatives are used in the conjecture of Riemann-Liouville and Caputo sense. Thus, we develop the Galerkin weighted residual formulation, in matrix form, to the linear fractional order boundary value problems, in details, which is easy to understand. The accuracy and applicability of the present method are demonstrated through few numerical examples. We observe that the approximate results converge monotonically to the exact solutions. In addition, we compare the approximate results with the exact solutions, and also with the existing solutions which are available in the literature. The absolute errors are depicted in tabular form as well as graphical representations, a reliable accuracy is achieved. The proposed method may be applied to fractional order partial differential equations also.