Document Type : Original Article

Authors

• Muhammad Abid ¹
• Madiha Bibi ²
• Nasir Yasin ³
• Muhammad Shahid ⁴

¹ Department of Mathematics, North Carolina State University, Raleigh, 27695 NC, United States

² Rawalpindi Women University, Rawalpindi, Punjab 46300, Pakistan

³ Departments of Mathematics & Statistics, Old Dominion University, VA 23529, Norfolk, USA

⁴ Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA

Abstract

Accurate numerical solution of parabolic and elliptic partial differential equations governing two-dimensional heat transfer is critical for engineering simulations but computationally challenging.This work employs key numerical techniques finite differences, conjugate gradients, and Crank-Nicolson time stepping to solve the heat diffusion equation and analyze method performance.The Poisson equation is discretized using second-order central finite differences and solved with the conjugate gradient approach to determine the steady state solution. The transient heat equation is integrated in time via the Crank-Nicolson implicit scheme, also utilizing conjugate gradients.The methods effectively compute solutions matching analytical and boundary conditions. Convergence and stability are achieved while capturing transient thermal evolution. Insights are gained into discretization and iteration parameter impacts.The numerical framework demonstrates accurate and efficient simulation of two-dimensional conductive heat transfer. It provides a template for extension to more complex geometries and multiphysics phenomena, contributing to advances in computational engineering.

Keywords

• Central Finite Differencing
• Conjugate Gradient Method
• Crank Nicolson Scheme
• Heat Generation