Analyze Second‐Order PDEs Using the Volterra–Fredholm Integral Equation

In this study, we propose a novel approach to address a particular second‐order partial differential equation along with its boundary value conditions (SPDEs). In this process, we transfer the SPDEs problem into Volterra–Fredholm integral equation (VFIE), and we perform the Tau method bases on orthogonal Legendre polynomials directly, for solution of SPDEs and also for solution of VFIE problem. The numerical results obtained from applying the spectral Tau method in the two specified cases demonstrate the impressive results of using the integral form VFIE, which this conclusion clarifies the high numerical stability integral form VFIE respect to SPDEs form. Moreover, the convergence analysis of the proposed approximate procedure is presented in the Hilbert spaces of and the Sobolev spaces .