Skip to main content
SpringerLink
Log in

A computational study of variable coefficients fractional advection–diffusion–reaction equations via implicit meshless spectral algorithm

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this article, a meshless spectral radial point interpolation method is proposed for the numerical solutions of a class of time-fractional advection–diffusion–reaction equations. The current approach utilizes meshless shape functions, having Kronecker delta function property, for approximation of spatial operators. Forward difference along with quadrature formula is used for tempered fractional derivative approximation in the framework of implicit time marching scheme. Assessment of the proposed method is made by applying to various concrete test problems having constant and variable coefficients. Approximation and function reproduction quality are measured via \({E}_{\infty }\), \({E}_{2}\) and \({E}_{\text {rms}}\) error norms. Comparison of simulated results is also made with available exact solutions as well as earlier reported works. Stability analysis of the proposed method is thoroughly discussed and computationally affirmed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. LHilfer R (2000) Applications of fractional calculus in physics. World Scientific, River Edge

    Google Scholar 

  2. Podlubny I (1999) Fractional differential equations. Academic, San Diego, p 198

    MATH  Google Scholar 

  3. Fallahgoul HA, Focardi SM, Fabozzi FJ (2017) Fractional calculus and fractional processes with applications to financial economics: Theory and Applications. Elsevier, Oxford

    MATH  Google Scholar 

  4. Mainardi F (1997) Fractional calculus: Some basic problems in continuum and statistical mechanics. In: carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics. Springer, New York, pp 291–348

    MATH  Google Scholar 

  5. Fujita Y (1990) Cauchy problems of fractional order and stable processes. Jpn J Appl Math 7(3):459–476

    MathSciNet  MATH  Google Scholar 

  6. Hilfer R (1995) Foundations of fractional dynamics. Fractals 3(3):549–556

    MathSciNet  MATH  Google Scholar 

  7. Hilfer R (2000) Fractional diffusion based on Riemann-Liouville fractional derivative. J Phys Chem 104:3914–3917

    Google Scholar 

  8. Caputo M (1967) Linear models of dissipation whose \(Q\) is almost frequency independent, Part II. J R Astral Soc 13:529–539

    Google Scholar 

  9. Metzler JKR (2000) The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77

    MathSciNet  MATH  Google Scholar 

  10. Schumer R, Benson DA, Meerschaert MM et al (2003) Fractal mobile/immobile solute transport. Water Resour Res 39(10):1296

    Google Scholar 

  11. Schumer R, Meerschaert MM, Baeumer B (2009) Fractional advection-dispersion equations for modeling transport at the Earth surface. J Geophys Res 114:F00A07. https://doi.org/10.1029/2008JF001246

    Article  Google Scholar 

  12. Akrami MH, Erjaee GH (2015) Examples of analytical solutions by means of Mittag-Leffler function of fractional Black-Scholes option pricing equations. Fract Calc Appl Anal 18(1):38–47

    MathSciNet  MATH  Google Scholar 

  13. Liu F, Zhuang P, Anh V et al (2007) Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl Math Comput 191:12–20

    MathSciNet  MATH  Google Scholar 

  14. Tadjeran C, Meerschaert MM, Scheffler H-P (2006) A second-order accurate numerical approximation for the fractional diffusion equation. J Comput Phys 213:205–213

    MathSciNet  MATH  Google Scholar 

  15. Yuste S (2006) Weighted average finite difference metods for fractional diffusion equations. J Comput Phys 216(1):264–274

    MathSciNet  MATH  Google Scholar 

  16. Gao G-H, Sun H-W (2015) Three-point combined compact difference schemes for time-fractional advectiondiffusion equations with smooth solutions. J Comput Phys 298:520–538

    MathSciNet  MATH  Google Scholar 

  17. Cui M (2015) Compact exponential scheme for the time fractional convectiondiffusion reaction equation with variable coefficients. J Comput Phys 280:143–163

    MathSciNet  MATH  Google Scholar 

  18. Chen Y, Wu Y, Cui Y et al (2010) Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. J Comput Sci 1:146–149

    Google Scholar 

  19. Saadatmandi A, Dehghan M, Azizi M-R (2012) The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun Nonlinear Sci Numer Simulat 17:4125–4136

    MathSciNet  MATH  Google Scholar 

  20. Liu F, Zhuang P, Burrage K (2012) Numerical methods and analysis for a class of fractional advection dispersion models. Comput Math Appl 64:2990–3007

    MathSciNet  MATH  Google Scholar 

  21. Kansa EJ (1990) Multiquadrics-A scattered data approximation scheme with application to computation fluid dynamics, II. Solutions to hyperbolic, parabolic, and elliptic partial differential equations. Comput Math Appl 19:149–161

    MathSciNet  Google Scholar 

  22. Uddin M, Haq S (2011) RBF approximation method for time fractional partial differential equations. Commun Nonlinear Sci Numer Simulat 16(11):4208–4214

    MathSciNet  MATH  Google Scholar 

  23. Haq S, Hussain M (2018) Selection of shape parameter in radial basis functions for solution of time-fractional Black-Scholes models. Appl Math Comput 335:248–263

    MathSciNet  MATH  Google Scholar 

  24. Haq S, Hussain M (2018) The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients. RACSAM. https://doi.org/10.1007/s13398-018-0593-x

  25. Liu L, Yang H (2007) A paralleled element-free Galerkin analysis for structures with cyclic symmetry. Eng Comput 23(2):137–144

    Google Scholar 

  26. Kia AD, Fallah N (2018) Comparison of enriched meshless finite volume and element free Galerkin methods for the analysis of heterogeneous media. Eng Comput 34(4):787–799

    Google Scholar 

  27. Karami B, Janghorban M, Tounsi A (2018) Galerkins approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions. Eng Comput. https://doi.org/10.1007/s00366-018-0664-9

  28. Gu YT, Liu GR (2001) A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput Mech 27:188–198

    MATH  Google Scholar 

  29. Ling L, Opfer R, Schaback R (2006) Results on meshless collocation techniques. Eng Anal Bound Elem 30:247–253

    MATH  Google Scholar 

  30. Ling L, Schaback R (2008) Stable and convergent unsymmetric meshless collocation methods. SIAM J Numer Anal 46:1097–1115

    MathSciNet  MATH  Google Scholar 

  31. Kansa EJ, Hon YC (2000) Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput Math Appl 39:123137

    MathSciNet  MATH  Google Scholar 

  32. Dehghan M, Abbaszadeh M (2017) Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross-Pitaevskii equation. Eng Comput 33(4):983–996

    Google Scholar 

  33. Assari P, Cuomo S (2018) The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines. Eng Comput. https://doi.org/10.1007/s00366-018-0671-x

  34. Eshaghi J, Kazem S, Adibi H (2018) The local discontinuous Galerkin method for 2D nonlinear time-fractional advection-diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-018-0665-8

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

    MathSciNet  MATH  Google Scholar 

  36. Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer, Berlin

    Google Scholar 

  37. Shivanian E (2016) More accurate results for two-dimensional heat equation with Neumanns and non-classical boundary conditions. Eng Comput 32(4):729–743

    Google Scholar 

  38. Rad JA, Rashedi K, Parand K et al (2017) The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem. Eng Comput 33:547–571

    Google Scholar 

  39. Shivanian E, Jafarabadi A (2018) Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation. Eng Comput 34(1):77–90

    Google Scholar 

  40. Shivanian E, Jafarabadi A (2018) Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation. Eng Comput 34(3):603–619

    Google Scholar 

  41. Hussain M, Haq S (2019) Meshless spectral method for solution of time-fractional coupled KdV equations. Appl Math Comput 341:321–334

    MathSciNet  MATH  Google Scholar 

  42. Hussain Manzoor, Haq Sirajul, Ghafoor Abdul, Ali Ihteram. Numerical solution of time-fractional coupled viscous Burgers equations using meshfree spectral method. Comp Appl Math. Accepted (to appear)

  43. Hussain M, Haq S (2019) Weighted meshless spectral method for the solutions of multi-term time fractional advection-diffusion problems arising in heat and mass transfer. Int J Heat Mass Transf 129:1305–1316

    Google Scholar 

  44. Micchelli CA (1986) Interpolation of scattered data: distance matrix and conditionally positive definite functions. Construct Approx 2:11–22

    MATH  Google Scholar 

  45. Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, River Edge

    MATH  Google Scholar 

  46. Shivanian E (2014) Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations. Ocean Eng 89:173–188

    Google Scholar 

  47. Fatahi H, Nadjafi JS, Shivanian E (2016) A new spectral meshless radial point interpolation (SMRPI) method for the two-dimensional Fredholm integral equations on general domains with error analysis. J Comput Appl Math 294:196–209

    MathSciNet  MATH  Google Scholar 

  48. Shivanian E (2016) Spectral meshless radial point interpolation (SMRPI) method to twodimensional fractional telegraph equation. Math Methods Appl Sci 39(7):1820–1835

    MathSciNet  MATH  Google Scholar 

  49. Shivanian E, Jafarabadi A (2017) Inverse Cauchy problem of annulus domains in the framework of spectral meshless radial point interpolation. Eng Comput 33(3):431–442

    MATH  Google Scholar 

  50. 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 Elem 72:42–54

    MathSciNet  MATH  Google Scholar 

  51. Shivanian E, Jafarabadi A (2017) An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2D fractional evolution equation. J Comput Appl Math 325:18–33

    MathSciNet  MATH  Google Scholar 

  52. Assari P, Dehghan M (2017) A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions. Appl Math Comput 315:424–444

    MathSciNet  MATH  Google Scholar 

  53. Assari P, Dehghan M (2018) Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method. Appl Numer Math 123:137–158

    MathSciNet  MATH  Google Scholar 

  54. Assari P, Dehghan M (2019) Application of thin plate splines for solving a class of boundary integral equations arisen from Laplaces equations with nonlinear boundary conditions. Int J Comput Math 96:170–198

    MathSciNet  Google Scholar 

  55. Assari P, Dehghan M (2018) A meshless Galerkin scheme for the approximate solution of nonlinear logarithmic boundary integral equations utilizing radial basis functions. J Comput Appl Math 333:362–381

    MathSciNet  MATH  Google Scholar 

  56. Mardani A, Hooshmandasl MR, Heydari MH et al (2018) A meshless method for solving the time fractional advection-diffusion equation with variable coefficients. Comput Math Appl 75(1):122–133

    MathSciNet  MATH  Google Scholar 

  57. Zhuang P, Gu YT, Liu F et al (2011) Time-dependent fractional advectiondiffusion equations by an implicit MLS meshless method. Int J Numer Methods Eng 88(13):1346–1362

    MATH  Google Scholar 

  58. Mehrdoust F, Sheikhani AHR, Mashoof M et al (2017) Block-pulse operational matrix method for solving fractional Black-Scholes equation. J Econ Stud 44(3):489–502

    Google Scholar 

  59. Elsheikh AM, Elzaki TM (2015) Variation iteration method for solving porous medium equation. Int J Dev Res 5(6):4677–4680

    Google Scholar 

Download references

Acknowledgements

The authors are thankful to the anonymous reviewers for their helpful remarks that have improved quality of the paper.

Author information

Authors and Affiliations

  1. FES, GIK Institute, Topi, KP, 23640, Pakistan

    Sirajul Haq & Manzoor Hussain

  2. Institute of Numerical Sciences KUST, Kohat, KP, 26000, Pakistan

    Abdul Ghafoor

Authors
  1. Sirajul Haq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manzoor Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdul Ghafoor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manzoor Hussain.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haq, S., Hussain, M. & Ghafoor, A. A computational study of variable coefficients fractional advection–diffusion–reaction equations via implicit meshless spectral algorithm. Engineering with Computers 36, 1243–1263 (2020). https://doi.org/10.1007/s00366-019-00760-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-019-00760-x

Keywords

Mathematics Subject Classification

Search

Navigation