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

  • * Corresponding author: Xiaozhong Yang

    * Corresponding author: Xiaozhong Yang 

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)

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  • 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.

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

    Citation:

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  • Figure 1.  Curved surface of exact solution

    Figure 2.  Curved surface of implicit scheme solution

    Figure 3.  Curved surface of E-I scheme solution

    Figure 4.  Curved surface of I-E scheme solution

    Figure 5.  SRET of numerical solutions of three schemes

    Figure 6.  The distribution of DTE in spatial grid

    Figure 7.  Comparison of computing time among three schemes

    Figure 8.  Curved surface of exact solution

    Figure 9.  Curved surface of implicit scheme solution

    Figure 10.  Curved surface of E-I scheme solution

    Figure 11.  Curved surface of I-E scheme solution

    Figure 12.  SRET of numerical solutions of two schemes

    Figure 13.  The distribution of DTE in spatial grid

    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
     | Show Table
    DownLoad: CSV

    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
    $ 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
     | Show Table
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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
    $ 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
    $ 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
     | Show Table
    DownLoad: CSV

    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
    $ 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
     | Show Table
    DownLoad: CSV
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