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Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation

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Abstract

In the present paper, a spectral meshless radial point interpolation (SMRPI) technique is applied to solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in two dimensional case. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in \(H^1\) and convergent with the order of convergence \(\mathcal {O}(\delta t^{1+\beta })\), \(0<\beta <1\). In the current work, the thin plate splines (TPS) are used as the basis functions. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme. Two numerical examples show that the SMRPI has reliable accuracy in general shape domains.

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References

  1. Oldham KB, Spanier J (1974) The fractional calculus, vol. 111 of mathematics in science and engineering, Academic Press, New York, London

  2. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic Press, Cambridge

    MATH  Google Scholar 

  3. Dehghan M, Abbaszadeh M (2017) Two meshless procedures: moving kriging interpolation and element-free Galerkin for fractional PDEs. Appl Anal 96(6):936–969

    Article  MathSciNet  MATH  Google Scholar 

  4. Uchaikin VV (2013) Fractional derivatives for physicists and engineers. Springer, Berlin

    Book  MATH  Google Scholar 

  5. Metzler Ralf, Klafter Joseph (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A Math Gen 37(31):R161

    Article  MathSciNet  MATH  Google Scholar 

  6. Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol (1978–present) 27(3):201–210

  7. Tenreiro Machado J, Kiryakova V, Mainardi F (2011) Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 16(3):1140–1153

    Article  MathSciNet  MATH  Google Scholar 

  8. Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408

  9. Alkahtani BST (2016) Chua’s circuit model with Atangana–Baleanu derivative with fractional order. Chaos Solitons Fractals 89:547–551

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen W, Ye L, Sun H (2010) Fractional diffusion equations by the Kansa method. Comput Math Appl 59(5):1614–1620

    Article  MathSciNet  MATH  Google Scholar 

  11. Wei S, Chen W, Hon Y-C (2015) Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations. Therm Sci 19(suppl. 1):59–67

    Article  Google Scholar 

  12. Chen W, Pang G (2016) A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J Comput Phys 309:350–367

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhuo-Jia F, Chen W, Yang H-T (2013) Boundary particle method for Laplace transformed time fractional diffusion equations. J Comput Phys 235:52–66

    Article  MathSciNet  MATH  Google Scholar 

  14. Tan W, Masuoka T (2005) Stokes first problem for a second grade fluid in a porous half-space with heated boundary. Int J Non-Linear Mech 40(4):515–522

    Article  MATH  Google Scholar 

  15. Tan W, Masuoka T (2005) Stokes first problem for an Oldroyd-B fluid in a porous half space. Phys Fluids 17(2):023101

    Article  MathSciNet  MATH  Google Scholar 

  16. Fetecu C, Fetecu C (2002) The Rayleigh–Stokes problem for heated second grade fluids. Int J Non-Linear Mech 37(6):1011–1015

    Article  Google Scholar 

  17. Shen F, Tan W, Zhao Y, Masuoka T (2006) The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Anal Real World Appl 7(5):1072–1080

    Article  MathSciNet  MATH  Google Scholar 

  18. Rajagopal KR (1982) A note on unsteady unidirectional flows of a non-Newtonian fluid. Int J Non-Linear Mech 17(5–6):369–373

    Article  MathSciNet  MATH  Google Scholar 

  19. Bandelli R, Rajagopal KR (1995) Start-up flows of second grade fluids in domains with one finite dimension. Int J Non-Linear Mech 30(6):817–839

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhuang P, Liu Q (2009) Numerical method of rayleigh-stokes problem for heated generalized second grade fluid with fractional derivative. Appl Math Mech 30(12):1533

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen C-M, Liu F, Anh V (2008) Numerical analysis of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl Math Comput 204(1):340–351

    MathSciNet  MATH  Google Scholar 

  22. Mohebbi A, Abbaszadeh M, Dehghan M (2013) Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Comput Methods Appl Mech Eng 264:163–177

    Article  MathSciNet  MATH  Google Scholar 

  23. Dehghan M, Abbaszadeh M (2016) A finite element method for the numerical solution of Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng Comput. doi:10.1007/s00366-016-0491-9

  24. Fornberg B, Flyer N (2015) Solving PDEs with radial basis functions. Acta Numerica 24:215–258

    Article  MathSciNet  MATH  Google Scholar 

  25. Fornberg B, Flyer N (2015) A primer on radial basis functions with applications to the geosciences. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611974041

  26. Dehghan M, Haghjoo-Saniji M (2017) The local radial point interpolation meshless method for solving Maxwell equations. Eng Comput 1–22. doi:10.1007/s00366-017-0505-2

  27. Fili A, Naji A, Duan Y (2010) Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions. J Comput Appl Math 234(8):2456–2468

    Article  MathSciNet  MATH  Google Scholar 

  28. Peng M, Li D, Cheng Y (2011) The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems. Eng Struct 33(1):127–135

    Article  Google Scholar 

  29. Fu-Nong B, Dong-Ming L, Jian-Fei W, Yu-Min C (2012) An improved complex variable element-free Galerkin method for two-dimensional elasticity problems. Chin Phys B 21(2):020204

    Article  Google Scholar 

  30. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307–318

    Article  MATH  Google Scholar 

  31. Shivanian E, Khodabandehlo HR (2015) Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem. Ain Shams Eng J

  32. Shivanian E (2013) Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Eng Anal Bound Elements 37(12):1693–1702

    Article  MathSciNet  MATH  Google Scholar 

  33. Dehghan M, Mirzaei D (2009) Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl Numer Math 59(5):1043–1058

    Article  MathSciNet  MATH  Google Scholar 

  34. Shivanian E (2015) Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Bound Elements 50:249–257

    Article  MathSciNet  Google Scholar 

  35. Shirzadi A, Takhtabnoos F (2016) A local meshless method for Cauchy problem of elliptic pdes in annulus domains. Inverse Probl Sci Eng 24(5):729–743

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen W, Fu Z-J, Chen C-S (2014) Recent advances in radial basis function collocation methods. Springer, Berlin

    Book  MATH  Google Scholar 

  37. Shivanian E, Abbasbandy S, Alhuthali MS, Alsulami HH (2015) Local integration of 2-D fractional telegraph equation via moving least squares approximation. Eng Anal Bound Elements 56:98–105

    Article  MathSciNet  Google Scholar 

  38. Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130(2):1–21

    Article  Google Scholar 

  39. Aslefallah M, Shivanian E (2015) Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions. Eur Phys J Plus 130(3):1–9

    Article  Google Scholar 

  40. Shivanian E (2016) On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations. Int J Numer Methods Eng 105(2):83–110

    Article  MathSciNet  MATH  Google Scholar 

  41. Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput Phys Commun 181(4):772–786

    Article  MathSciNet  MATH  Google Scholar 

  42. Tadeu A, Chen CS, António J, Simoes N (2011) A boundary meshless method for solving heat transfer problems using the Fourier transform. Adv Appl Math Mech 3(05):572–585

    Article  MathSciNet  MATH  Google Scholar 

  43. Abbasbandy S, Roohani Ghehsareh H, Hashim I, Alsaedi A (2014) A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng Anal Bound Elements 47:10–20

    Article  MathSciNet  MATH  Google Scholar 

  44. Shivanian E (2015) A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms. Eng Anal Bound Elements 54:1–12

    Article  MathSciNet  Google Scholar 

  45. Shivanian E, Jafarabadi A (2016) More accurate results for nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) problem through spectral meshless radial point interpolation (SMRPI). Eng Anal Bound Elements 72:42–54

    Article  MathSciNet  Google Scholar 

  46. Deng W, Li C (2011) Finite difference methods and their physical constraints for the fractional Klein–Kramers equation. Numer Methods Partial Differ Equ 27(6):1561–1583

    Article  MathSciNet  MATH  Google Scholar 

  47. Lin Y, Chuanju X (2007) Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys 225(2):1533–1552

    Article  MathSciNet  MATH  Google Scholar 

  48. Lin Y, Li X, Chuanju X (2011) Finite difference/spectral approximations for the fractional cable equation. Math Comput 80(275):1369–1396

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang N, Deng W, Yujiang W (2012) Finite difference/element method for a two-dimensional modified fractional diffusion equation. Adv Appl Math Mech 4(04):496–518

    Article  MathSciNet  MATH  Google Scholar 

  50. Mohebbi A, Abbaszadeh M, Dehghan M (2014) Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method. Eng Anal Bound Elements 38:72–82

    Article  MathSciNet  MATH  Google Scholar 

  51. Dehghan M, Abbaszadeh M, Mohebbi A (2016) Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation. Appl Math Model 40(5):3635–3654

    Article  MathSciNet  Google Scholar 

  52. Wendland H (2005) Scattered Data Approximation, cambridge monographs on applied and computational mathematics, Cambridge University Press, Cambridge, UK

  53. Mohebbi A, Abbaszadeh M, Dehghan M (2013) A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term. J Comput Phys 240:36–48

    Article  MathSciNet  MATH  Google Scholar 

  54. Cui M (2009) Compact finite difference method for the fractional diffusion equation. J Comput Phys 228(20):7792–7804

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to three anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.

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  1. Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran

    Elyas Shivanian & Ahmad Jafarabadi

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  1. Elyas Shivanian
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  2. Ahmad Jafarabadi
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Correspondence to Elyas Shivanian.

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Shivanian, E., Jafarabadi, A. Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation. Engineering with Computers 34, 77–90 (2018). https://doi.org/10.1007/s00366-017-0522-1

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