Lie symmetries, homotopy, non-standard finite difference method for solution of the linear space-fractional telegraph equation
محورهای موضوعی : Numerical analysisA. Mohammadi 1 , S. R. Hejazi 2
1 - Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 36199-95161, Shahrood, Iran
2 - Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 36199-95161, Shahrood, Iran
کلید واژه: Lie symmetry, numerical solution, homotopy, non-standard finite difference,
چکیده مقاله :
The main purpose of this paper is to use the homotopy analytical method in order to solve the linear space-fractional telegraph differential equation. Also, non-standard finite difference method has been used to solve this equation numerically. Next, the concepts of Lie symmetry is established and the symmetries of the equation are calculated. In this article, Matlab software was used for simulation. Numerical results are presented to evaluate the efficiency and usefulness of the proposed method.
The main purpose of this paper is to use the homotopy analytical method in order to solve the linear space-fractional telegraph differential equation. Also, non-standard finite difference method has been used to solve this equation numerically. Next, the concepts of Lie symmetry is established and the symmetries of the equation are calculated. In this article, Matlab software was used for simulation. Numerical results are presented to evaluate the efficiency and usefulness of the proposed method.
[1] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Application of Symmetry Methods to Partial Differential Equations, Springer-Verlage, New York, 2000.
[2] E. Botton, J. B. Greenberg, A. Arad, D. Katoshevski, V. Vaikuntanathan, M. Ibach, B. Weigand, An investigation of grouping of two falling dissimilar droplets using the homotopy analysis method, Appl. Math. Model. 104 (2) (2022), 486-498.
[3] S. E. Fadugba, Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation, Chaos Solitons Fractals. 141 (2020), 2:110351.
[4] M. Ganjiani, Solution of nonlinear fractional differential equations using homotopy analysis method, Appl. Math. Model. 34 (6) (2010), 1634-1641.
[5] N. H. Ibragimov, Transformation Group Applied to Mathematical Physics, Riedel, Dordrecht 1985.
[6] N. H. Ibragimov, A. V. Aksenov, V. A. Baikov, Vladimir A. Chugunov, R. K. Gazizov, A. G. Meshkov, CRC Handbook of Lie Group Analysis of Differential Equations, Applications in Engineering and Physical Sciences, CRC Press, 1995.
[7] C. Kailash, P. Kapil, K. Sharma, ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay, Appl. Math. Comput. 175 (1) (2006), 864-890.
[8] M. Kashif, K. D. Dwivedi, T. Som, Numerical solution of coupled type fractional order Burgers' equation using finite difference and Fibonacci collocation method, Chinese J. Phys., 77 (2022), 2314-2323.
[9] S. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul. 2 (2) (1997), 95-100.
[10] R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, 2000.
[11] R. E. Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numer. Methods Partial Differential Equations, 5 (4) (1989), 313-325.
[12] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World scientific, 1994.
[13] K. S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
[14] K. Moaddy, S. Momani, I. Hashim, The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics, Comput. Math. Appl. 61 (4) (2011), 1209-1216.
[15] S. Momani, Analytic and approximate solutions of the space and time-fractional telegraph equations, Appl. Math. Comput. 170 (2) (2005), 1126-1134.
[16] Z. Odibat, On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations, Appl. Numer. Math. 137 (4) (2019), 203-212.
[17] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlage, New York, 1993.
[18] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
[19] H. Rezazadeh, M. S. Osman, M. Eslami, M. Ekici, A. Sonmezoglu, M. Asma, W. A. M. Othman, B. R. Wong, M. Mirzazadeh, Q. Zhou, A. Biswas, M. Belick, Mitigating internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity, Optik. 164 (2018), 84-92.
[20] A. M. Vargas, Finite difference method for solving fractional differential equations at irregular meshes, Math. Comput. Simul. 193 (2022), 204-216.
