Ferdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713320230901An optimal control approach for solving an inverse heat source problem applying shifted Legendre polynomials4604804368510.22067/ijnao.2023.81202.1221ENT.ShojaeizadehDepartment of Mathematics, Qom Branch, Islamic Azad University, Qom, Iran.M.DarehmirakiDepartment of Mathematics, Behbahan Khatam Alanbia University of Technology, Khouzestan, Iran.Journal Article20230218This 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 objective<br />function, 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 methodâ€™s efficacy and precision, two examples are provided.https://ijnao.um.ac.ir/article_43685_1ef940d5e57278e43eeb2e0828c0532d.pdf