# American Institute of Mathematical Sciences

October  2021, 14(10): 3401-3417. doi: 10.3934/dcdss.2020423

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

 Siirt University, Art and Science Faculty, Department of Mathematics, TR-56100 Siirt, Turkey

* Corresponding author: Ali Akgül

Received  October 2019 Revised  February 2020 Published  October 2021 Early access  September 2020

Fund Project: 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.

Citation: Ali Akgül. Analysis and new applications of fractal fractional differential equations with power law kernel. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3401-3417. doi: 10.3934/dcdss.2020423
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##### References:
Exact Solutions (ES) of the second problem for $\alpha = \beta = 0.1$ and $\alpha = \beta = 0.9$
Exact Solutions (ES) of the second problem for $\alpha=\beta=0.5$ and $\alpha=\beta=0.9$
Exact Solutions (ES) of the second problem for $\alpha = \beta = 1.0$ and $\alpha = \beta = 0.9$
The dynamical behavior of the chaotic attractor for $\alpha = 1 = \beta.$
The dynamical behavior of the chaotic attractor for $\alpha = 0.98$ and $\beta = 0.99$
The dynamical behavior of the chaotic attractor for $\alpha = 0.1$ and different values of $\beta.$
The dynamical behavior of the chaotic attractor for $\beta = 1$ and different values of $\alpha.$
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
 $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.5$ $\alpha=\beta=0.9$ 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
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
 $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.55$ $\alpha=\beta=0.95$ 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
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
 $x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$ 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
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
 $x$ $AS$ $ES$ $AE$ 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
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
 $x$ $AS$ 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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