Numerical solution of elliptic type of inverse problems by Pascal polynomial method

The Pascal polynomial approach is presented in this paper as an effective method for solving inverse partial differential equations (PDEs) of second order. Two different approaches are suggested. In the first method, Pascal polynomials are used to approximate the source term and the unknown dependent variable. A system of algebraic equations that can be solved is then produced by substituting these approximations and their derivatives into the governing equations and boundary conditions. By imposing a requirement that the source term satisfy Laplace’s equation, the second approach reformulate the inverse problem as a direct one, which is subsequently solved via the Pascal polynomial method. We examine a number of benchmark problems to assess the suggested strategy, showing that it produces high accuracy when given accurate input data. Even though input data noise lowers accuracy, the numerical results are still reliable and acceptable. Despite its sensitivity to noise, these results demonstrate the potential of the Pascal polynomial approach for solving direct and inverse PDEs, especially when the input data is reliable.