# American Institute of Mathematical Sciences

July  2021, 14(7): 2025-2039. doi: 10.3934/dcdss.2020402

## Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme

 1 Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran, Iran 2 Department of Mathematics, University of Mazandaran, Babolsar, Iran, Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa 3 Department of Mathematics, Cankaya University, Ankara, Turkey, Institute of Space Sciences, Magurele-Bucharest, Romania

* Corresponding author: Hamid Safdari

Received  December 2019 Revised  January 2020 Published  July 2021 Early access  June 2020

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order $\mathcal{O}(\delta \tau ^{2})$ is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.

Citation: Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402
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##### References:
Plots of the approximate solution (left side) and absolute error (right side) of Example 5.1 at $T = 1$, $M = 400$ and $N = 5$
The maximum absolute error and error norm $L_{2}$ of Example 5.1 at $T = 1$, $N = 5$ and $M = 200,400,600, \ldots, 3000$
Error histories of Example 5.1 at $T = 1$, $N = 5$ and $M = 100,200,400,800, 1600$
Error histories of Example 5.1 at $T = 1$, $M = 400$ and $N = 3, 5, 7, 9$
Error histories of Example 5.2 at $T = 1$, with $M = 100,200,400,800, 1600,$ $N = 5$ (left side) and $N = 7$ (right side)
The absolute error of Example 5.1 at $T = 1$
 $x$ with $N=7$ in with $N=7$ in with $N=3$ in our method with $N=3$ [15] [27] [30] $0$ $2.81\times 10^{-5}$ $0$ $0$ $4.77\times 10^{-17}$ $0.1$ $4.26\times 10^{-5}$ $4.66\times 10^{-5}$ $5.46\times 10^{-6}$ $3.17\times 10^{-9}$ $0.2$ $5.39\times 10^{-5}$ $7.74\times 10^{-5}$ $8.51\times 10^{-6}$ $5.85\times 10^{-9}$ $0.3$ $6.12\times 10^{-5}$ $5.00\times 10^{-5}$ $9.60\times 10^{-6}$ $7.97\times 10^{-9}$ $0.4$ $6.48\times 10^{-5}$ $2.30\times 10^{-5}$ $9.18\times 10^{-6}$ $9.44\times 10^{-9}$ $0.5$ $6.45\times 10^{-5}$ $2.74\times 10^{-5}$ $7.69\times 10^{-6}$ $1.02\times 10^{-8}$ $0.6$ $5.98\times 10^{-5}$ $4.38\times 10^{-5}$ $5.60\times 10^{-6}$ $1.01\times 10^{-8}$ $0.7$ $5.23\times 10^{-5}$ $3.87\times 10^{-5}$ $3.33\times 10^{-6}$ $9.12\times 10^{-9}$ $0.8$ $4.48\times 10^{-5}$ $1.01\times 10^{-5}$ $1.34\times 10^{-6}$ $7.17\times 10^{-9}$ $0.9$ $3.91\times 10^{-5}$ $3.35\times 10^{-5}$ $8.39\times 10^{-8}$ $4.16\times 10^{-9}$ $1.0$ $2.81\times 10^{-5}$ $0$ $0$ $7.55\times 10^{-17}$
 $x$ with $N=7$ in with $N=7$ in with $N=3$ in our method with $N=3$ [15] [27] [30] $0$ $2.81\times 10^{-5}$ $0$ $0$ $4.77\times 10^{-17}$ $0.1$ $4.26\times 10^{-5}$ $4.66\times 10^{-5}$ $5.46\times 10^{-6}$ $3.17\times 10^{-9}$ $0.2$ $5.39\times 10^{-5}$ $7.74\times 10^{-5}$ $8.51\times 10^{-6}$ $5.85\times 10^{-9}$ $0.3$ $6.12\times 10^{-5}$ $5.00\times 10^{-5}$ $9.60\times 10^{-6}$ $7.97\times 10^{-9}$ $0.4$ $6.48\times 10^{-5}$ $2.30\times 10^{-5}$ $9.18\times 10^{-6}$ $9.44\times 10^{-9}$ $0.5$ $6.45\times 10^{-5}$ $2.74\times 10^{-5}$ $7.69\times 10^{-6}$ $1.02\times 10^{-8}$ $0.6$ $5.98\times 10^{-5}$ $4.38\times 10^{-5}$ $5.60\times 10^{-6}$ $1.01\times 10^{-8}$ $0.7$ $5.23\times 10^{-5}$ $3.87\times 10^{-5}$ $3.33\times 10^{-6}$ $9.12\times 10^{-9}$ $0.8$ $4.48\times 10^{-5}$ $1.01\times 10^{-5}$ $1.34\times 10^{-6}$ $7.17\times 10^{-9}$ $0.9$ $3.91\times 10^{-5}$ $3.35\times 10^{-5}$ $8.39\times 10^{-8}$ $4.16\times 10^{-9}$ $1.0$ $2.81\times 10^{-5}$ $0$ $0$ $7.55\times 10^{-17}$
The absolute error of Example 5.1 at $T = 2$
 $x$ with $N=5$ in with $N=5$ in with $N=3$ in our method with $N=3$  [15] [27] [30] $0$ $2.74\times 10^{-5}$ $0$ $0$ $1.86\times 10^{-17}$ $0.1$ $4.20\times 10^{-5}$ $4.47\times 10^{-6}$ $3.33\times 10^{-6}$ $1.28\times 10^{-8}$ $0.2$ $3.76\times 10^{-5}$ $2.78\times 10^{-7}$ $5.65\times 10^{-6}$ $2.05\times 10^{-8}$ $0.3$ $8.44\times 10^{-5}$ $5.81\times 10^{-6}$ $7.05\times 10^{-6}$ $2.40\times 10^{-8}$ $0.4$ $3.27\times 10^{-5}$ $1.02\times 10^{-5}$ $7.64\times 10^{-6}$ $2.40\times 10^{-8}$ $0.5$ $3.61\times 10^{-5}$ $1.17\times 10^{-5}$ $7.52\times 10^{-6}$ $2.15\times 10^{-8}$ $0.6$ $1.94\times 10^{-5}$ $1.08\times 10^{-5}$ $6.80\times 10^{-6}$ $1.72\times 10^{-8}$ $0.7$ $2.95\times 10^{-5}$ $8.54\times 10^{-6}$ $5.59\times 10^{-6}$ $1.21\times 10^{-8}$ $0.8$ $4.92\times 10^{-5}$ $6.06\times 10^{-6}$ $3.98\times 10^{-6}$ $6.93\times 10^{-9}$ $0.9$ $2.83\times 10^{-5}$ $3.67\times 10^{-6}$ $2.08\times 10^{-6}$ $2.62\times 10^{-9}$ $1.0$ $7.73\times 10^{-5}$ $0$ $0$ $8.24\times 10^{-18}$
 $x$ with $N=5$ in with $N=5$ in with $N=3$ in our method with $N=3$  [15] [27] [30] $0$ $2.74\times 10^{-5}$ $0$ $0$ $1.86\times 10^{-17}$ $0.1$ $4.20\times 10^{-5}$ $4.47\times 10^{-6}$ $3.33\times 10^{-6}$ $1.28\times 10^{-8}$ $0.2$ $3.76\times 10^{-5}$ $2.78\times 10^{-7}$ $5.65\times 10^{-6}$ $2.05\times 10^{-8}$ $0.3$ $8.44\times 10^{-5}$ $5.81\times 10^{-6}$ $7.05\times 10^{-6}$ $2.40\times 10^{-8}$ $0.4$ $3.27\times 10^{-5}$ $1.02\times 10^{-5}$ $7.64\times 10^{-6}$ $2.40\times 10^{-8}$ $0.5$ $3.61\times 10^{-5}$ $1.17\times 10^{-5}$ $7.52\times 10^{-6}$ $2.15\times 10^{-8}$ $0.6$ $1.94\times 10^{-5}$ $1.08\times 10^{-5}$ $6.80\times 10^{-6}$ $1.72\times 10^{-8}$ $0.7$ $2.95\times 10^{-5}$ $8.54\times 10^{-6}$ $5.59\times 10^{-6}$ $1.21\times 10^{-8}$ $0.8$ $4.92\times 10^{-5}$ $6.06\times 10^{-6}$ $3.98\times 10^{-6}$ $6.93\times 10^{-9}$ $0.9$ $2.83\times 10^{-5}$ $3.67\times 10^{-6}$ $2.08\times 10^{-6}$ $2.62\times 10^{-9}$ $1.0$ $7.73\times 10^{-5}$ $0$ $0$ $8.24\times 10^{-18}$
The absolute error of Example 5.1 at $T = 10$
 $x$ $N=3$ $N=5$ $N=7$ $0$ $5.82\times 10^{-21}$ $5.93\times 10^{-22}$ $4.43\times 10^{-21}$ $0.2$ $1.01\times 10^{-9}$ $4.74\times 10^{-9}$ $2.28\times 10^{-9}$ $0.4$ $8.21\times 10^{-9}$ $8.11\times 10^{-9}$ $4.21\times 10^{-9}$ $0.6$ $1.28\times 10^{-9}$ $1.17\times 10^{-9}$ $1.15\times 10^{-9}$ $0.8$ $3.76\times 10^{-9}$ $7.93\times 10^{-10}$ $2.71\times 10^{-10}$ $1.0$ $4.34\times 10^{-21}$ $3.78\times 10^{-21}$ $1.14\times 10^{-22}$
 $x$ $N=3$ $N=5$ $N=7$ $0$ $5.82\times 10^{-21}$ $5.93\times 10^{-22}$ $4.43\times 10^{-21}$ $0.2$ $1.01\times 10^{-9}$ $4.74\times 10^{-9}$ $2.28\times 10^{-9}$ $0.4$ $8.21\times 10^{-9}$ $8.11\times 10^{-9}$ $4.21\times 10^{-9}$ $0.6$ $1.28\times 10^{-9}$ $1.17\times 10^{-9}$ $1.15\times 10^{-9}$ $0.8$ $3.76\times 10^{-9}$ $7.93\times 10^{-10}$ $2.71\times 10^{-10}$ $1.0$ $4.34\times 10^{-21}$ $3.78\times 10^{-21}$ $1.14\times 10^{-22}$
The convergence order, the errors $L_{2}$ and $L_{\infty}$ for Example 5.1 with $T = 1$ and $N = 3$
 $\delta \tau$ $L_{\infty}$ $C_{\delta \tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.62773\times 10^{-7}$ $3.76647\times 10^{-7}$ $\frac{1}{200}$ $4.06928\times 10^{-8}$ $2.00002$ $9.41607\times 10^{-8}$ $2.00002$ $\frac{1}{400}$ $1.01732\times 10^{-8}$ $2.00000$ $2.35401\times 10^{-8}$ $2.00000$ $\frac{1}{800}$ $2.54329\times 10^{-9}$ $2.00000$ $5.88503\times 10^{-9}$ $2.00000$ $\frac{1}{1600}$ $6.35828\times 10^{-10}$ $1.99999$ $1.47127\times 10^{-9}$ $1.99999$ $\mathrm{TCO}$ $2$ $2$
 $\delta \tau$ $L_{\infty}$ $C_{\delta \tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.62773\times 10^{-7}$ $3.76647\times 10^{-7}$ $\frac{1}{200}$ $4.06928\times 10^{-8}$ $2.00002$ $9.41607\times 10^{-8}$ $2.00002$ $\frac{1}{400}$ $1.01732\times 10^{-8}$ $2.00000$ $2.35401\times 10^{-8}$ $2.00000$ $\frac{1}{800}$ $2.54329\times 10^{-9}$ $2.00000$ $5.88503\times 10^{-9}$ $2.00000$ $\frac{1}{1600}$ $6.35828\times 10^{-10}$ $1.99999$ $1.47127\times 10^{-9}$ $1.99999$ $\mathrm{TCO}$ $2$ $2$
The convergence order, the errors $L_{2}$ and $L_{\infty}$ for Example 5.1 with $T = 10$ and $N = 3$
 $\delta \tau$ $L_{\infty}$ $C_{\delta\tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.63402\times 10^{-7}$ $3.10926\times 10^{-7}$ $\frac{1}{200}$ $4.08673\times 10^{-8}$ $1.99941$ $7.77632\times 10^{-8}$ $1.99941$ $\frac{1}{400}$ $1.02179\times 10^{-8}$ $1.99985$ $1.94428\times 10^{-8}$ $1.99985$ $\frac{1}{800}$ $2.55453\times 10^{-9}$ $1.99996$ $4.86082\times 10^{-9}$ $1.99996$ $\frac{1}{1600}$ $6.38636\times 10^{-10}$ $1.99999$ $1.21521\times 10^{-9}$ $1.99999$ $\mathrm{TCO}$ $2$ $2$
 $\delta \tau$ $L_{\infty}$ $C_{\delta\tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.63402\times 10^{-7}$ $3.10926\times 10^{-7}$ $\frac{1}{200}$ $4.08673\times 10^{-8}$ $1.99941$ $7.77632\times 10^{-8}$ $1.99941$ $\frac{1}{400}$ $1.02179\times 10^{-8}$ $1.99985$ $1.94428\times 10^{-8}$ $1.99985$ $\frac{1}{800}$ $2.55453\times 10^{-9}$ $1.99996$ $4.86082\times 10^{-9}$ $1.99996$ $\frac{1}{1600}$ $6.38636\times 10^{-10}$ $1.99999$ $1.21521\times 10^{-9}$ $1.99999$ $\mathrm{TCO}$ $2$ $2$
The convergence order, the errors $L_{2}$ and $L_{\infty}$ for Example 5.2 with $N = 7$ at $T = 1$
 $\delta\tau$ $L_{\infty}$ $C_{\delta\tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.71816\times 10^{-6}$ $3.73349\times 10^{-6}$ $\frac{1}{200}$ $4.29538\times 10^{-7}$ $2.00000$ $9.33372\times 10^{-7}$ $2.00000$ $\frac{1}{400}$ $1.07384\times 10^{-7}$ $2.00000$ $2.33343\times 10^{-7}$ $2.00000$ $\frac{1}{800}$ $2.68460\times 10^{-8}$ $2.00000$ $5.83360\times 10^{-8}$ $2.00000$ $\frac{1}{1600}$ $6.71143\times 10^{-9}$ $2.00002$ $1.45842\times 10^{-8}$ $1.99998$ $\mathrm{TCO}$ $2$ $2$
 $\delta\tau$ $L_{\infty}$ $C_{\delta\tau}$ $L_{2}$ $C_{\delta \tau}$ $\frac{1}{100}$ $1.71816\times 10^{-6}$ $3.73349\times 10^{-6}$ $\frac{1}{200}$ $4.29538\times 10^{-7}$ $2.00000$ $9.33372\times 10^{-7}$ $2.00000$ $\frac{1}{400}$ $1.07384\times 10^{-7}$ $2.00000$ $2.33343\times 10^{-7}$ $2.00000$ $\frac{1}{800}$ $2.68460\times 10^{-8}$ $2.00000$ $5.83360\times 10^{-8}$ $2.00000$ $\frac{1}{1600}$ $6.71143\times 10^{-9}$ $2.00002$ $1.45842\times 10^{-8}$ $1.99998$ $\mathrm{TCO}$ $2$ $2$
The comparison of maximum error of our proposed method and [32] for Example 5.2, at $T = 1$
 Max error-CN [32] Max error-ext CN [32] the present method with N=3 $6.84895\times 10^{-4}$ $2.82750 \times 10^{-5}$ $9.95930\times 10^{-8}$
 Max error-CN [32] Max error-ext CN [32] the present method with N=3 $6.84895\times 10^{-4}$ $2.82750 \times 10^{-5}$ $9.95930\times 10^{-8}$
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