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doi: 10.3934/naco.2021026
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The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative

1. 

Laboratory for Pure and Applied Mathematics, University of M'sila, Bp 166 M'sila, 28000, Algeria

2. 

Department of Mathematics, Laboratory for Pure and Applied Mathematics, University of M'sila, Bp 166 M'sila, 28000, Algeria

*Corresponding author: Yacine Arioua

Received  May 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was financially supported by the General Direction of Scientific Research and Technological Development (DGRSDT)-Algeria, PRFU(Grant No. C00L03UN280120180010)

In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.

Citation: Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021026
References:
[1]

R. B AlbadarnehaI. M Batihab and M Zurigatb, Numerical solutions for linear fractional differential equations of order $1<\alpha <2$ using finite difference method (FFDM), Int. J. Math., 16 (2016), 103-111.   Google Scholar

[2]

R. Almeida, T. Odzijewicz and Agnieszka B. Malinowska, Fractional differential Equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dyn, 11 (2016), 11. Google Scholar

[3]

J. F Cheng and Y. M Chu, Solution to the linear fractional differential equation using Adomian decomposition method, Math. Probl. Eng., 2011 (2011), 14. doi: 10.1155/2011/587068.  Google Scholar

[4]

K. Diethelm, The Analysis of Fractional Differential Equations, Springer Science Berlin, New York, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[5]

N. FarazY. Khan and D. S. Sankar, Decomposition-transform method for fractional differential equations, Int. J. Nonl. Sci. Num. Sim., 11 (2010), 305-310.   Google Scholar

[6]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/3779.  Google Scholar

[7]

M. Janaki, K. Kanagarajan and D. Vivek, Analytic study on fractional implicit differential equations with impulses via Katugampola fractional derivative, Int. J. Math. And Appl., 55 (2018), 7. Google Scholar

[8]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[9]

U. N. Katugampola, A new approach to generalized fractional derivatives, J. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar

[10]

A. A. Kilbas, H. H. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. doi: 10.1016/s0304-0208(06)x8001-5.  Google Scholar

[11]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models Imperial, College Press, Singapore, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[13]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.  Google Scholar

[14]

C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-Order Systems and Controls, Advances in Industrial Control, Springer, 2010. doi: 10.1007/978-1-84996-335-0.  Google Scholar

[15]

I. Petras, Fractional-Order Nonlinear Systems, Springer, New York, 2011. doi: 10.1007/978-3-642-18101-6_3.  Google Scholar

[16] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.   Google Scholar
[17]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar

[18]

S. G. Samko, A. A. Kilbas and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.  Google Scholar

[19]

S. ShenF. Liu and V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. Algorithms, 56 (2011), 383-403.  doi: 10.1007/s11075-010-9393-x.  Google Scholar

[20]

H. Sheng, Y. Q. Chen and T. S. Qiu, Fractional Processes and Fractional-order Signal Processing, Springer, London, 2012. doi: 10.1007/978-1-4471-2233-3.  Google Scholar

[21]

X. C. ShiL. L. Huang and Y. Zeng, Fast Adomian decomposition method for the Cauchy problem of the time–fractional reaction diffusion equation, Adv. Mech. Eng, 8 (2016), 1-5.   Google Scholar

[22]

B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators, 1$^{st}$ edition, Springer, New York, 2003. doi: 10.1007/978-0-387-21746-8.  Google Scholar

[23]

S. ZengD. BaleanuY. Bai and G. Wu, Fractional differential equations of Caputo–Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.  doi: 10.1016/j.amc.2017.07.003.  Google Scholar

show all references

References:
[1]

R. B AlbadarnehaI. M Batihab and M Zurigatb, Numerical solutions for linear fractional differential equations of order $1<\alpha <2$ using finite difference method (FFDM), Int. J. Math., 16 (2016), 103-111.   Google Scholar

[2]

R. Almeida, T. Odzijewicz and Agnieszka B. Malinowska, Fractional differential Equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dyn, 11 (2016), 11. Google Scholar

[3]

J. F Cheng and Y. M Chu, Solution to the linear fractional differential equation using Adomian decomposition method, Math. Probl. Eng., 2011 (2011), 14. doi: 10.1155/2011/587068.  Google Scholar

[4]

K. Diethelm, The Analysis of Fractional Differential Equations, Springer Science Berlin, New York, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[5]

N. FarazY. Khan and D. S. Sankar, Decomposition-transform method for fractional differential equations, Int. J. Nonl. Sci. Num. Sim., 11 (2010), 305-310.   Google Scholar

[6]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/3779.  Google Scholar

[7]

M. Janaki, K. Kanagarajan and D. Vivek, Analytic study on fractional implicit differential equations with impulses via Katugampola fractional derivative, Int. J. Math. And Appl., 55 (2018), 7. Google Scholar

[8]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[9]

U. N. Katugampola, A new approach to generalized fractional derivatives, J. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar

[10]

A. A. Kilbas, H. H. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. doi: 10.1016/s0304-0208(06)x8001-5.  Google Scholar

[11]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models Imperial, College Press, Singapore, 2010.  doi: 10.1142/9781848163300.  Google Scholar
[13]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.  Google Scholar

[14]

C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue and V. Feliu, Fractional-Order Systems and Controls, Advances in Industrial Control, Springer, 2010. doi: 10.1007/978-1-84996-335-0.  Google Scholar

[15]

I. Petras, Fractional-Order Nonlinear Systems, Springer, New York, 2011. doi: 10.1007/978-3-642-18101-6_3.  Google Scholar

[16] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.   Google Scholar
[17]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. doi: 10.1007/978-1-4020-6042-7.  Google Scholar

[18]

S. G. Samko, A. A. Kilbas and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.  Google Scholar

[19]

S. ShenF. Liu and V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. Algorithms, 56 (2011), 383-403.  doi: 10.1007/s11075-010-9393-x.  Google Scholar

[20]

H. Sheng, Y. Q. Chen and T. S. Qiu, Fractional Processes and Fractional-order Signal Processing, Springer, London, 2012. doi: 10.1007/978-1-4471-2233-3.  Google Scholar

[21]

X. C. ShiL. L. Huang and Y. Zeng, Fast Adomian decomposition method for the Cauchy problem of the time–fractional reaction diffusion equation, Adv. Mech. Eng, 8 (2016), 1-5.   Google Scholar

[22]

B.J. West, M. Bologna and P. Grigolini, Physics of Fractal Operators, 1$^{st}$ edition, Springer, New York, 2003. doi: 10.1007/978-0-387-21746-8.  Google Scholar

[23]

S. ZengD. BaleanuY. Bai and G. Wu, Fractional differential equations of Caputo–Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.  doi: 10.1016/j.amc.2017.07.003.  Google Scholar

Figure 1.  Graphical comparison of the numerical and the exact solution with $ h = 0.001 $, $ k = 0.1 $, $ \rho = 2 $, $ \alpha = 0.7, $ $ n = 20 $ and $ m = 25 $
Figure 2.  Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 2 $, $ \alpha = 0.6, $ $ \beta = 1.8, $ $ n = 30 $ and $ m = 25 $
Figure 3.  Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 2, $ $ \alpha = 0.9 $, $ (a)\ \beta = 1 $, $ (b)\ \beta = 2 $ and $ m = 25 $
Figure 4.  Graphical comparison of the numerical and the exact solution with $ h = 0.005 $, $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.7, $ $ n = 20 $ and $ m = 15 $
Figure 5.  Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.8, $ $ \beta = 1.8, $ $ n = 40 $ and $ m = 15 $
Figure 6.  Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.9, $ $ (a)\ \beta = 1 $, $ (b)\ \beta = 2 $ and $ m = 15 $
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