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 OJMSi  Vol.10 No.3 , July 2022
Computational Analysis for Solving the Linear Space-Fractional Telegraph Equation
Zaki Mrzog Alaofi1,2, Talaat Sayed El-Danaf3, Adel Hadhoud4, Silvestru Sever Dragomir2
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Abstract: Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.
Keywords: Fractional Differential Equations, Quadratic Spline Functions, Linear Space-Fractional Telegraph Equation, Von Neumann Stability
Cite this paper: Alaofi, Z. , El-Danaf, T. , Hadhoud, A. and Dragomir, S. (2022) Computational Analysis for Solving the Linear Space-Fractional Telegraph Equation. Open Journal of Modelling and Simulation, 10, 267-282. doi: 10.4236/ojmsi.2022.103014.
References

[1]   Bagley, R.L. and Torvik, P.J. (1983) A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity. Journal of Rheology, 27, 201-210.
https://doi.org/10.1122/1.549724

[2]   Debnath, L. (1997) Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhauser, Boston.
https://doi.org/10.1007/978-1-4899-2846-7

[3]   Metaxas, A.C. and Meredith, R.J. (1993) Industrial Microwave Heating. Peter Peregrinus, London.

[4]   Odibat, Z. and Momani, S. (2008) A Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order. Applied Mathematics Letters, 21, 194-199.
https://doi.org/10.1016/j.aml.2007.02.022

[5]   Garg, M., Manohar, P. and Kalla, S.L. (2011) Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation. International Journal of Differential Equations, 2011, Article ID: 548982.
https://doi.org/10.1155/2011/548982

[6]   Zhao, Z. and Li, C. (2012) Fractional Difference/Finite Element Approximations for the Time-Space Fractional Telegraph Equation. Applied Mathematics and Computation, 219, 2975-2988.
https://doi.org/10.1016/j.amc.2012.09.022

[7]   Gómez Aguilar, J.F. and Baleanu, D. (2014) Solutions of the Telegraph Equations Using a Fractional Calculus Approach. Proceedings of the Romanian Academy. Series A, 15, 27-34.

[8]   Abedi-Varaki, M., Rajabi, S., Ghorbani, V. and Hosseinzadeh, F. (2015) Solution of the Space-Fractional Telegraph Equations by Using HAM. Ciência e Naturalium, 37, 320.
https://doi.org/10.5902/2179460X20789

[9]   Lopushanska, H. and Rapita, V. (2014) Inverse Coefficient Problem for the Semi-Linear Fractional Telegraph Equation. Electronic Journal of Differential Equations, 2015, 1-13.

[10]   Khan, H., Tunç, C. and Khan, R.A. (2018) Approximate Analytical Solutions of Space-Fractional Telegraph Equations by Sumudu Adomian Decomposition Method. Applications and Applied Mathematics, 13, 781-802.

[11]   Uddin, K.M. and Ali, A. (2018) On the Approximation of Time-Fractional Telegraph Equations Using Localized Kernel-Based Method. Advances in Difference Equations, 2018, Article No. 305.
https://doi.org/10.1186/s13662-018-1775-8

[12]   Wei, L., Liu, L. and Sun, H. (2018) Numerical Methods for Solving the Time-Fractional Telegraph Equation. Taiwanese Journal of Mathematics, 22, 1509-1528.
https://doi.org/10.11650/tjm/180503

[13]   Liu, R. (2018) Fractional Difference Approximations for Time-Fractional Telegraph Equation. Journal of Applied Mathematics and Physics, 6, 301-309.
https://doi.org/10.4236/jamp.2018.61029

[14]   Mohammadian, S., Mahmoudi, Y. and Saei, F.D. (2019) Solution of Fractional Telegraph Equation with Riesz Space-Fractional Derivative. AIMS Mathematics, 4, 1664-1683.
https://doi.org/10.3934/math.2019.6.1664

[15]   Akram, T., Abbas, M., Ismail, A.I., Ali, N.H.M. and Baleanu, D. (2019) Extended Cubic B-Splines in the Numerical Solution of Time Fractional Telegraph Equation. Advances in Difference Equations, 2019, Article No. 365.
https://doi.org/10.1186/s13662-019-2296-9

[16]   Kumar, K., Pandey, R.K. and Yadav, S. (2019) Finite Difference Scheme for a Fractional Telegraph Equation with Generalized Fractional Derivative Terms. Physica A: Statistical Mechanics and Its Applications, 535, Article ID: 122271.
https://doi.org/10.1016/j.physa.2019.122271

[17]   Ali, A. and Ali, N.H.M. (2019) On Skewed Grid Point Iterative Method for Solving 2D Hyperbolic Telegraph Fractional Differential Equation. Advances in Difference Equations, 2019, Article No. 303.
https://doi.org/10.1186/s13662-019-2238-6

[18]   Hosseininia, M. and Heydari, M.H. (2019) Meshfree Moving Least Squares Method for Nonlinear Variable-Order Time Fractional 2D Telegraph Equation Involving Mittag-Leffler Non-Singular Kernel. Chaos, Solitons and Fractals, 127, 389-399.
https://doi.org/10.1016/j.chaos.2019.07.015

[19]   Bouaouid, M., Hilal, K. and Melliani, S. (2019) Nonlocal Telegraph Equation in Frame of the Conformable Time-Fractional Derivative. Advances in Mathematical Physics, 2019, Article ID: 7528937.
https://doi.org/10.1186/s13662-019-1954-2

[20]   Mohammadian, S., Mahmoudi, Y. and Saei, F.D. (2020) Analytical Approximation of Time-Fractional Telegraph Equation with Riesz Space-Fractional Derivative. Difference Equations and Applications, 12, 243-258.
https://doi.org/10.7153/dea-2020-12-16

[21]   Wu, L., Pan, Y. and Yang, X. (2021) An Efficient Alternating Segment Parallel Finite Difference Method for Multi-Term Time Fractional Diffusion-Wave Equation. Computational and Applied Mathematics, 40, Article No. 67.
https://doi.org/10.1007/s40314-021-01455-0

[22]   Hamada, Y.M. (2020) Solution of a New Model of Fractional Telegraph Point Reactor Kinetics Using Differential Transformation Method. Applied Mathematical Modelling, 78, 297-321.
https://doi.org/10.1016/j.apm.2019.10.001

[23]   Devi, A. and Jakhar, M. (2021) A New Computational Approach for Solving Fractional Order Telegraph Equations. Journal of Scientific Research, 13, 715-732.
https://doi.org/10.3329/jsr.v13i3.50659

[24]   Hamza, A.E., Mohamed, M.Z., Abd Elmohmoud, E.M. and Magzoub, M. (2021) Conformable Sumudu Transform of Space-Time Fractional Telegraph Equation. Abstract and Applied Analysis, 2021, Article ID: 6682994.
https://doi.org/10.1155/2021/6682994

[25]   Azhar, O.F., Naeem, M., Mofarreh, F. and Kafle, J. (2021) Numerical Analysis of the Fractional-Order Telegraph Equations. Journal of Function Spaces, 2021, Article ID: 2295804.
https://doi.org/10.1155/2021/2295804

[26]   Ibrahim, W. and Bijiga, L.K. (2021) Neural Network Method for Solving Time-Fractional Telegraph Equation. Mathematical Problems in Engineering, 2021, Article ID: 7167801.
https://doi.org/10.1155/2021/7167801

[27]   Vieira, N., Rodrigues, M.M. and Ferreira, M. (2021) Time-Fractional Telegraph Equation of Distributed Order in Higher Dimensions. Communications in Nonlinear Science and Numerical Simulation, 102, Article ID: 105925.
https://doi.org/10.1016/j.cnsns.2021.105925

[28]   Khater, M.M.A., Park, C., Lee, J.R., Mohamed, M.S. and Attia, R.A.M. (2021) Five Semi Analytical and Numerical Simulations for the Fractional Nonlinear Space-Time Telegraph Equation. Advances in Difference Equations, 2021, Article No. 227.
https://doi.org/10.1186/s13662-021-03387-9

[29]   Nikan, O., Avazzadeh, Z. and Machado, J.A.T. (2021) Numerical Approximation of the Nonlinear Time-Fractional Telegraph Equation Arising in Neutron Transport. Communications in Nonlinear Science and Numerical Simulation, 99, Article ID: 105755.
https://doi.org/10.1016/j.cnsns.2021.105755

[30]   Ahlberg, J., Nilson, E. and Walsh, J. (1967) The Theory of Splines and Their Applications. Academic Press, New York.

[31]   Wai, K.K.S. and Tint, S.S. (2019) Von Neumann Stability Analysis for Time-Dependent Diffusion Equation. IRE Journals, 3, 279-283.

 
 
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