Generalized fractional-order Legendre wavelet method for two dimensional distributed order fractional optimal control problem

This paper is concerned with a two-dimensional fractional optimal control problem whose governing equations are distributed order fractional differential equations in the Caputo sense. A generalized fractional-order Legendre wavelet method has been used to solve the two-dimensional distributed-order fractional optimal control problem. An exact formula for the Riemann-Liouville integration of generalized fractional-order Legendre wavelet has been derived by using regularized beta functions. This formula and the two-dimensional Gauss-Legendre integration formula have been used to solve the two-dimensional distributed order fractional optimal control problem. Moreover, an L-2-error estimate in the approximation of an unknown function with a generalized fractional-order Legendre wavelet has been derived and the estimated order has been verified for a given function. Furthermore, convergence analysis for the proposed method has been presented. In the last, two test problems have been considered to illustrate the efficiency of the proposed method.