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A numerical solution to Klein-Gordon equation with Dirichlet boundary condition. (English) Zbl 1126.65090
Summary: The Klein-Gordon equation arises in relativistic quantum mechanics and field theory, so it is of a great importance for the high energy physicists. In this paper, we establish the existence and uniqueness of the solution and in the second part a numerical scheme is developed based on the finite element method. For the one space dimensional case, a complete numerical algorithm for the numerical solutions using the quadratic interpolation functions is constructed. The one-dimensional model equation is formulated over an arbitrary element, applying the assembly process on the elements of the domain, employing numerical schemes to integrate the nonlinear terms and solving the system of equations numerically. Finally, the obtained results of simulations are visualized, which show the overflow of the solution as expected.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI
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