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