ABSTRACT: This study presents a comparative investigation of two widely used numerical techniques—Finite Difference Method (FDM) and Finite Element Method (FEM)—for solving one-dimensional wave equation boundary value problems. The Finite Difference Method approximates spatial and temporal derivatives using discrete grid points, providing a computationally efficient approach for problems with simple geometries and regular domains. In contrast, the Finite Element Method employs a variational formulation and approximates the solution through suitable basis functions over a collection of finite elements, making it particularly effective for complex geometries and non-uniform meshes. The study examines the formulation, discretization procedures, boundary conditions, stability considerations, and numerical accuracy of both methods. Numerical solutions are compared with analytical solutions wherever available to evaluate approximation errors and convergence behavior. The analysis demonstrates that both methods can provide reliable solutions when appropriate mesh sizes, time steps, and boundary treatments are selected. However, FEM offers greater flexibility for irregular domains and complex boundary conditions, whereas FDM is generally simpler to implement and computationally economical for structured problems. The comparative results highlight the strengths and limitations of each approach and provide useful guidance for selecting an appropriate numerical method for wave equation boundary value problems.
KEYWORDS: Wave Equation, Finite Difference Method, Boundary Value Problems, Numerical Methods, Discretization, Stability, Numerical Approximation.
FINITE DIFFERENCE AND FINITE ELEMENT METHODS FOR SOLVING WAVE EQUATION BOUNDARY VALUE PROBLEMS
ANNU KUMARI, DR. VINEETA BASOTIA