# American Institute of Mathematical Sciences

July  2021, 26(7): 3921-3942. doi: 10.3934/dcdsb.2020269

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

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

* Corresponding author: Xiaozhong Yang

Received  December 2019 Revised  July 2020 Published  July 2021 Early access  September 2020

Fund Project: 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)

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.

Citation: Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269
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##### References:
Curved surface of exact solution
Curved surface of implicit scheme solution
Curved surface of E-I scheme solution
Curved surface of I-E scheme solution
SRET of numerical solutions of three schemes
The distribution of DTE in spatial grid
Comparison of computing time among three schemes
Curved surface of exact solution
Curved surface of implicit scheme solution
Curved surface of E-I scheme solution
Curved surface of I-E scheme solution
SRET of numerical solutions of two schemes
The distribution of DTE in spatial grid
Comparison of numerical and exact solutions
 21 41 61 81 Exact solution 0.032816 0.038767 0.0458 0.054104 Implicit scheme solution 0.032727 0.038662 0.045673 0.053957 E-I scheme solution 0.032238 0.03796 0.044992 0.053152 I-E scheme solution 0.032168 0.038002 0.044895 0.053037
 21 41 61 81 Exact solution 0.032816 0.038767 0.0458 0.054104 Implicit scheme solution 0.032727 0.038662 0.045673 0.053957 E-I scheme solution 0.032238 0.03796 0.044992 0.053152 I-E scheme solution 0.032168 0.038002 0.044895 0.053037
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 $E_{2}(h, \tau)$ 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
 $\alpha$ M Implicit scheme E-I scheme I-E scheme $E_{2}(h, \tau)$ 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
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 $E_{2}(h, \tau)$ 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
 $\alpha$ N Implicit scheme E-I scheme I-E scheme $E_{2}(h, \tau)$ 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
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
 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
Spatial convergence orders and numerical errors of implicit scheme and E-I scheme ($\tau = h^{2})$
 $\alpha$ M Implicit scheme E-I scheme $E_{2}(h, \tau)$ 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
 $\alpha$ M Implicit scheme E-I scheme $E_{2}(h, \tau)$ 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
Time convergence orders and numerical errors of implicit scheme and E-I scheme ($h = \frac{1}{40}$)
 $\alpha$ N Implicit scheme E-I scheme $E_{2}(h, \tau)$ 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
 $\alpha$ N Implicit scheme E-I scheme $E_{2}(h, \tau)$ 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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