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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.

MSC:
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
Software:
dverk
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References:
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press · Zbl 0186.19101
[2] Bishop, A.R.; Fesser, K.; Lomdahl, P.S., Influence of solitons in the initial state on chaos in the driven damped sine-Gordon system, Physica D, 7, 259-279, (1983) · Zbl 1194.35344
[3] Dautray, R.; Lions, J.L., Mathematical analysis and numerical methods for science and technology, Evolution problems, vol. 5, (1992), Springer-Verlag
[4] Hartman, P., Ordinary differential equations, (1964), Wiley New York · Zbl 0125.32102
[5] T.E. Hull, W.H. Enright, K.R. Jackson, User’s guide for DVERK–A subroutine for solving non-stiff ODEs, Department of Computer Science Technical Report 100, University of Toronto, 1976
[6] Lions, J.L., Quelques methód des Vésolution des problèmes aus limites non lineaires, (1969), Dunod Paris · Zbl 0189.40603
[7] Lions, J.L.; Magenes, E., Non-homogeneous boundary value problems and applications I, II, (1972), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0227.35001
[8] Reddy, J.N., An introduction to the finite element method, second rev. ed, (1993), McGraw-Hill, Inc
[9] Rubino, B., Weak solutions to quasilinear wave equations of klein – gordon or sine-Gordon type and relaxation to reaction – diffusion equations, Nonlinear differential appl, 4, (1997) · Zbl 0893.35115
[10] Tanabe, H., Equations of evolution, (1979), Pitman London
[11] Temam, R., Navier – stokes equations, theory and numerical analysis, third rev. ed, (1984), North-Holland Amsterdam · Zbl 0568.35002
[12] Temam, R., Infinite-dimensional dynamical systems in mechanics and physis, Appl. math. sci, vol. 68, (1988), Springer-Verlag
[13] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in FortRan90
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