High-precision Numerical Algorithms For Optimal Control Problems Of Fractional Partial Differential Equation

Fractional order partial differential equations are widely used in many engineering problems,and they can better describe the system than integer order partial differential equations.In the application process of various fields,since the expected goal is required to reach an optimal situation,the optimal control problem needs to be considered for research and analysis.Therefore,the optimal control problem of fractional order partial differential equations has become a research hot-spot.This paper mainly studies the high-order numerical algorithm for optimal control of fractional order partial differential equations.It is to be divided into two main parts.The first part examines the high-order numerical scheme of the right Caputo fractional partial differential equation.The finite difference method is first used to construct a high-order numerical scheme for the right Caputo fractional ordinary differential equation,and analyzing its local truncation error and stability,it is obtained that the convergence order of the scheme is 3-.Secondly,based on the resulting numerical scheme,the second-order central difference method is used to solve the high-order numerical scheme of the one-dimensional and two-dimensional cases of the right-Caputo fractional partial differential equations,and the resulting spatial convergence order is2.Finally,several examples are given for numerical simulation,through which the solution is verified the validity of the scheme and the correctness of the theoretical analysis.The second part studies the construction of a high-order numerical scheme for the optimal control problem of the Caputo fractional partial differential equation.according to the idea of optimization before discrete,firstly,the Lagrange multiplier is introduced to change the constrained condition into an unconstrained condition,so as to solve the optimality condition of the optimal control problem.and secondly,the spectral method is used spatially and the finite difference is adopted in time,to discretize the optimality conditions,finally,a strict error analysis is given for the scheme after discreteness.