# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020223
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##### References:
Schematics of central local supported domain in 2D geometry for $n_i = 5$
Schematic of upwind local supported domain in 1D (row 1) and 2D (row 2) geometry for $n_i = 3$ and $n_i = 5$ respectively
Numerical solutions of the GMM for Test Problem 2
Numerical solutions of the LMM with out upwind technique (left), with upwind technique (right) for Test Problem 2
Numerical solutions of the LMM with upwind technique for Test Problem 2
Computational domain and absolute error for Test Problem 3
Computational domain and absolute error for Test Problem 3
Computational domain and absolute error for Test Problem 3
Computational domain and numerical solution for Test Problem 3
Computational domain, approximate and exact solution for Test Problem 3
Numerical solution of the GMM for $Re = 200$ (left) and $Re = 300$ (right) for Test Problem 4
Numerical solution of the LMM for $Re = 100$ (left) and $Re = 150$ (right) for Test Problem 4
Results of the LMM using upwind technique for $Re = 150$ (left) and $Re = 1000$ (right) for Test Problem 4
Results of the LMM using upwind technique for $Re = 10^{10}$ (left) and $Re = 10^{17}$ (right) for Test Problem 4
CPU time comparison of the LMM and the GMM for Test Problem 4
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
$Ave.L_{abs}$ error norms of the LMM for Test Problem 3
 N 5 10 15 20 $\alpha=1.5$ 0.0027911 0.00033331 6.3412e-05 1.1145e-05 $\alpha=1.8$ 0.0028124 0.0003587 8.4685e-05 2.7917e-05
 N 5 10 15 20 $\alpha=1.5$ 0.0027911 0.00033331 6.3412e-05 1.1145e-05 $\alpha=1.8$ 0.0028124 0.0003587 8.4685e-05 2.7917e-05
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
$Ave.L_{abs}$ 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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