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  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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
[1]

Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439

[2]

Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543

[3]

Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453

[4]

Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951

[5]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[6]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793

[7]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

[8]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569

[9]

Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057

[10]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675

[11]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[12]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

[13]

Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495

[14]

Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317

[15]

Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033

[16]

Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial & Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050

[17]

Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40

[18]

Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 443-466. doi: 10.3934/dcdss.2020025

[19]

Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1393-1408. doi: 10.3934/dcds.2009.24.1393

[20]

Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849

2018 Impact Factor: 0.545

Article outline

Figures and Tables

[Back to Top]