Algorithmic solution for (3+1) dimensional Klein-Gordon equation using non-polynomial cubic spline technique

Rabia Sattar; Mohd Khalid; Muhammad Ozair Ahmad; Anjum Pervaiz; Nauman Ahmed; Ali Akgül; Saleem Yousuf Lone

doi:10.5269/bspm.76893

Rabia Sattar
Mohd Khalid Maulana Azad National Urdu University, Hyderabad-500032, India.
Muhammad Ozair Ahmad
Anjum Pervaiz
Nauman Ahmed
Ali Akgül
Saleem Yousuf Lone

Abstract

The algorithmic solution of problems based on the Klein-Gordon Equation in (3+1) dimensions has been discussed in this study. Time variable t is discretized by using the central difference formula. Cubic Spline function involving trigonometric functions is used for all three spacial variables and Suitable parametric values provide the accuracy of order O(h^2+k^2+σ^2+τ^2 h^2+τ^2 k^2+τ^2 σ^2 ) in our proposed scheme. The stability of this scheme is also discussed in this paper and the truncation error too. This method elucidates numerical problems and compares them to the results obtained from the literature.

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References

Torre, C. G., 09 The Wave Equation in 3 Dimensions, 2014.

Baskonus, H. M., Sulaiman, T. A. and Bulut, H., On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics, Indian Journal of Physics, 93 (2019), 393–399.

He, J. H., Variational iteration method – a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 34(4) (1999), 699–708.

Maireche, A., Solutions of Klein-Gordon equation for the modified central complex potential in the symmetries of noncommutative quantum mechanics, Sri Lankan Journal of Physics, 22 (2021), 1–18.

Deresse, A. T., Mussa, Y. O. and Gizaw, A. K., Solutions of two-dimensional nonlinear sine-Gordon equation via triple Laplace transform coupled with iterative method, Journal of Applied Mathematics, 2021 (2021), 1–15.

Ibrahim, W. and Tamiru, M., Solutions of Three-Dimensional Nonlinear Klein-Gordon Equations by Using Quadruple Laplace Transform, International Journal of Differential Equations, 2022 (2022).

Maireche, A., A new asymptotic study to the 3-dimensional radial Schrodinger equation under modified quark-antiquark interaction potential, Journal of Nanoscience: Current Research, 4(1) (2019).

Hamil, B. and Lutfuoglu, B. C., Dunkl–Klein–Gordon Equation in Three-Dimensions: The Klein–Gordon Oscillator and Coulomb Potential, Few-Body Systems, 63(4) (2022), 74.

Strauss, W. A., Nonlinear Wave Equations, American Mathematical Society, No. 73 (1990).

Sabri, M. and Rasheed, M., On the solutions of wave equation in three dimensions using D’Alembert formula, International Journal of Mathematics Trends and Technology, 49(5) (2017), 311–315.

Mohanty, R. K., An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation, Applied Mathematics Letters, 17(1) (2004), 101–105.

Gao, F. and Chi, C., Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation, Applied Mathematics and Computation, 187(2) (2007), 1272–1276.

Mohebbi, A. and Dehghan, M., High order compact solution of the one-space-dimensional linear hyperbolic equation, Numerical Methods for Partial Differential Equations: An International Journal, 24(5) (2008), 1222–1235.

Zadvan, H. and Rashidinia, J., Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term, Numerical Algorithms, 74 (2017), 289–306.

Raggett, G. F. and Wilson, P. D., A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique, IMA Journal of Applied Mathematics, 14(1) (1974), 75–78.

Rashidinia, J., Jalilian, R. and Kazemi, V., Spline methods for the solutions of hyperbolic equations, Applied Mathematics and Computation, 190(1) (2007), 882–886.

Ding, H. and Zhang, Y., A new unconditionally stable compact difference scheme of O(τ 2 + h4) for the 1D linear hyperbolic equation, Applied Mathematics and Computation, 207(1) (2009), 236–241.

Korkmaz, A., Ersoy, O. and Dag, I., Trigonometric Cubic B-spline Collocation Method for Solitons of the Klein-Gordon Equation, 2016.