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September  2022, 12(3): 621-636. doi: 10.3934/naco.2021026

## 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 Published  September 2022 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 and Optimization, 2022, 12 (3) : 621-636. doi: 10.3934/naco.2021026
##### References:
 [1] R. B Albadarneha, I. 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. [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. [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. [4] K. Diethelm, The Analysis of Fractional Differential Equations, Springer Science Berlin, New York, 2010. doi: 10.1007/978-3-642-14574-2. [5] N. Faraz, Y. Khan and D. S. Sankar, Decomposition-transform method for fractional differential equations, Int. J. Nonl. Sci. Num. Sim., 11 (2010), 305-310. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/3779. [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. [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. [9] U. N. Katugampola, A new approach to generalized fractional derivatives, J. Math. Anal. Appl., 6 (2014), 1-15. [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. [11] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models Imperial, College Press, Singapore, 2010.  doi: 10.1142/9781848163300. [13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [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. [15] I. Petras, Fractional-Order Nonlinear Systems, Springer, New York, 2011. doi: 10.1007/978-3-642-18101-6_3. [16] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [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. [18] S. G. Samko, A. A. Kilbas and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. [19] S. Shen, F. 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. [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. [21] X. C. Shi, L. 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. [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. [23] S. Zeng, D. Baleanu, Y. 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.

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##### References:
 [1] R. B Albadarneha, I. 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. [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. [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. [4] K. Diethelm, The Analysis of Fractional Differential Equations, Springer Science Berlin, New York, 2010. doi: 10.1007/978-3-642-14574-2. [5] N. Faraz, Y. Khan and D. S. Sankar, Decomposition-transform method for fractional differential equations, Int. J. Nonl. Sci. Num. Sim., 11 (2010), 305-310. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/3779. [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. [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. [9] U. N. Katugampola, A new approach to generalized fractional derivatives, J. Math. Anal. Appl., 6 (2014), 1-15. [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. [11] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. [12] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models Imperial, College Press, Singapore, 2010.  doi: 10.1142/9781848163300. [13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [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. [15] I. Petras, Fractional-Order Nonlinear Systems, Springer, New York, 2011. doi: 10.1007/978-3-642-18101-6_3. [16] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [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. [18] S. G. Samko, A. A. Kilbas and O.I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. [19] S. Shen, F. 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. [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. [21] X. C. Shi, L. 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. [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. [23] S. Zeng, D. Baleanu, Y. 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.
Graphical comparison of the numerical and the exact solution with $h = 0.001$, $k = 0.1$, $\rho = 2$, $\alpha = 0.7,$ $n = 20$ and $m = 25$
Graphical comparison of the numerical and the exact solution with $k = 0.1$, $\rho = 2$, $\alpha = 0.6,$ $\beta = 1.8,$ $n = 30$ and $m = 25$
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 $m = 25$
Graphical comparison of the numerical and the exact solution with $h = 0.005$, $k = 0.1$, $\rho = 3$, $\alpha = 0.7,$ $n = 20$ and $m = 15$
Graphical comparison of the numerical and the exact solution with $k = 0.1$, $\rho = 3$, $\alpha = 0.8,$ $\beta = 1.8,$ $n = 40$ and $m = 15$
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 $m = 15$