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  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, 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.  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]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[2]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007

[3]

Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160

[4]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[5]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[6]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[7]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[8]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[9]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[10]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[11]

Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282

[12]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069

[13]

Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314

[14]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[15]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001

[16]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[17]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[18]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[19]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[20]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (234)
  • HTML views (334)
  • Cited by (4)

Other articles
by authors

[Back to Top]