Skip to main content
SpringerLink
Log in

On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods

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

Abstract

In this paper, a Lie-group integrator based on \(GL_4(\mathbb {R})\) and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Using the truncation rule for the differential equations over the infinite or semi-infinite intervals, we set \(\eta ^f=\eta _\infty \) as the final value of interval \([0,\infty )\).

  2. \(gl_4(\mathbb {R})\) is the Lie algebra associated with the Lie group \(GL_4(\mathbb {R}).\)

References

  1. Aronszajn N (1950) Theory of reproducing Kernels. Trans Am Math Soc 68:337–404

    MathSciNet  MATH  Google Scholar 

  2. Abbasbandy S, Azarnavid B, Alhuthali MS (2015) A shooting reproducing Kernel Hilbert space method for multiple solutions of nonlinear boundary value problems. J Comput Appl Math 279:293–305

    MathSciNet  MATH  Google Scholar 

  3. Arqub OA (2017) Fitted reproducing Kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and neumann boundary conditions. Comput Math Appl 73(6):1243–1261

    MathSciNet  MATH  Google Scholar 

  4. Akgül A (2014) A new method for approximate solutions of fractional order boundary value problems. Neural Parallel Sci Comput 22(1–2):223–237

    MathSciNet  Google Scholar 

  5. Akgül A (2015) New reproducing kernel functions. Math Probl Eng Art. ID 158134:10

  6. Abbasbandy S, Azarnavid B (2016) Some error estimates for the reproducing Kernel Hilbert spaces method. J Comput Appl Math 296:789–797

    MathSciNet  MATH  Google Scholar 

  7. Arqub OA, Al-Smadi M, Shawagfeh N (2013) Solving Fredholm integro-differential equations using reproducing Kernel Hilbert space method. Appl Math Comput 219(17):8938–8948

    MathSciNet  MATH  Google Scholar 

  8. Azarnavid B, Parand K (2018) An iterative reproducing Kernel method in Hilbert space for the multi-point boundary value problems. J Comput Appl Math 328:151–163

    MathSciNet  MATH  Google Scholar 

  9. Babolian E, Javadi S, Moradi E (2016) Error analysis of reproducing Kernel Hilbert space method for solving functional integral equations. J Comput Appl Math 300:300–311

    MathSciNet  MATH  Google Scholar 

  10. Bushnaq S, Momani S, Zhou Y (2013) A reproducing Kernel Hilbert space method for solving integro-differential equations of fractional order. J Optim Theory Appl 156(1):96–105

    MathSciNet  MATH  Google Scholar 

  11. Bushnaq S, Maayah B, Ahmad M (2016) Reproducing kernel Hilbert space method for solving Fredholm integro-differential equations of fractional order. Ital J Pure Appl Math 36:307–318

    MathSciNet  MATH  Google Scholar 

  12. Gumah G, Moaddy K, Al-Smadi M, Hashim I (2016) Solutions to uncertain Volterra integral equations by fitted reproducing kernel Hilbert space method. J Funct Sp. Art. ID 2920463:11

  13. Sahihi H, Abbasbandy S, Allahviranloo T (2018) Reproducing Kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space. J Comput Appl Math 328:30–43

    MathSciNet  MATH  Google Scholar 

  14. Dorodnitsyn V (2010) Applications of Lie groups to difference equations. CRC Press, London

    MATH  Google Scholar 

  15. Hairer E (2001) Geometric integration of ordinary differential equations on manifolds. BIT Numer Math 41(5):996–1007

    MathSciNet  Google Scholar 

  16. Hairer E, Lubich C, Wanner G et al (2003) Geometric numerical integration illustrated by the Stormer–Verlet method. Acta Numer 12(12):399–450

    MathSciNet  MATH  Google Scholar 

  17. Liu C-S (2001) Cone of non-linear dynamical system and group preserving schemes. Int J Non-Linear Mech 36:1047–1068

    MathSciNet  MATH  Google Scholar 

  18. Hashemi MS (2015) Constructing a new geometric numerical integration method to the nonlinear heat transfer equations. Commun Nonlinear Sci Numer Simul 22(1):990–1001

    MathSciNet  MATH  Google Scholar 

  19. Hashemi MS (2017) A novel simple algorithm for solving the magneto-hemodynamic flow in a semi-porous channel. Eur J Mech B Fluids 65:359–367

    MathSciNet  MATH  Google Scholar 

  20. Hajiketabi M, Abbasbandy S, Casas F (2018) The lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains. Appl Math Comput 321:223–243

    MathSciNet  MATH  Google Scholar 

  21. Hajiketabi M, Abbasbandy S (2018) The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation. Eng Anal Boundary Elem 87:36–46

    MathSciNet  MATH  Google Scholar 

  22. Abbasbandy S, Van Gorder R, Hajiketabi M (2015) The lie-group shooting method for radial symmetric solutions of the yamabe equation. CMES 104(4):329–351

    Google Scholar 

  23. Ali F, Sheikh N, Khan I, Saqib M (2017) Magnetic field effect on blood flow of casson fluid in axisymmetric cylindrical tube: a fractional model. J Magn Magn Mater 423:327–336

    Google Scholar 

  24. Ali F, Jan S, Khan I, Gohar M, Sheikh N (2016) Solutions with special functions for time fractional free convection flow of brinkman type fluid. The European Physical Journal Plus 131:310

    Google Scholar 

  25. Ali F, Saqib M, Khan I, Sheikh N (2016) Application of Caputo–Fabrizio derivatives to MHD free convection flow of generalized walters’-b fluid model. Eur Phys J Plus 131:377

    Google Scholar 

  26. Ali F, Sheikh N, Saqib M, Khan A (2017) Hidden phenomena of an MHD unsteady flow in porous medium with heat transfer. Nonlinear Sci. Lett. A. 8:110–116

    Google Scholar 

  27. Sheikh N, Ali F, Khan I, Saqib M (2017) A modern approach of Caputo-Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Comput Appl. 1–11

  28. Saqib M, Ali F, Khan I, Sheikh N (2017) Heat and mass transfer phenomena in the flow of casson fluid over an infinite oscillating plate in the presence of first-order chemical reaction and slip effect. Neural Comput and Applic. 1–14

  29. Saqib M, Ali F, Khan I, Sheikh N, Khan A (2017) MHD flow of micropolar fluid over an oscillating vertical plate embedded in porous media with constant temperature and concentration. Math Probl Eng 20

  30. Zin N, Khan I, Shafie S, Exact and numerical solutions for unsteady heat and mass transfer problem of Jeffrey fluid with MHD and Newtonian heating effects. Neural Comput Appl

  31. Khan I (2017) Shape effects of MoS2 nanoparticles on MHD slip flow of molybdenum disulphide nanofluid in a porous medium. J Mol Liq 233:442–451

    Google Scholar 

  32. Khan I, Gul A, Shaife S (2017) Effects of magnetic field on molybdenum disulfide nanofluids in mixed convection flow inside a channel filled with a saturated porous medium. J Porous Media 435–448

  33. Khan N, Gul T, Islam S, Khan I, Alqahtani A, Alshomrani A (2017) Magnetohydrodynamic nanoliquid thin film sprayed on a stretching cylinder with heat transfer. Appl Sci 7:271

    Google Scholar 

  34. 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 

  35. Shivanian E, Jafarabadi A (2017) An efficient numerical technique for solution of two-dimensional cubic nonlinear schrödinger equation with error analysis. Eng Anal Boundary Elem 83:74–86

    MathSciNet  MATH  Google Scholar 

  36. Fatahi H, Saberi-Nadjafi J, 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 

  37. Bhattacharyya K, Layek G (2012) Similarity solution of mhd boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing. Meccanica 47(4):1043–1048

    MathSciNet  MATH  Google Scholar 

  38. Abbasbandy S, Hayat T, Ghehsareh H, Alsaedi A (2013) Mhd Falkner–Skan flow of maxwell fluid by rational Chebyshev collocation method. Appl Math Mech 34(8):921–930

    MathSciNet  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. Mukhopadhyay S, Mandal IC (2015) Magnetohydrodynamic (mhd) mixed convection slip flow and heat transfer over a vertical porous plate. Eng Sci Technol Int J 18(1):98–105

    Google Scholar 

  41. Bhattacharyya K, Arif MG, Pramanik WA (2012) Mhd boundary layer stagnation-point flow and mass transfer over a permeable shrinking sheet with suction/blowing and chemical reaction. Acta Tech 57(1):1–15

    MathSciNet  Google Scholar 

  42. Akgül A, Khan Y, Karatas Akgül E, Baleanu D, Al Qurashi M M (2017) Solutions of nonlinear systems by reproducing Kernel method. J Nonlinear Sci Appl 10(8):4408–4417

    MathSciNet  MATH  Google Scholar 

  43. Al-Smadi M, Arqub OA, Shawagfeh N, Momani S (2016) Numerical investigations for systems of second-order periodic boundary value problems using reproducing Kernel method. Appl Math Comput 291:137–148

    MathSciNet  MATH  Google Scholar 

  44. Babolian E, Hamedzadeh D (2017) A splitting iterative method for solving second kind integral equations in reproducing Kernel spaces. J Comput Appl Math 326:204–216

    MathSciNet  MATH  Google Scholar 

  45. Cheng RJ, Liew KM (2012) Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing Kernel particle Ritz method. Comput Methods Appl Mech Eng 245(246):132–143

    MathSciNet  MATH  Google Scholar 

  46. Geng FZ, Qian SP (2015) Modified reproducing Kernel method for singularly perturbed boundary value problems with a delay. Appl Math Model 39(18):5592–5597

    MathSciNet  MATH  Google Scholar 

  47. Jiang W, Chen Z (2013) Solving a system of linear Volterra integral equations using the new reproducing Kernel method. Appl Math Comput 219(20):10225–10230

    MathSciNet  MATH  Google Scholar 

  48. Yang C-T (2013) Application of reproducing Kernel particle method and element-free Galerkin method on the simulation of the membrane of capacitive micromachined microphone in viscothermal air. Comput Mech 51(3):295–308

    MathSciNet  Google Scholar 

  49. Akgül A, Hashemi MS, Inc M, Raheem SA (2017) Constructing two powerful methods to solve the Thomas–Fermi equation. Nonlinear Dyn 87(2):1435–1444

    MathSciNet  MATH  Google Scholar 

  50. Hashemi MS, Darvishi E, Baleanu D (2016) A geometric approach for solving the density dependent diffusion Nagumo equation. Adv Differ Equ 1:1–13

    MathSciNet  MATH  Google Scholar 

  51. Akgül A, Hashemi MS et al (2017) Group preserving scheme and reproducing Kernel method for the Poisson–Boltzmann equation for semiconductor devices. Nonlinear Dyn 88(4):2817–2829

    MathSciNet  MATH  Google Scholar 

  52. Hashemi MS, Abbasbandy S (2017) A geometric approach for solving Troesch’s problem. Bull Malay Math Sci Soc 40(1):97–116

    MathSciNet  MATH  Google Scholar 

  53. Hashemi MS, Baleanu D, Parto-Haghighi M (2015) A lie group approach to solve the fractional Poisson equation. Rom J Phys 60(9–10):1289–1297

    Google Scholar 

  54. Hashemi MS, Baleanu D, Parto-Haghighi M, Darvishi E (2015) Solving the time-fractional diffusion equation using a lie group integrator. Rom J Phys 19:S77–S83

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for the helpful remarks and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bonab, Bonab, 55517, Iran

    M. S. Hashemi

  2. Department of Mathematics, Art and Science Faculty, Siirt University, 56100, Siirt, Turkey

    A. Akgül

Authors
  1. M. S. Hashemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Akgül
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. S. Hashemi.

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

Hashemi, M.S., Akgül, A. On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods. Engineering with Computers 37, 1147–1158 (2021). https://doi.org/10.1007/s00366-019-00876-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-019-00876-0

Keywords

Search

Navigation