PAPER ON NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS - Rathod, Chetan (ShodhKosh: Journal of Visual and Performing Arts, 2024)
Partial differential equations (PDEs) are fundamental in describing various physical phenomena, such as fluid dynamics, heat conduction, and wave propagation. However, analytical solutions to these equations are often difficult or impossible to obtain due to their complexity and the boundary conditions involved. Numerical methods provide an effective alternative by approximating solutions through discretization techniques. This paper explores various numerical methods for solving PDEs, including finite difference, finite element, and finite volume methods. We discuss their theoretical foundations, implementation strategies, and advantages in handling different types of PDEs, such as elliptic, parabolic, and hyperbolic equations. Moreover, the paper addresses key challenges such as stability, convergence, and computational efficiency, and reviews the use of high-performance computing in tackling large-scale problems. The applications of these methods in scientific computing and engineering are highlighted, demonstrating their versatility and importance in solving real-world problems. The numerical solution of partial differential equations (PDEs) plays a crucial role in solving real-world problems across various fields, including physics, engineering, and finance. Exact analytical solutions to PDEs are often not feasible due to their complexity and the nature of boundary conditions. As a result, numerical methods such as the finite difference, finite element, and finite volume methods are widely employed to approximate solutions. This paper provides an overview of these methods, emphasizing their formulation, implementation, and application to different types of PDEs, including elliptic, parabolic, and hyperbolic equations. Key considerations such as stability, convergence, and accuracy are discussed, along with strategies for improving computational efficiency. The paper also highlights the use of advanced computational techniques and parallel computing in addressing large-scale and complex PDE systems. Overall, numerical methods offer powerful tools for solving PDEs and are essential for simulating and analyzing complex phenomena in science and engineering.