Numerical Solution of Parabolic Partial Differential Equations via Conjugate Gradient Technique

Abstract

Parabolic partial differential equations (PPDEs) arise in many areas of science and engineering, including heat transfer, diffusion, and fluid dynamics. Analytical solutions to these PPDEs are often difficult or impossible to obtain, so numerical methods are needed to approximate the solution. In this research, we investigate the use of the conjugate gradient technique for numerically solving parabolic PDEs. The technique involves discretizing the PPDE with regard to both space and time. The parabolic partial differential equations are then transformed into systems of linear algebraic equations using the Crank-Nicholson centred difference approach. Then, these equations are solved to yield the unknown points in the grids, which are subsequently substituted into the assumed solution to obtain the required estimated solution, which is reported in tabular format. A comparison was made between the conjugate gradient solutions and those produced using the Jacobi preconditioned conjugate gradient technique in terms of the time required and rate of convergence at that point. Results indicate that conjugate gradient techniques are suitable for solving parabolic-type partial differential equations, with Jacobi-preconditioned conjugate gradient technique converging faster. This research has potential applications in various areas of science and engineering where parabolic PDEs arise.