October  2020, 13(10): 2641-2654. doi: 10.3934/dcdss.2020223

Local meshless differential quadrature collocation method for time-fractional PDEs

1. 

Department of Mathematics, University of Swabi, Khyber Pakhtunkhwa 23430, Pakistan

2. 

Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

3. 

Shaheed Benazir Bhutto Women University, Peshawar, Pakistan

* Corresponding authors: siraj.islam@gmail.com (Siraj-ul-Islam)

* Corresponding authors: imtiazkakakhil@gmail.com (Imtiaz Ahmad)

Received  February 2019 Revised  July 2019 Published  October 2020 Early access  December 2019

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of $ \alpha \in (0,1) $ and $ \alpha\in(1,2) $. To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

Citation: Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223
References:
[1]

I. AhmadM. RiazM. AyazM. ArifS. Islam and P. Kumam, Numerical simulation of partial differential equations via local meshless method, Symmetry, 11 (2019), 257 pp.  doi: 10.3390/sym11020257.

[2]

I. AhmadM. AhsanZaheer-ud-DinM. Ahmad and P. Kumam, An efficient local formulation for time-dependent PDEs, Mathematics, 7 (2019), 216 pp.  doi: 10.3390/math7030216.

[3]

I. AhmadSiraj-ul-Islam and A. Q. M. Khaliq, Local RBF method for multi-dimensional partial differential equations, Comput. Math. Appl., 74 (2017), 292-324.  doi: 10.1016/j.camwa.2017.04.026.

[4]

I. AhmadM. AhsanI. HussainP. Kumam and W. Kumam, Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11 (2019), 394 pp.  doi: 10.3390/sym11030394.

[5]

W. CaoQ. XuQinwu and Z. Zheng, Solution of two-dimensional time-fractional Burgers' equation with high and low Reynolds numbers, Advances in Difference Equations, 338 (2017), 14 pp.  doi: 10.1186/s13662-017-1398-5.

[6]

S. ChenF. LiuP. Zhuang and V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), 256-273.  doi: 10.1016/j.apm.2007.11.005.

[7]

K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[8]

T. S. El-Danaf and A. R. Hadhoud, Parametric spline functions for the solution of the one time fractional Burgers' equation, Appl. Math. Model., 36 (2012), 4557-4564.  doi: 10.1016/j.apm.2011.11.035.

[9]

Y. T. Gu and G. R. Liu, Meshless techniques for convection dominated problems, Comput. Mech., 38 (2006), 171-182.  doi: 10.1007/s00466-005-0736-8.

[10]

V. R. HosseiniE. Shivanian and W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312 (2016), 307-332.  doi: 10.1016/j.jcp.2016.02.030.

[11]

M. Inc, The approximate and exact solutions of the space-and time-fractional Burgers' equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476-484.  doi: 10.1016/j.jmaa.2008.04.007.

[12]

H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1962-1969.  doi: 10.1016/j.cnsns.2008.06.019.

[13]

D. LiC. Zhang and M. Ran, A linear finite difference scheme for generalized time fractional Burgers' equation, Appl. Math. Model., 40 (2016), 6069-6081.  doi: 10.1016/j.apm.2016.01.043.

[14]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[15]

G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Berlin, Springer-Verlag 2005.

[16]

A. MohebbiM. Abbaszadeh and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37 (2013), 475-485.  doi: 10.1016/j.enganabound.2012.12.002.

[17]

M. D. Ortigueira, The fractional quantum derivative and its integral representations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 956-962.  doi: 10.1016/j.cnsns.2009.05.026.

[18]

A. PrakashM. Kumar and K. K. Sharma, Numerical method for solving fractional coupled Burgers equations, Appl. Math. Comput., 260 (2015), 314-320.  doi: 10.1016/j.amc.2015.03.037.

[19]

Y. Sanyasiraju and C. Satyanarayana, Upwind strategies for local RBF scheme to solve convection dominated problems, Eng. Anal. Bound. Elem., 48 (2014), 1-13.  doi: 10.1016/j.enganabound.2014.06.008.

[20]

A. Saravanan and N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, J. Egyptian Math. Soc., 21 (2013), 259-265.  doi: 10.1016/j.joems.2013.03.004.

[21]

E. ScalasR. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Physica A, 284 (2000), 376-384.  doi: 10.1016/S0378-4371(00)00255-7.

[22]

Q. Shen, Local RBF-based differential quadrature collocation method for the boundary layer problems, Eng. Anal. Bound. Elem., 34 (2010), 213-228.  doi: 10.1016/j.enganabound.2009.10.004.

[23]

C. Shu, Differential Quadrature and Its Application in Engineering, Springer-Verlag London, Ltd., London, 2000. doi: 10.1007/978-1-4471-0407-0.

[24]

B. K. Singh and P. Kumar, Numerical computation for time-fractional gas dynamics equations by fractional reduced differential transforms method, Journal of Mathematics and System Science, 6 (2016), 248-259. 

[25]

Siraj-ul-Islam and I. Ahmad, A comparative analysis of local meshless formulation for multi-asset option models, Eng. Anal. Bound. Elem., 65 (2016), 159-176.  doi: 10.1016/j.enganabound.2015.12.020.

[26]

Siraj-ul-Islam and I. Ahmad, Local meshless method for PDEs arising from models of wound healing, Appl. Math. Model., 48 (2017), 688-710.  doi: 10.1016/j.apm.2017.04.015.

[27]

P. ThounthongM. N. KhanI. HussainI. Ahmad and P. Kumam, Symmetric radial basis function method for simulation of elliptic partial differential equations, Mathematics, 6 (2018), 327 pp.  doi: 10.3390/math6120327.

[28]

J. Y. YangY. M. ZhaoN. LiuW. P. BuT. L. Xu and Y. F. Tang, An implicit MLS meshless method for 2-D time dependent fractional diffusion–wave equation, Appl. Math. Model., 39 (2015), 1229-1240.  doi: 10.1016/j.apm.2014.08.005.

[29]

G. YaoSiraj-ul-Islam and B. Sarler, Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions, Eng. Anal. Bound. Elem., 36 (2012), 1640-1648.  doi: 10.1016/j.enganabound.2012.04.012.

[30]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput., 215 (2009), 524-529.  doi: 10.1016/j.amc.2009.05.018.

show all references

References:
[1]

I. AhmadM. RiazM. AyazM. ArifS. Islam and P. Kumam, Numerical simulation of partial differential equations via local meshless method, Symmetry, 11 (2019), 257 pp.  doi: 10.3390/sym11020257.

[2]

I. AhmadM. AhsanZaheer-ud-DinM. Ahmad and P. Kumam, An efficient local formulation for time-dependent PDEs, Mathematics, 7 (2019), 216 pp.  doi: 10.3390/math7030216.

[3]

I. AhmadSiraj-ul-Islam and A. Q. M. Khaliq, Local RBF method for multi-dimensional partial differential equations, Comput. Math. Appl., 74 (2017), 292-324.  doi: 10.1016/j.camwa.2017.04.026.

[4]

I. AhmadM. AhsanI. HussainP. Kumam and W. Kumam, Numerical simulation of PDEs by local meshless differential quadrature collocation method, Symmetry, 11 (2019), 394 pp.  doi: 10.3390/sym11030394.

[5]

W. CaoQ. XuQinwu and Z. Zheng, Solution of two-dimensional time-fractional Burgers' equation with high and low Reynolds numbers, Advances in Difference Equations, 338 (2017), 14 pp.  doi: 10.1186/s13662-017-1398-5.

[6]

S. ChenF. LiuP. Zhuang and V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), 256-273.  doi: 10.1016/j.apm.2007.11.005.

[7]

K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[8]

T. S. El-Danaf and A. R. Hadhoud, Parametric spline functions for the solution of the one time fractional Burgers' equation, Appl. Math. Model., 36 (2012), 4557-4564.  doi: 10.1016/j.apm.2011.11.035.

[9]

Y. T. Gu and G. R. Liu, Meshless techniques for convection dominated problems, Comput. Mech., 38 (2006), 171-182.  doi: 10.1007/s00466-005-0736-8.

[10]

V. R. HosseiniE. Shivanian and W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312 (2016), 307-332.  doi: 10.1016/j.jcp.2016.02.030.

[11]

M. Inc, The approximate and exact solutions of the space-and time-fractional Burgers' equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476-484.  doi: 10.1016/j.jmaa.2008.04.007.

[12]

H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1962-1969.  doi: 10.1016/j.cnsns.2008.06.019.

[13]

D. LiC. Zhang and M. Ran, A linear finite difference scheme for generalized time fractional Burgers' equation, Appl. Math. Model., 40 (2016), 6069-6081.  doi: 10.1016/j.apm.2016.01.043.

[14]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[15]

G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Berlin, Springer-Verlag 2005.

[16]

A. MohebbiM. Abbaszadeh and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37 (2013), 475-485.  doi: 10.1016/j.enganabound.2012.12.002.

[17]

M. D. Ortigueira, The fractional quantum derivative and its integral representations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 956-962.  doi: 10.1016/j.cnsns.2009.05.026.

[18]

A. PrakashM. Kumar and K. K. Sharma, Numerical method for solving fractional coupled Burgers equations, Appl. Math. Comput., 260 (2015), 314-320.  doi: 10.1016/j.amc.2015.03.037.

[19]

Y. Sanyasiraju and C. Satyanarayana, Upwind strategies for local RBF scheme to solve convection dominated problems, Eng. Anal. Bound. Elem., 48 (2014), 1-13.  doi: 10.1016/j.enganabound.2014.06.008.

[20]

A. Saravanan and N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, J. Egyptian Math. Soc., 21 (2013), 259-265.  doi: 10.1016/j.joems.2013.03.004.

[21]

E. ScalasR. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Physica A, 284 (2000), 376-384.  doi: 10.1016/S0378-4371(00)00255-7.

[22]

Q. Shen, Local RBF-based differential quadrature collocation method for the boundary layer problems, Eng. Anal. Bound. Elem., 34 (2010), 213-228.  doi: 10.1016/j.enganabound.2009.10.004.

[23]

C. Shu, Differential Quadrature and Its Application in Engineering, Springer-Verlag London, Ltd., London, 2000. doi: 10.1007/978-1-4471-0407-0.

[24]

B. K. Singh and P. Kumar, Numerical computation for time-fractional gas dynamics equations by fractional reduced differential transforms method, Journal of Mathematics and System Science, 6 (2016), 248-259. 

[25]

Siraj-ul-Islam and I. Ahmad, A comparative analysis of local meshless formulation for multi-asset option models, Eng. Anal. Bound. Elem., 65 (2016), 159-176.  doi: 10.1016/j.enganabound.2015.12.020.

[26]

Siraj-ul-Islam and I. Ahmad, Local meshless method for PDEs arising from models of wound healing, Appl. Math. Model., 48 (2017), 688-710.  doi: 10.1016/j.apm.2017.04.015.

[27]

P. ThounthongM. N. KhanI. HussainI. Ahmad and P. Kumam, Symmetric radial basis function method for simulation of elliptic partial differential equations, Mathematics, 6 (2018), 327 pp.  doi: 10.3390/math6120327.

[28]

J. Y. YangY. M. ZhaoN. LiuW. P. BuT. L. Xu and Y. F. Tang, An implicit MLS meshless method for 2-D time dependent fractional diffusion–wave equation, Appl. Math. Model., 39 (2015), 1229-1240.  doi: 10.1016/j.apm.2014.08.005.

[29]

G. YaoSiraj-ul-Islam and B. Sarler, Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions, Eng. Anal. Bound. Elem., 36 (2012), 1640-1648.  doi: 10.1016/j.enganabound.2012.04.012.

[30]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput., 215 (2009), 524-529.  doi: 10.1016/j.amc.2009.05.018.

Figure 1.  Schematics of central local supported domain in 2D geometry for $ n_i = 5 $
Figure 2.  Schematic of upwind local supported domain in 1D (row 1) and 2D (row 2) geometry for $ n_i = 3 $ and $ n_i = 5 $ respectively
Figure 3.  Numerical solutions of the GMM for Test Problem 2
Figure 4.  Numerical solutions of the LMM with out upwind technique (left), with upwind technique (right) for Test Problem 2
Figure 5.  Numerical solutions of the LMM with upwind technique for Test Problem 2
Figure 6.  Computational domain and absolute error for Test Problem 3
Figure 7.  Computational domain and absolute error for Test Problem 3
Figure 8.  Computational domain and absolute error for Test Problem 3
Figure 9.  Computational domain and numerical solution for Test Problem 3
Figure 10.  Computational domain, approximate and exact solution for Test Problem 3
Figure 11.  Numerical solution of the GMM for $ Re = 200 $ (left) and $ Re = 300 $ (right) for Test Problem 4
Figure 12.  Numerical solution of the LMM for $ Re = 100 $ (left) and $ Re = 150 $ (right) for Test Problem 4
Figure 13.  Results of the LMM using upwind technique for $ Re = 150 $ (left) and $ Re = 1000 $ (right) for Test Problem 4
Figure 14.  Results of the LMM using upwind technique for $ Re = 10^{10} $ (left) and $ Re = 10^{17} $ (right) for Test Problem 4
Figure 15.  CPU time comparison of the LMM and the GMM for Test Problem 4
Table 1.  Comparison of the LMM with different local sub-domain $ n_i $ and the method reported in [8] for Test Problem 1
Time t=1 t=2 t=2.5 t=3
Max. abs. error[8] 4.632e-03 5.267e-03 5.569e-03 5.857e-03
LMM ($ L_{\infty} $)
$ n_i=3 $ 3.6104e-04 6.3774e-04 7.8362e-04 9.2358e-04
$ n_i=5 $ 1.6807e-05 2.5918e-05 2.8962e-05 3.3891e-05
$ n_i=7 $ 7.4546e-06 1.1882e-05 1.3669e-05 1.5306e-05
$ n_i=9 $ 6.5724e-06 1.0753e-05 1.2321e-05 1.3640e-05
$ n_i=11 $ 6.4933e-06 1.0749e-05 1.2303e-05 1.3355e-05
Time t=1 t=2 t=2.5 t=3
Max. abs. error[8] 4.632e-03 5.267e-03 5.569e-03 5.857e-03
LMM ($ L_{\infty} $)
$ n_i=3 $ 3.6104e-04 6.3774e-04 7.8362e-04 9.2358e-04
$ n_i=5 $ 1.6807e-05 2.5918e-05 2.8962e-05 3.3891e-05
$ n_i=7 $ 7.4546e-06 1.1882e-05 1.3669e-05 1.5306e-05
$ n_i=9 $ 6.5724e-06 1.0753e-05 1.2321e-05 1.3640e-05
$ n_i=11 $ 6.4933e-06 1.0749e-05 1.2303e-05 1.3355e-05
Table 2.  $ Ave.L_{abs} $ error norms of the LMM for Test Problem 3
N 5 10 15 20
$ \alpha=1.5 $ 2.7911e-03 3.3331e-04 6.3412e-05 1.1145e-05
$ \alpha=1.8 $ 2.8124e-03 3.5870e-04 8.4685e-05 2.7917e-05
N 5 10 15 20
$ \alpha=1.5 $ 2.7911e-03 3.3331e-04 6.3412e-05 1.1145e-05
$ \alpha=1.8 $ 2.8124e-03 3.5870e-04 8.4685e-05 2.7917e-05
Table 3.  Numerical results of the LMM and the method reported in [28] for Test Problem 3
$ \alpha=1.5 $ $ \alpha=1.8 $
$ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28] $ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28]
1/10 4.3826e-02 1.2550e-02 1/10 2.9099e-02 2.0496e-02
1/20 1.2592e-02 6.6277e-03 1/20 9.2610e-03 1.0696e-02
1/30 6.3054e-03 4.5292e-03 1/30 4.4599e-03 7.2811e-03
1/40 3.5999e-03 3.4518e-03 1/40 2.3587e-03 5.5407e-03
1/50 2.1402e-03 2.7951e-03 1/50 1.1993e-03 4.4822e-03
1/60 1.2811e-03 2.3526e-03 1/60 5.8970e-04 3.7688e-03
1/70 8.3729e-04 2.0338e-03 1/70 4.7926e-04 3.2548e-03
$ \alpha=1.5 $ $ \alpha=1.8 $
$ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28] $ \tau $ $ Ave.L_{abs} $ $ Ave.L_{abs} $ [28]
1/10 4.3826e-02 1.2550e-02 1/10 2.9099e-02 2.0496e-02
1/20 1.2592e-02 6.6277e-03 1/20 9.2610e-03 1.0696e-02
1/30 6.3054e-03 4.5292e-03 1/30 4.4599e-03 7.2811e-03
1/40 3.5999e-03 3.4518e-03 1/40 2.3587e-03 5.5407e-03
1/50 2.1402e-03 2.7951e-03 1/50 1.1993e-03 4.4822e-03
1/60 1.2811e-03 2.3526e-03 1/60 5.8970e-04 3.7688e-03
1/70 8.3729e-04 2.0338e-03 1/70 4.7926e-04 3.2548e-03
Table 4.  $ Ave.L_{abs} $ of the LMM for Test Problem 3
$ \alpha $ Regular nodes Chebyshev nodes
Explicit CN Implicit Explicit CN Implicit
1.5 8.0527e-05 2.2050e-04 4.1275e-04 2.4177e-04 3.4777e-04 4.5997e-04
1.6 8.1357e-05 2.2263e-04 4.1881e-04 2.3932e-04 3.4587e-04 4.5902e-04
1.7 8.2287e-05 2.2266e-04 4.2312e-04 2.3691e-04 3.4459e-04 4.5928e-04
1.8 8.3535e-05 2.2109e-04 4.2679e-04 2.3405e-04 3.4367e-04 4.6080e-04
$ \alpha $ Regular nodes Chebyshev nodes
Explicit CN Implicit Explicit CN Implicit
1.5 8.0527e-05 2.2050e-04 4.1275e-04 2.4177e-04 3.4777e-04 4.5997e-04
1.6 8.1357e-05 2.2263e-04 4.1881e-04 2.3932e-04 3.4587e-04 4.5902e-04
1.7 8.2287e-05 2.2266e-04 4.2312e-04 2.3691e-04 3.4459e-04 4.5928e-04
1.8 8.3535e-05 2.2109e-04 4.2679e-04 2.3405e-04 3.4367e-04 4.6080e-04
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