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

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

Ali Akgül ,

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.

Keywords: Fractal fractional differential equations, power law kernel, reproducing kernel Hilbert space method, Malkus waterwheel model, numerical simulations.
Mathematics Subject Classification: Primary: 28A80; 26A33; and 46E22.
Citation: Ali Akgül. Analysis and new applications of fractal fractional differential equations with power law kernel. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3401-3417. doi: 10.3934/dcdss.2020423
References:
[1]

A. Akgül and D. Grow, Existence of solutions to the telegraph equation in binary reproducing kernel Hilbert spaces, Differential Equations and Dynamical Systems, (2019). doi: 10.1007/s12591-019-00453-3.

[2]

A. Akgül, M. Inc and E. Karatas, Reproducing kernel functions for difference equations, Discret. Contin. Dyn. Syst. Ser. S, 8 (2015), 1055-1064.  doi: 10.3934/dcdss.2015.8.1055.

[3]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.2307/1990404.

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons and Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[5]

A. Atangana and A. Akgül, On solutions of fractal fractional differential equations, Discret. Contin. Dyn. Syst. Ser. S. doi: 10.3934/dcdss.2020421.

[6]

P. Bouboulis and M. Mavroforakis, Reproducing kernel Hilbert spaces and fractal interpolation, J. Comput. Appl. Math., 235 (2011), 3425-3434.  doi: 10.1016/j.cam.2011.02.003.

[7]

V. F. M. Delgado, J. F. G. Aguilar, H. Y. Martínez, D. Baleanu, R. F. E. Jimenez and V. H. O. Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equ., (2016), 164. doi: 10.1186/s13662-016-0891-6.

[8]

J. Fahd and T. A. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results Nonlinear Anal., 2 (2018), 88-98. 

[9]

J. Fan and J. He, Fractal derivative model for air permeability in hierarchic porous media, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/354701.

[10]

H. K. Jassim, C. Ünlü, S. P. Moshokoa and C. M. Khalique, Local fractional Laplace variational iteration method for solving diffusion and wave equations on Cantor sets within local fractional operators, Math. Probl. Eng., (2015), 309870. doi: 10.1155/2015/309870.

[11]

J. Liouville, Mémoire sur quelques qustions de géomerie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions, J.d'École Polytechnique, 1 (1832), 1–69.

[12]

A. Talbot, The accurate numerical inversion of laplace transforms, IMA J. Appl. Math., 23 (1979), 97-120. 

[13]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444.

[14]

S. Xiang, Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind, Appl. Math. Comput., 232 (2014), 944-954.  doi: 10.1016/j.amc.2014.01.054.

[15]

L. M. Yan, Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations, Therm. Sci., 17 (2013), 1409-1414. 

[16]

X. J. Yang, Advanced Local Fractional Calculus and its Applications, World Science, New York, NY, USA, 2012.

show all references

References:
[1]

A. Akgül and D. Grow, Existence of solutions to the telegraph equation in binary reproducing kernel Hilbert spaces, Differential Equations and Dynamical Systems, (2019). doi: 10.1007/s12591-019-00453-3.

[2]

A. Akgül, M. Inc and E. Karatas, Reproducing kernel functions for difference equations, Discret. Contin. Dyn. Syst. Ser. S, 8 (2015), 1055-1064.  doi: 10.3934/dcdss.2015.8.1055.

[3]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.2307/1990404.

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons and Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[5]

A. Atangana and A. Akgül, On solutions of fractal fractional differential equations, Discret. Contin. Dyn. Syst. Ser. S. doi: 10.3934/dcdss.2020421.

[6]

P. Bouboulis and M. Mavroforakis, Reproducing kernel Hilbert spaces and fractal interpolation, J. Comput. Appl. Math., 235 (2011), 3425-3434.  doi: 10.1016/j.cam.2011.02.003.

[7]

V. F. M. Delgado, J. F. G. Aguilar, H. Y. Martínez, D. Baleanu, R. F. E. Jimenez and V. H. O. Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equ., (2016), 164. doi: 10.1186/s13662-016-0891-6.

[8]

J. Fahd and T. A. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results Nonlinear Anal., 2 (2018), 88-98. 

[9]

J. Fan and J. He, Fractal derivative model for air permeability in hierarchic porous media, Abstract and Applied Analysis, (2012). doi: 10.1155/2012/354701.

[10]

H. K. Jassim, C. Ünlü, S. P. Moshokoa and C. M. Khalique, Local fractional Laplace variational iteration method for solving diffusion and wave equations on Cantor sets within local fractional operators, Math. Probl. Eng., (2015), 309870. doi: 10.1155/2015/309870.

[11]

J. Liouville, Mémoire sur quelques qustions de géomerie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions, J.d'École Polytechnique, 1 (1832), 1–69.

[12]

A. Talbot, The accurate numerical inversion of laplace transforms, IMA J. Appl. Math., 23 (1979), 97-120. 

[13]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, The European Physical Journal Plus, 132 (2017), 444.

[14]

S. Xiang, Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind, Appl. Math. Comput., 232 (2014), 944-954.  doi: 10.1016/j.amc.2014.01.054.

[15]

L. M. Yan, Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations, Therm. Sci., 17 (2013), 1409-1414. 

[16]

X. J. Yang, Advanced Local Fractional Calculus and its Applications, World Science, New York, NY, USA, 2012.

Figure 1.  Exact Solutions (ES) of the second problem for $ \alpha = \beta = 0.1 $ and $ \alpha = \beta = 0.9 $
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Figure 2.  Exact Solutions (ES) of the second problem for $ \alpha=\beta=0.5 $ and $ \alpha=\beta=0.9 $
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Figure 3.  Exact Solutions (ES) of the second problem for $ \alpha = \beta = 1.0 $ and $ \alpha = \beta = 0.9 $
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Figure 4.  The dynamical behavior of the chaotic attractor for $ \alpha = 1 = \beta. $
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Figure 5.  The dynamical behavior of the chaotic attractor for $ \alpha = 0.98 $ and $ \beta = 0.99 $
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Figure 6.  The dynamical behavior of the chaotic attractor for $ \alpha = 0.1 $ and different values of $ \beta. $
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Figure 7.  The dynamical behavior of the chaotic attractor for $ \beta = 1 $ and different values of $ \alpha. $
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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
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$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.5$ $\alpha=\beta=0.9$
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
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$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.55$ $\alpha=\beta=0.95$
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
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$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$
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
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$x$ $AS$ $ES$ $AE$
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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$x$ $AS$
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
[1]

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