ABSTRACT: This study presents both analytical and numerical solutions of the one-dimensional wave equation under mixed boundary conditions, where different types of boundary constraints are imposed at opposite ends of the spatial domain. The analytical solution is obtained using the method of separation of variables and eigenfunction expansion, providing an exact representation of the wave behavior for prescribed initial and boundary conditions. To complement the analytical approach, a finite difference method (FDM) is employed to develop a numerical approximation of the governing equation. The stability, convergence, and accuracy of the numerical scheme are evaluated through comparative analysis with the analytical solution. Numerical simulations demonstrate that the finite difference method accurately captures wave propagation characteristics, including displacement, reflection, and oscillatory behavior, while maintaining low computational error under appropriate discretization parameters. The results indicate excellent agreement between the analytical and numerical solutions, validating the effectiveness of the proposed numerical approach for solving wave equations with mixed boundary conditions. This study highlights the importance of combining exact mathematical techniques with computational methods to address complex boundary value problems encountered in engineering, applied mathematics, and physics. The findings provide a reliable framework for modeling wave phenomena in practical applications where analytical solutions alone may not be feasible due to complex geometries or varying boundary conditions.
KEYWORDS: One-dimensional wave equation, Mixed boundary conditions, Analytical solution, Numerical solution, Finite difference method, Partial differential equations
ANALYTICAL AND NUMERICAL SOLUTIONS OF THE ONE-DIMENSIONAL WAVE EQUATION WITH MIXED BOUNDARY CONDITIONS
ANNU KUMARI, DR. VINEETA BASOTIA