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Analysis and new applications of fractal fractional differential equations with power law kernel

  • * Corresponding author: Ali Akgül

    * Corresponding author: Ali Akgül

The first author is supported by 2020-SIUFEB-022

Abstract Full Text(HTML) Figure(7) / Table(5) Related Papers Cited by
  • We obtain the solutions of fractal fractional differential equations with the power law kernel by reproducing kernel Hilbert space method in this paper. We also apply the Laplace transform to get the exact solutions of the problems. We compare the exact solutions with the approximate solutions. We demonstrate our results by some tables and figures. We prove the efficiency of the proposed technique for fractal fractional differential equations.

    Mathematics Subject Classification: Primary: 28A80; 26A33; and 46E22.

    Citation:

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  • Figure 1.  Exact Solutions (ES) of the second problem for $ \alpha = \beta = 0.1 $ and $ \alpha = \beta = 0.9 $

    Figure 2.  Exact Solutions (ES) of the second problem for $ \alpha=\beta=0.5 $ and $ \alpha=\beta=0.9 $

    Figure 3.  Exact Solutions (ES) of the second problem for $ \alpha = \beta = 1.0 $ and $ \alpha = \beta = 0.9 $

    Figure 4.  The dynamical behavior of the chaotic attractor for $ \alpha = 1 = \beta. $

    Figure 5.  The dynamical behavior of the chaotic attractor for $ \alpha = 0.98 $ and $ \beta = 0.99 $

    Figure 6.  The dynamical behavior of the chaotic attractor for $ \alpha = 0.1 $ and different values of $ \beta. $

    Figure 7.  The dynamical behavior of the chaotic attractor for $ \beta = 1 $ and different values of $ \alpha. $

    Table 1.  Approximate solutions of the first problem

    $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.5$ $\alpha=\beta=0.9$
    0.1 0.03854888333 0.0307043774 0.0078792973
    0.2 0.05333186893 0.0614101561 0.0259058607
    0.3 0.06270054818 0.0921156879 0.0516560594
    0.4 0.06958768905 0.1228211511 0.0837457401
    0.5 0.07504844747 0.1535265840 0.1212235367
    0.6 0.07958273648 0.1842320012 0.1633740132
    0.7 0.08346678875 0.2149374096 0.2096297969
    0.8 0.08686918110 0.2456428118 0.2595266634
    0.9 0.08990026700 0.2763482087 0.3126792521
    1.0 0.09263619896 0.3070536026 0.3687589104
     | Show Table
    DownLoad: CSV

    Table 2.  Absolute Errors for the second problem

    $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.55$ $\alpha=\beta=0.95$
    0.1 0.00000062152 0.00000676818 0.00000856466
    0.2 0.00000032204 0.00000490756 0.00001243897
    0.3 0.00000022407 0.00000407914 0.00002174689
    0.4 0.00000017321 0.00000358476 0.00003698863
    0.5 0.00000014600 0.00000324696 0.00005597480
    0.6 0.00000012155 0.00000299770 0.00008452815
    0.7 0.00000010765 0.00000282100 0.00009461450
    0.8 0.00000009145 0.00000269290 0.00012236140
    0.9 0.00000018475 0.00001399970 0.00021363610
    1.0 0.00000018475 0.00001399970 0.00021363610
     | Show Table
    DownLoad: CSV

    Table 3.  Relative Errors for the second problem

    $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$
    0.1 0.0001037114955 0.00247111550600 0.019860952690
    0.2 0.0000233907536 0.00041794915520 0.003864445499
    0.3 0.0000100048043 0.00014826368870 0.002084659184
    0.4 0.0000054760999 0.00007121228610 0.001539514613
    0.5 0.0000035331934 0.00004037018634 0.001219745100
    0.6 0.0000023623478 0.00002541507333 0.001085556125
    0.7 0.0000017388686 0.00001730286530 0.000777074981
    0.8 0.0000012705894 0.00001247820391 0.000682296305
    0.9 0.0000009831650 0.00000965486512 0.000773284787
    1.0 0.0000019451577 0.00004060126398 0.000623683400
     | Show Table
    DownLoad: CSV

    Table 4.  Approximate Solution (AS), Exact Solution (ES), Absolute Error (AE) and Relative Error (RE) for the second problem for $ \alpha = 0.5 $ and $ \beta = 1 $

    $x$ $AS$ $ES$ $AE$
    0.1 0.0019195134 0.00190306572 0.00001644769
    0.2 0.0107895265 0.01076536543 0.00002416107
    0.3 0.0296830910 0.02966585872 0.00001723228
    0.4 0.0610499710 0.06089810315 0.00015186785
    0.5 0.1065047070 0.10638460810 0.00012009890
    0.6 0.1680971830 0.16781543890 0.00028174410
    0.7 0.2469466021 0.24671689310 0.00022970900
    0.8 0.3444606528 0.34449169370 0.00003104090
    0.9 0.4653131081 0.46244497090 0.00339997550
    1.0 0.6052021980 0.60180222250 0.00339997550
     | Show Table
    DownLoad: CSV

    Table 5.  Approximate Solution (AS) for the third problem for $ \alpha = \beta = 0.5 $

    $x$ $AS$
    0.1 0.00048062670
    0.2 0.00395841402
    0.3 0.01287525177
    0.4 0.02728111814
    0.5 0.04281856069
    0.6 0.05461350578
    0.7 0.05592662854
    0.8 0.03911508097
    0.9 -0.0027967424
    1.0 -0.0752782621
     | Show Table
    DownLoad: CSV
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