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

• * Corresponding author: Ali Akgül

The first author is supported by 2020-SIUFEB-022

• 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:

• 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

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

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

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

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
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