In this paper, we develop an Explicit Finite difference method to solve the ill-posed Cauchy problem for the three-dimensional acoustic wave equation with the data given on a part of the boundary (continuation problem) in a cube. These problems are defined by the combination of space and time. The solution of wave propagation problems in continuous acoustic medium is of great interest in various science areas, for example, the acoustic wave equation with a non-zero point source function has been widely used to model the wave propagation in Geophysics, in medicine and engineering, among others. Imaging these waves in the field of medicine were shown to provide very objective information about the biological tissue being examined. Finite difference method is one of the numerical methods that is used to compute the solutions of Hyperbolic PDEs by discretizing the given domain into finite number of regions and consequent reduction of a given PDEs into a system of linear algebraic equations. Due to its high accuracy, low memory and fast computing speed, the finite difference method is a powerful tool for acoustic or seismic wave simulations, especially for models with complex geological structures.
In addition, for three-dimensional problems, it usually results in a series of large-scale sparse linear systems, which needs iteration on each time step. We present a theory and a MATLAB code is developed to implement numerical solution of this approach and discuss about efficient solution of dense system of equations using iterative solvers. We extend the formulation of the Jacobi, Gauss-Seidel and SOR$(w_{opt})$ iterative methods in solving the linear system to improve computational efficiency and establish the convergence properties of the proposed method. Numerical experiments are conducted and we compare the analytical solution and numerical solution for different time phenomena. The conclusion of this study finds that the SOR$(w_{opt})$ iterative method is more efficient in terms of less number of iterations and less execution time when compared with the other two iterative methods.