Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature

Jing Li; Linlin Dai; Kamran; Waqas Nazeer; Jing Li; Linlin Dai; Kamran; Waqas Nazeer

doi:10.3934/math.2020373

AIMS Mathematics

2020, Volume 5, Issue 6: 5813-5838. doi: 10.3934/math.2020373

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Research article

Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature

1.
School of Data Science, Qingdao Huanghai University, Qingdao 266427, China
2.
School of Data Science, Qingdao Huanghai University, Qingdao 266427, China
3.
Department of Mathematics, Islamia College Peshawar, Khyber Pakhtoon Khwa, Pakistan
4.
Department of mathematics, GC University, Lahore Pakistan

Received: 18 April 2020 Accepted: 20 June 2020 Published: 14 July 2020
MSC : 26A33, 65M12, 65R10, 81Q05

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.
- Laplace transform,
- multi-term time fractional wave-diffusion equations,
- local meshless method,
- stability,
- convergence
Citation: Jing Li, Linlin Dai, Kamran, Waqas Nazeer. Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature[J]. AIMS Mathematics, 2020, 5(6): 5813-5838. doi: 10.3934/math.2020373

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Abstract

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.

References

[1]	K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.
[2]	F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Soliton. Fract., 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5
[3]	A. Carpinteri, F. Mainardi, Fractals and Fractional Calculas Continuum Mechanics, SpringerVerlag Wien, 1997.
[4]	F. Liu, M. M. Meerschaert, R. J. McGough, et al. Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal., 16 (2013), 9-25.
[5]	H. Ding, C. Li, Numerical algorithms for the fractional diffusion-wave equation with reaction term, Abstr. Appl. Anal., 2013 (2013), 1-15.
[6]	K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194
[7]	H. Ye, F. Liu, V. Anh, et al. Maximum principle and numerical method for the multi-term timespace riesz-caputo fractional differential equations, Appl. Math. Model., 227 (2014), 531-540.
[8]	C. M. Chen, F. Liu, I. Turner, et al. A fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012
[9]	M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792-7804. doi: 10.1016/j.jcp.2009.07.021
[10]	F. Liu, C. Yang, K. Burrage, Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231 (2009), 160-176. doi: 10.1016/j.cam.2009.02.013
[11]	H. Ding, C. Li, High-order numerical methods for riesz space fractional turbulent diffusion equation, Fract. Calc. Appl. Anal., 19 (2016), 19-55.
[12]	A. Chen, C. Li, Numerical solution of fractional diffusion-wave equation, Numer. Func. Anal. Opt., 37 (2016), 19-39. doi: 10.1080/01630563.2015.1078815
[13]	M. Garg, P. Manohar, Numerical solution of fractional diffusion-wave equation with two space variables by matrix method, Fract. Calc. Appl. Anal., 13 (2010), 191-207.
[14]	M. Dehghan, M. Safarpoor, M. Abbaszadeh, Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations, J. Comput. Appl. Math., 290 (2015), 174-195. doi: 10.1016/j.cam.2015.04.037
[15]	J. Huang, Y. Tang, L. Vázquez, et al. Two finite difference schemes for time fractional diffusionwave equation, Numer. Algorithms, 64 (2013), 707-720. doi: 10.1007/s11075-012-9689-0
[16]	A. H. Bhrawy, E. H. Doha, D. Baleanu, et al. A spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293 (2015), 142-156. doi: 10.1016/j.jcp.2014.03.039
[17]	M. H. Heydari, M. R. Hooshmandasl, F. M. M. Ghaini, et al, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A, 379 (2015), 71-76. doi: 10.1016/j.physleta.2014.11.012
[18]	Y. N. Zhang, Z. Z. Sun, X. Zhao, Compact alternating direction implicit scheme for the twodimensional fractional diffusion-wave equation, SIAM J. Numer. Anal., 50 (2012),1535-1555. doi: 10.1137/110840959
[19]	J. Ren, Z. Z. Sun, Efficient numerical solution of the multi-term time fractional diffusion-wave equation, E. Asian J. Appl. Math., 5 (2015),1-28. doi: 10.4208/eajam.080714.031114a
[20]	J. Y. Yang, J. F. Huang, D. M.Liang, et al. Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model., 38 (2014), 3652-3661. doi: 10.1016/j.apm.2013.11.069
[21]	Y. Yang, Y. Chen, Y. Huang, et al. Spectral collocation method for the time-fractional diffusionwave equation and convergence analysis, Comput. Math. Appl., 73 (2017), 1218-1232. doi: 10.1016/j.camwa.2016.08.017
[22]	T. Belytschko, Y. Y. Lu, L. Gu, Element free galerkin methods, Int. J. Numer. Meth. Eng., 37 (1994), 229-256. doi: 10.1002/nme.1620370205
[23]	W. K. Liu, Y. Chen, S. Jun, et al. Overview and applications of the reproducing kernel particle methods, Arch. Comput. Method. E., 3 (1996), 3-80. doi: 10.1007/BF02736130
[24]	W. Chen, Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method, Chin. J. Solid Mech., 30 (2009), 592-599.
[25]	Z. Fu, W. Chen, C. Zhang, Boundary particle method for cauchy inhomogeneous potential problems, Inverse probl. Sci. En., 20 (2012), 189-207. doi: 10.1080/17415977.2011.603085
[26]	V. R. Hosseini, E. Shivanian, W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312 (2016), 307- 332.
[27]	M. Dehghan, M. Abbaszadeh, A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. algorithms, 73 (2016), 445-476. doi: 10.1007/s11075-016-0103-1
[28]	Y. Gu, P. Zhuang, F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES-Comp. Model. Eng., 56 (2010), 303-334.
[29]	Q. Liu, Y. T. Gu, P. Zhuang, et al. An implicit rbf meshless approach for time fractional diffusion equations, Comput. Mech., 48 (2011), 1-12. doi: 10.1007/s00466-011-0573-x
[30]	M. Abbaszadeh, M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate, Numer. Algorithms, 75 (2017), 173-211. doi: 10.1007/s11075-016-0201-0
[31]	R. Salehi, A meshless point collocation method for 2-d multi-term time fractional diffusion-wave equation, Numer. Algorithms, 74 (2017), 1145-1168. doi: 10.1007/s11075-016-0190-z
[32]	M. Uddin, Kamran, A. Ali, A localized transform-based meshless method for solving time fractional wave-diffusion equation, Eng. Anal. Bound. Elem., 92 (2018), 108-113. doi: 10.1016/j.enganabound.2017.10.021
[33]	W. Chen, L. Ye, H. Sun, Fractional diffusion equations by the kansa method, Comput. Math. Appl., 59 (2010), 1614-1620. doi: 10.1016/j.camwa.2009.08.004
[34]	P. Zhuang, Y. T. Gu, F. Liu, et al. Time-dependent fractional advection-diffusion equations by an implicit mls meshless method, Int. J. Numer. Meth. Eng., 88 (2011), 1346-1362. doi: 10.1002/nme.3223
[35]	J. Y. Yang, Y. M. Zhao, N. Liu, et al. 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
[36]	G. J. Moridis, E. J. Kansa, The Laplace transform multiquadric method: a highly accurate scheme for the numerical solution of partial differential equations, J. Appl. sci. comput., 1 (1994) 375-475.
[37]	W. McLean, V. Thomee, Time discretization of an evolution equation via laplace transforms, IMA J. Numer. Anal., 24 (2004), 439 -463.
[38]	W. McLean, V. Thomée, Numerical solution via laplace transforms of a fractional order evolution equation, J. Integral Equ. Appl., 22 (2010), 57-94. doi: 10.1216/JIE-2010-22-1-57
[39]	B. A. Jacobs, High-order compact finite difference and Laplace transform method for the solution of time fractional heat equations with Dirichlet and Neumann boundary conditions, Numer. Meth. Part. D. E., 32 (2016), 1184-1199. doi: 10.1002/num.22046
[40]	Q. T. L. Gia, W. Mclean, Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions, Adv. Comput. Math., 40 (2014), 353-375. doi: 10.1007/s10444-013-9311-6
[41]	A. Talbot, The accurate numerical inversion of laplace transform, IMA J. Appl. Math., 23 (1979), 97-120. doi: 10.1093/imamat/23.1.97
[42]	D. G. Duffy, On the numerical inversion of laplace transforms: comparison of three new methods on characteristic problems from applications, ACM T. Math. Software, 19 (1993), 333-359. doi: 10.1145/155743.155788
[43]	J. A. C. Weideman, Optimizing talbot's contours for the inversion of the laplace transform, SIAM J. Numer. Anal., 44 (2006), 2342-2362. doi: 10.1137/050625837
[44]	C. Lubich, A. Schädle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput., 24 (2002), 161-182. doi: 10.1137/S1064827501388741
[45]	F. Zeng, I. Turner, K. Burrage, et al. A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations, SIAM J. Sci. Comput., 40 (2018), A2986-A3011.
[46]	B. Dingfelder, J. Weideman, An improved talbot method for numerical laplace transform inversion, Numer. Algorithms, 68 (2015), 167-183. doi: 10.1007/s11075-014-9895-z
[47]	R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3 (1995), 251-264. doi: 10.1007/BF02432002

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