Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

Abstract

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.