Numerical analysis of two new finite difference methods for time-fractional telegraph equation

Xiaozhong Yang; Xinlong Liu

doi:10.3934/dcdsb.2020269

Article Contents

2021, Volume 26, Issue 7: 3921-3942. Doi: 10.3934/dcdsb.2020269

This issue Previous Article On the reducibility of a class of almost periodic Hamiltonian systems Next Article On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

Xiaozhong Yang^, and
Xinlong Liu

School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

^* Corresponding author: Xiaozhong Yang
^* Corresponding author: Xiaozhong Yang

Received: December 1, 2019

Revised: July 1, 2020

Early access: September 2, 2020

Published online: July 1, 2021

The work was supported in part by the Subproject of Major Science and Technology Program of China (No.2017ZX07101001-01) and the National Natural Science Foundation of China (No.11371135).

Abstract / Introduction Full Text(HTML) Figure(13) / Table(6) Related Papers Cited by

Abstract

Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals. Aiming at the one-dimensional time-fractional telegraph equation, a class of explicit-implicit (E-I) difference methods and implicit-explicit (I-E) difference methods are proposed. The two methods are based on a combination of the classical implicit difference method and the classical explicit difference method. Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, $ 2-\alpha $ order time accuracy, and have significant time-saving, their calculation efficiency is higher than the classical implicit scheme. The research shows that the E-I and I-E difference methods constructed in this paper are effective for solving the time-fractional telegraph equation.

Keywords:

Time-fractional telegraph equation,
E-I difference scheme and I-E difference scheme,
unconditional stability,
convergence,
numerical experiments.

Mathematics Subject Classification: Primary: 65M06; Secondary: 65M12.

Citation:

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Figure 1. Curved surface of exact solution

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Figure 2. Curved surface of implicit scheme solution

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Figure 3. Curved surface of E-I scheme solution

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Figure 4. Curved surface of I-E scheme solution

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Figure 5. SRET of numerical solutions of three schemes

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Figure 6. The distribution of DTE in spatial grid

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Figure 7. Comparison of computing time among three schemes

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Figure 8. Curved surface of exact solution

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Figure 9. Curved surface of implicit scheme solution

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Figure 10. Curved surface of E-I scheme solution

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Figure 11. Curved surface of I-E scheme solution

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Figure 12. SRET of numerical solutions of two schemes

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Figure 13. The distribution of DTE in spatial grid

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Table 1. Comparison of numerical and exact solutions

	21	41	61	81
Exact solution	0.032816	0.038767	0.045800	0.054104
Implicit scheme solution	0.032727	0.038662	0.045673	0.053957
E-I scheme solution	0.032238	0.037960	0.044992	0.053152
I-E scheme solution	0.032168	0.038002	0.044895	0.053037

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Table 2. Spatial convergence orders and numerical errors of implicit scheme, E-I scheme and I-E scheme($ \tau = h^{2}) $

$ \alpha $	M	Implicit scheme		E-I scheme		I-E scheme
$ \alpha $	M	$ E_{2}(h, \tau) $	Order1	$ E_{2}(h, \tau) $	Order1	$ E_{2}(h, \tau) $	Order1
0.6	6	0.601261438	—	0.670908875	—	0.676867797	—
	12	0.171331703	1.811200301	0.185496548	1.854724481	0.185742355	1.865571243
	24	0.027626239	1.861354661	0.020497637	1.880049324	0.056380674	1.888092124
0.7	6	0.572754131	—	0.625245364	—	0.632403830	—
	12	0.154051855	1.894499914	0.167207909	1.902779364	0.167706812	1.914904807
	24	0.043125187	1.968989813	0.042627245	1.954591537	0.042269721	1.953396613
0.8	6	0.331546455	—	0.384936527	—	0.392207783	—
	12	0.093281787	1.829543269	0.000168779	1.942187657	0.000592926	1.963089311
	24	0.025790914	1.887189262	0.042627245	1.954591537	0.042269721	1.953396613
0.9	6	0.312856198	—	0.351172503	—	0.354347245	—
	12	0.079460269	1.977194091	0.084117921	2.061694794	0.084447694	2.069033890
	24	0.020221416	1.978009907	0.021169310	2.057999271	0.021427164	2.056110578

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Table 3. Time convergence orders and numerical errors of implicit scheme, E-I scheme and I-E scheme($ h = \frac{1}{40} $)

$ \alpha $	N	Implicit scheme		E-I scheme		I-E scheme
$ \alpha $	N	$ E_{2}(h, \tau) $	Order2	$ E_{2}(h, \tau) $	Order2	$ E_{2}(h, \tau) $	Order2
0.6	300	0.001748225	—	0.001257729	—	0.001257729	—
	600	0.000647496	1.432945906	0.000769857	1.408158851	0.000795404	1.415571243
	1200	0.000238734	1.437531875	0.000300146	1.406435732	0.000300015	1.406473635
0.7	300	0.003765612	—	0.002220764	—	0.002081502	—
	600	0.001462733	1.364217584	0.000902026	1.299814239	0.000902020	1.206383164
	1200	0.000662609	1.342435234	0.000608222	1.325685702	0.000608231	1.322349016
0.8	300	0.002921061	—	0.001592343	—	0.001453911	—
	600	0.001224089	1.254076687	0.000647496	1.298205933	0.000647496	1.236914092
	1200	0.000532251	1.205163582	0.000267776	1.296425166	0.000264108	1.293815168
0.9	300	0.003178454	—	0.002285670	—	0.002149714	—
	600	0.001481610	1.101159256	0.000973810	1.230904340	0.000910832	1.238886806
	1200	0.000791962	1.052349016	0.000348563	1.116311302	0.000451147	1.116552467

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Table 4. Comparison of numerical and exact solutions

	21	41	61	81
Exact solution	-0.03079	-0.04618	-0.04618	-0.03079
Implicit scheme solution	-0.03008	-0.04505	-0.04505	-0.03008
E-I scheme solution	-0.03001	-0.04492	-0.04492	-0.03001
I-E scheme solution	-0.03001	-0.04492	-0.04492	-0.03001

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Table 5. Spatial convergence orders and numerical errors of implicit scheme and E-I scheme ($ \tau = h^{2}) $

$ \alpha $	M	Implicit scheme		E-I scheme
$ \alpha $	M	$ E_{2}(h, \tau) $	Order1	$ E_{2}(h, \tau) $	Order1
	8	0.007395715	—	0.006928145	—
0.2	16	0.001926582	1.940537064	0.001786161	1.955606884
	32	0.000486619	1.985178414	0.000447580	1.996642455
	64	0.000121970	1.996305831	0.000110261	2.004338515
	8	0.007399299	—	0.006730926	—
0.3	16	0.001926975	1.944911724	0.001748227	1.944911722
	32	0.000486657	1.996642453	0.000440517	1.988623313
	64	0.000121970	1.996374655	0.000110261	1.998266445
	8	0.007406310	—	0.006655592	—
0.4	16	0.001927758	1.941830430	0.001736738	1.938186701
	32	0.000486747	1.985359045	0.000438647	1.988623313
	64	1.988623313	1.996518415	0.000109876	1.996547989
	8	0.007418404	—	0.006638599	—
0.5	16	0.001929358	1.942987829	0.001735124	1.935839557
	32	0.000486961	1.986241089	0.000438424	1.984638177
	64	0.000122001	1.996814720	0.000109876	1.996450948

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Table 6. Time convergence orders and numerical errors of implicit scheme and E-I scheme ($ h = \frac{1}{40} $)

$ \alpha $	N	Implicit scheme		E-I scheme
$ \alpha $	N	$ E_{2}(h, \tau) $	Order2	$ E_{2}(h, \tau) $	Order2
	150	0.003247111	—	0.003127877	—
0.2	300	0.0038902781	0.260726667	0.003593341	0.200141963
	600	0.0046878951	0.269069826	0.000447580	0.211580117
	150	0.003053496	—	0.002880601	—
0.3	300	0.003751789	0.297116513	0.003603101	0.322869023
	600	0.004664256	0.314069826	0.004535791	0.332115801
	150	0.002847062	—	0.002645836	—
0.4	300	0.003604515	0.340331208	0.003490906	0.399877794
	600	0.004639253	0.359191478	0.004623847	0.405493914
	150	0.002620013	—	0.002391991	—
0.5	300	0.003442873	0.394038915	0.003330367	0.477464645
	600	0.004561314	0.405837085	0.004690116	0.493943176

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References

[1]	T. Akram, M. Abbas, A. I. Ismail, N. Hj. M. Ali and D. Baleanu, Extended cubic B-splines in the numerical solution of time fractional telegraph equation, Adv. Differ. Equ., (2019), Paper No. 365, 20 pp. doi: 10.1186/s13662-019-2296-9.
[2]	D. J. Arrigo and S. G. Krantz, Analytical Techniques for Solving Nonlinear Partial Differential Equations, Morgan & Claypool Publishers, 2019. doi: 10.2200/S00907ED1V01Y201903MAS025.
[3]	A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043.
[4]	R. C. Cascaval, E. C. Eckstein, C. L. Frota and J. A. Goldstein, Fractional telegraph equations, J. Math. Anal. Appl., 276 (2002), 145-159. doi: 10.1016/S0022-247X(02)00394-3.
[5]	J. Chen, F. Liu and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377. doi: 10.1016/j.jmaa.2007.06.023.
[6]	S. Das, K. Vishal, P. K. Gupta and A. Yildirim, An approximate analytical solution of time-fractional telegraph equation, Appl. Math. Comput., 217 (2011), 7405-7411. doi: 10.1016/j.amc.2011.02.030.
[7]	W. Deng and Z. Zhang, High Accuracy Algorithm for the Differentail Equations Governing Anomalous Diffusion, Algorithm and Models for Anomalous Diffusion, World Scientific, Singapore, 2019.
[8]	K. Diethelm, The Analysis of Fraction Differential Equations, Springer, 2010. doi: 10.1007/978-3-642-14574-2.
[9]	M. Ferreira, M. M. Rodrigues and N. Vieira, First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac oprators, Adv. Appl. Clifford Algebr, 28 (2018), Art. 42, 14 pp. doi: 10.1007/s00006-018-0858-7.
[10]	N. J. Ford, M. M. Rodrigues, J. Xiao and Y. Yan, Numerical analysis of a two-parameter fractional telegraph equation, J. Comput. Appl. Math., 249 (2013), 95-106. doi: 10.1016/j.cam.2013.02.009.
[11]	B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, 2011. doi: 10.1142/9543.
[12]	L. Hervé and L. Brigitte, Partial Differential Equations: Modeling, Analysis and Numerical Approximation, Springer International Publishing, Switzerland, 2016.
[13]	M. H. Heydari, M. R. Hooshmandasl and F. Mohammadi, Two-Dimensional legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6 (2014), 247-260. doi: 10.4208/aamm.12-m12132.
[14]	V. R. Hosseini, W. Chen and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem, 38 (2014), 31-39. doi: 10.1016/j.enganabound.2013.10.009.
[15]	W. Jiang and Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3639-3645. doi: 10.1016/j.cnsns.2010.12.019.
[16]	K. Kumar, R. K. Pandey and S. Yadav, Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms, Physica A, 535 (2019), Art. 122271, 15 pp. doi: 10.1016/j.physa.2019.122271.
[17]	C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015.
[18]	C. Li, Z. Zhao and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875. doi: 10.1016/j.camwa.2011.02.045.
[19]	Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001.
[20]	F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191 (2007), 17-20. doi: 10.1016/j.amc.2006.08.162.
[21]	F. Liu, P. Zhuang and Q. Liu, Numerical Solutions of Fractional Order Partial Differential Equations and its Applications, Science Press, Beijing, 2015.
[22]	M. O. Mamchuev, Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method, Fract. Calc. Appl. Anal., 20 (2017), 190-211. doi: 10.1515/fca-2017-0010.
[23]	S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170 (2005), 1126-1134. doi: 10.1016/j.amc.2005.01.009.
[24]	E. Orsingher and L. Beghin, Time-fractional telegraph equations and telegraph processes with brownian time, Probab. Theory Relat. Fields, 128 (2004), 141-160. doi: 10.1007/s00440-003-0309-8.
[25]	A. Saadatmandi and M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155-166.
[26]	J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, World Book Incorporated Beijing, 2014. doi: 10.1007/978-1-4020-6042-7.
[27]	M. Stynes, E. O' Riordan and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. doi: 10.1137/16M1082329.
[28]	Z. Sun and G. Gao, Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015.
[29]	V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Volume II Applications, Higher Education Press; Springer, Heidelberg, Beijing, 2013. doi: 10.1007/978-3-642-33911-0.
[30]	S. Vong, P. Lyu and Z. Wang, A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under neumann boundary conditions, J. Sci. Comput., 66 (2016), 725-739. doi: 10.1007/s10915-015-0040-5.
[31]	H. Wang, A. Cheng and K. Wang, Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 1427-1441. doi: 10.3934/dcdsb.2015.20.1427.
[32]	L. Wei, H. Dai, D. Zhang and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192. doi: 10.1007/s10092-013-0084-6.
[33]	P. Xanthoulos and G. E. Zouraris, A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 239-263. doi: 10.3934/dcdsb.2008.10.239.
[34]	X. Yang, L. Wu, S. Sun and X. Zhang, A universal difference method for time-space fractional Black-Scholes equation, Adv. Differ. Equ., (2016), Paper No. 71, 14 pp. doi: 10.1186/s13662-016-0792-8.
[35]	A. Yildirim, He's homtopy perturbation method for solving the space and time fractional telegraph equations, Int. J. Comput. Math., 87 (2010), 2998-3006. doi: 10.1080/00207160902874653.
[36]	Z. Zhao and C. Li, Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975-2988. doi: 10.1016/j.amc.2012.09.022.