Numerical solution of partial differential equations using MATLAB: Applications to one-dimensional heat and wave equations

Abstract

Partial differential equations (PDEs) are powerful mathematical tools used to describe various physical phenomena in fields such as physics, engineering, and economics. In this study, numerical solution of PDEs was employed, focusing on one-dimensional heat and wave equations, using MATLAB. By employing finite difference methods, we discretize the PDEs and utilize MATLAB's computational capabilities to obtain numerical solutions. Surface plots are generated to visualize the behavior of the solutions over time and space. The article provides a comprehensive overview of the numerical techniques employed for solving PDEs in MATLAB, offering insights into the underlying mathematical principles and computational implementation. The significance of numerical solutions in understanding the behavior of physical systems and their applications in real-life scenarios was discussed. Specifically, we highlight the importance of PDEs in modeling heat transfer processes, such as diffusion and conduction, and wave propagation phenomena, including vibrations and oscillations. Furthermore, practical applications of PDEs in various fields, including engineering design, environmental science, and medical imaging are presented. The numerical solutions obtained using MATLAB enable researchers and practitioners to analyze complex systems, predict their behavior, and optimize design parameters. Additionally, the study contributes to the advancement of computational methods for solving PDEs, enhancing our ability to model and simulate diverse phenomena accurately. The study underscores the significance of numerical techniques in solving PDEs and their role in addressing real-world challenges. By leveraging MATLAB's computational capabilities, researchers can efficiently obtain solutions to complex PDEs, facilitating advancements in science, engineering, and technology.