Legendre wavelets optimization method for variable-order fractional Poisson equation

In this study, the Poisson equation is generalized with the concept of variable-order (V-O) fractional derivatives called variable-order fractional Poisson equation (V-OFPE). In order to find an accurate solution of this system, we establish an optimization method through the Legendre wavelets (LWs). To carry out the method, we firstly derive an operational matrix (OM) of V-O fractional derivative for the LWs to be employed in expanding the unknown solution. Then, the function of residual is applied to reform the V-OFPE to an optimization problem which leads to choose the unknown coefficients optimally. In the final step, we implement the constrained extremum method which adjoins the objective function implied from the two-norm of residual function and the constraints corresponded to the given boundary conditions by a set of Lagrange multipliers. Accordingly, the final optimal conditions are actually the algebraic equations including the expansion coefficients and Lagrange multipliers. Theoretical convergence and error analysis of the approximation procedure using the LWs are investigated. In addition, the applicability and computational efficiency are experimentally examined for some illustrative examples.