Combining the Least-Squares Method with Touchard Polynomials for Solving Mixed Integro-Differential Equation

Using the well-known least-squares weighted residual method (LSM) in coupling with various degrees of Touchard polynomials (TPs), found the numerical solutions of Volterra–Fredholm integro-differential equations (VFIDEs) and mixed Volterra–Fredholm integro-differential equations (MIDEs) of the second kind. There exist many approaches that have evaluated the approximate solution of the integro-differential equations (VFIDEs (MIDEs)) like the Adomian decomposition method and modified Adomian decomposition method, Homotopy analysis method, Taylor polynomials, power series expansion and cubic Legendre spline collocation method. In this work, we presented a method based on combining LSM with TPs is an essential component of the suggested approach. By implementing such a method, a system of algebraic equations can be generated that can be solved by employing well-known linear algebraic methods. Several VFIDEs (MIDEs) examples were solved with a comparatively minimal number of reiterations to show the accuracy and effectiveness of the presented approach when comparing the current method with other methods already accessible in the scientific literature, as well as from the approximate solutions of each of these situations, researchers found that there was an apparent agreement with the exact solutions for some examples. The applicability of the proposed method was proven and the convergence analysis was discussed.