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  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, 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.  Google Scholar

[2]

A. AkgülM. 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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

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

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.  Google Scholar

[2]

A. AkgülM. 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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

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

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
$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.5$ $\alpha=\beta=0.9$
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
$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.55$ $\alpha=\beta=0.95$
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
$x$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$ $\alpha=\beta=0.1$
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
$x$ $AS$ $ES$ $AE$
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
$x$ $AS$
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]

Ali Akgül. A new application of the reproducing kernel method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020261

[2]

Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055

[3]

Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems & Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058

[4]

Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233

[5]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020048

[6]

Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221

[7]

Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020421

[8]

Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020428

[9]

Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031

[10]

Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541

[11]

Ying Lin, Rongrong Lin, Qi Ye. Sparse regularized learning in the reproducing kernel banach spaces with the $ \ell^1 $ norm. Mathematical Foundations of Computing, 2020, 3 (3) : 205-218. doi: 10.3934/mfc.2020020

[12]

Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842

[13]

Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188

[14]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289

[15]

Steven G. Krantz and Marco M. Peloso. New results on the Bergman kernel of the worm domain in complex space. Electronic Research Announcements, 2007, 14: 35-41. doi: 10.3934/era.2007.14.35

[16]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95

[17]

Ting Hu. Kernel-based maximum correntropy criterion with gradient descent method. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4159-4177. doi: 10.3934/cpaa.2020186

[18]

Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005

[19]

Badr Saad T. Alkahtani, Ilknur Koca. A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 429-442. doi: 10.3934/dcdss.2020024

[20]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (25)
  • HTML views (50)
  • Cited by (0)

Other articles
by authors

[Back to Top]