Engineering Transactions, 67, 1, pp. 21–34, 2019
10.24423/EngTrans.910.20190214

A Reliable Numerical Algorithm for the Fractional Klein-Gordon Equation

Harendra SINGH
School of Mathematical Sciences, National Institute of Science Education and Research(NISER)
India

Devendra KUMAR
JECRC University
India

Jagdev SINGH
JECRC University
India

C.S. SINGH
Department of Mathematics, Rajkiya Engineering College
India

The key purpose of the present work is to introduce a numerical algorithm for the solution of the fractional Klein-Gordon equation (FKGE). The numerical algorithm is based on the applications of the operational matrices of the Legendre scaling functions. The main advantage of the numerical algorithm is that it reduces the FKGE into Sylvester form of algebraic equations
which significantly simplify the problem. Numerical results derived by using suggested numerical scheme are compared with the exact solution. The results show that the suggested algorithm is very user friendly for solving FKGE and accurate.
Keywords: fractional Klein-Gordon equation; Legendre scaling functions; operational matrices
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References

Podlubny I., Fractional differential equations, Academic Press, San Diego, Calif, USA 198, 1999.

Caputo M., Elasticity and Dissipation [in Italy: Elasticità e Dissipazione], Zani-Chelli, Bologna, 1969.

KilbasA.A., Srivastava H.M., Trujillo, J.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam 204, 2006.

Hristov J., Transient Heat Diffusion with a Non-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2): 765-770, 2016.

Baskonus H.M., Mekkaoui T., Hammouch Z., Bulut H., Active Control of a Chaotic Fractional Order Economic System, Entropy, 17(8): 5771–5783, 2015.

Singh H., A new stable algorithm for fractional Navier-Stokes equation in polar coordinate, International Journal of Applied and Computational Mathematics, 2017, doi: 10.1007/s40819-017-0323-7.

Yang X.J., Machado J.A.T., Cattani C., Gao F., On a fractal LC-electric circuit modelled by local fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 47: 200–206, 2017.

Yang X.J., A new integral transform operator for solving the heat-diffusion problem, Applied Mathematics Letters, 64: 193–197, 2017.

Srivastava H.M., Kumar D., Singh J., An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45: 192–204, 2017.

Kumar D., Singh J., Baleanu D., A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Mathematical Methods in the Applied Sciences, 40(15): 5407–5682, 2017.

Kumar D., Agarwal R.P., Singh J., A modified numerical scheme and convergence analysis for fractional model of Lienard's equation, Journal of Computational and Applied Mathematics, 339: 405–413, 2018.

Singh J., Kumar D., Swroop R., Kumar S., An efficient computational approach for time-fractional Rosenau-Hyman equation, Neural Computing and Applications, 30(10):3063–3070, 2018.

Sumelka W., Thermoelasticity in the framework of the fractional continuum mechanics, Journal of Thermal Stresses, 37(6): 678–706, 2014.

Sumelka W., Non-local Kirchhoff–Love plates in terms of fractional calculus, Archives of Civil and Mechanical Engineering, 15(1): 231–242, 2015.

Béda P.B., Dynamical systems approach of integral length in fractional calculus, Engineering Transaction, 65(1): 209–215, 2017.

Lazopoulos A.K., On fractional peridynamic deformations, Archive of Applied Mechanics, 86(12): 1987–1994, 2016.

Ying Y.P., Lian Y.P., Tang S.Q., Liu W.K., High-order central difference scheme for Caputo fractional derivative, Computer Methods in Applied Mechanics and Engineering, 317: 42–54, 2017.

Abbasbandy S., Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, International Journal for Numerical Methods in Engineering, 70: 876–881, 2007.

Mohyud-Din S.T., Noor M.A., Noor K.I., Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering, 2009, doi:10.1155/2009/234849.

Wazwaz A.M., The modified decomposition method for analytic treatment of differential equations, Applied Mathematics and Computations, 173: 165–176, 2006.

Kumar D., Singh J., Kumar S., Sushila, Numerical computation of Klein-Gordon equations arising in quantum field theory by using homotopy analysis transform method, Alexandria Engineering Journal, 53: 469–474, 2014.

Golmankhaneh A.K., Baleanu D., On nonlinear fractional Klein-Gordon equation, Signal Processing, 91: 446–451, 2011.

Kurulay M., Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method, Advances in Difference Equations, 2012. doi: 10.1186/1687-1847-2012-187.

Gepreel K.A., Mohamed M.S., Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation, Chinese Physics B, 22, 2013, doi: 10.1088/1674-1056/22/1/010201.

Kumar D., Singh J., Baleanu D., A hybrid computational approach for Klein- Gordon equations on Cantor sets, Nonlinear Dynamics, 87: 511–517, 2017.

Khader M.M., Kumar S., An accurate numerical method for solving the linear fractional Klein–Gordon equation, Mathematical Methods in Applied Sciences, 2013, doi: 10.1002/mma.3035.

Yousif M. A., Mahmood B. A., Approximate solutions for solving the Klein-Gordon and sine-Gordon equations, Journal of the Association of Arab Universities for Basic and Applied Sciences, 22: 83–90, 2017.

Singh H., Srivastava H.M., Kumar D., A reliable numerical algorithm for the fractional vibration equation, Chaos Solitons & Fractals, 103: 131–138, 2017.

de la Hoz F., Vadillo F., The solution of two dimensional advection-diffusion equation via operational matrices, Applied Numerical Mathematics, 72: 172–187, 2013.

ur Rehman M., Khan R.A., Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Applied Mathematical Modelling, 37(7): 5233–5244, 2013, doi: 10.1016/j.apm.2012.10.045.

Singh H., Operational matrix approach for approximate solution of fractional model of Bloch equation, Journal of King Saud University–Science, 29: 235–240, 2017.

Bhrawy A.H., Zaky M.A., A method based on Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, Journal of Computational Physics, 281: 876–895, 2015.

Wu J.L., A wavelet operational method for solving fractional partial differential equations numerically, Applied Mathematics and Computations, 214: 31–40, 2009.

Wang L., Ma Y., Meng Z., Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computations, 227: 66–76, 2014.

Heydari M.H., Hooshmandasl M.R., Ghaini F.M.M., A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Applied Mathematical Modelling, 38: 1597–1606, 2014.

Singh H., Singh C.S., Stable numerical solutions of fractional partial differential equations using Legendre scaling functions operational matrix, Ain Shams Engineering Journal, 9(4): 717–725, 2018, http://dx.doi.org/10.1016/j.asej.2016.03.013.

Tohidi E., Bhrawy A.H., Erfani K., A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Applied Mathematical Modelling, 37: 4283–4294, 2013.

Kazem S., Abbasbandy S., Kumar S., Fractional order Legendre functions for solving fractional-order differential equations, Applied Mathematical Modelling, 37: 5498–5510, 2013.

Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, North-Holland Mathematical Studies, 204: Elsevier (North-Holland) Science Publishers, Amsterdam, 2006.

Singh H., A new numerical algorithm for fractional model of bloch equation in nuclear magnetic resonance, Alexandria Engineering Journal, 55: 2863–2869, 2016.

Singh C.S., Singh H., Singh V.K., Singh Om P., Fractional order operational matrix methods for fractional singular integro-differential equation, Applied Mathematical Modelling, 40: 10705–10718, 2016.




DOI: 10.24423/EngTrans.910.20190214

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