An optimal control approach for solving an inverse heat source problem applying shifted Legendre polynomials

This study addresses the inverse issue of identifying the space-dependent heat source of the heat equation, which is stated using the optimal con-trol framework. For the numerical solution of this class of problems, an approach based on shifted Legendre polynomials and the associated oper-ational matrix is presented. The approach turns the primary problem into the solution of a system of nonlinear algebraic equations. To do this, the temperature and heat source variables are enlarged in terms of the shifted Legendre polynomials with unknown coefficients employed in the objectivefunction, inverse problem, and initial and Neumann boundary conditions. When paired with their operational matrix, these basis functions provide a quadratic optimization problem with linear constraints, which is then solved using the Lagrange multipliers approach. To assess the method’s efficacy and precision, two examples are provided.