# American Institute of Mathematical Sciences

• Previous Article
Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition
• DCDS-S Home
• This Issue
• Next Article
Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays

## 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:

show all references

##### 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
 [1] Ali Akgül. A new application of the reproducing kernel method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2041-2053. 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, 2021, 15 (1) : 1-25. 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] Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053 [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] 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 [11] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2019 Impact Factor: 1.233

## Tools

Article outline

Figures and Tables