In this paper, an inverse problem of recovering the unknown space-dependent source term in a parabolic equation under a final overdetermination condition is considered. To keep matters simple, in our analysis the main problem has been considered in the one-dimensional case, however the proposed method can be applied for higher dimensional cases. The approximate solution of the inverse problem is implemented by the Ritz–Galerkin method. The shifted Legendre polynomials basis together with the Galerkin approach are employed to reduce the main problem to the solution of linear algebraic equations. To overcome the difficulties arising from solving the resultant ill-conditioned linear system, a type of regularization technique is utilized to obtain a stable solution. The convergence analysis of the suggested method using Gronwall’s inequality is studied. Finally, some numerical examples are provided to demonstrate the efficiency and applicability of the proposed algorithm in the presence of noise in input measured data.

本文考虑了在最终超确定条件下恢复抛物方程中未知的空间相关源项的反问题。为了简单起见，在我们的分析中，主要问题是在一维情况下考虑的，但是所提出的方法可以应用于高维情况。反问题的近似解是通过Ritz-Galerkin方法实现的。移位的勒让德多项式基础与Galerkin方法一起用于将主要问题简化为线性代数方程的解。为了克服因求解结果不良的线性系统而引起的困难，利用一种正则化技术来获得稳定的解。研究了使用Gronwall不等式的建议方法的收敛性分析。最后，提供了一些数值示例来说明在输入测量数据中存在噪声的情况下所提出算法的效率和适用性。