Skip to main content
SpringerLink
Log in

Reproducing kernel method to solve non-local fractional boundary value problem

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

This paper presents a numerical scheme to solve non-local fractional boundary value problems (NFBVPs) through a different implementation of the general form of the reproducing kernel method (RKM) similar to the method eliminating the Gram–Schmidt orthogonalization process to reduce the CPU time. The presented method provides a reliable technique to obtain a reproducing kernel applicable to non-local conditions of the fractional boundary value problems with the aim of increasing the accuracy of the approximate solutions. Therefore, it would be possible to provide a valid error analysis for NFBVP and the presented method. The accuracy of theoretical results is illustrated by solving two numerical examples.

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

Similar content being viewed by others

References

  1. Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Non. Anal. 72, 916–924 (2010)

    Article  MathSciNet  Google Scholar 

  2. Salem, H.A.H.: On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies. J. Comput. Appl. Math. 224, 565–572 (2009)

    Article  MathSciNet  Google Scholar 

  3. Zhong, W., Lin, W.: Nonlocal and multiple-point boundary value problem for fractional differential equations. Comput. Math. Appl. 59, 1345–1351 (2010)

    Article  MathSciNet  Google Scholar 

  4. Ur Rehman, M., Khan, R.A.: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. Appl. Math. Lett. 23, 1038–1044 (2010)

    Article  MathSciNet  Google Scholar 

  5. Ahmad, B., Sivasundaram, S.: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 217, 480–487 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Non. Anal. 71, 2391–2396 (2009)

    Article  MathSciNet  Google Scholar 

  7. Ahmeda, E., Elgazzar, A.S.: On fractional order differential equations model for nonlocal epidemics. Phys. A 379, 607–614 (2007)

    Article  Google Scholar 

  8. He, J.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Am. Soc. Inf. Sci. Technol. 15, 86–90 (1999)

    Google Scholar 

  9. He, J.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    Article  MathSciNet  Google Scholar 

  10. Mainardi, F.: 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 (1997)

    MATH  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud., vol. 204, Elsevier Science B.V., Amsterdam (2006)

  12. Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)

    MATH  Google Scholar 

  13. Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  14. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  Google Scholar 

  15. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives. Theory and Applications, Gordon and Breach, Switzerland (1993)

  16. Kilbas, A.A., Marzan, S.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differ. Equ. 1141, 84–89 (2005)

    Article  MathSciNet  Google Scholar 

  17. Nikan, O., Tenreiro Machado, J.A., Golbabai, A., Rashidinia, J.: Numerical evaluation of the fractional Klein–Kramers model arising in molecular dynamics. J. Comput. Phys. 428, 109983 (2021)

    Article  MathSciNet  Google Scholar 

  18. Nikan, O., Avazzadeh, Z., Tenreiro Machado, J.A.: A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer. J. Adv. Res. (2021)

  19. Nikan, O., Avazzadeh, Z.: An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem. Res. Phys. 23, 104048 (2021)

    Google Scholar 

  20. Nikan, O., Avazzadeh, Z., Tenreiro Machado, J.A.: Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun. Non. Sci. Numer. Simul. 99, 105755 (2021)

    Article  MathSciNet  Google Scholar 

  21. Safdari, H., Esmaeelzade Aghdam, Y., Gómez-Aguilar, J.F.: Shifted Chebyshev collocation of the fourth kind with convergence analysis for the space–time fractional advection-diffusion equation. Eng. Comput. (2020)

  22. Safdari, H., Mesgarani, H., Javidi, M., Esmaeelzade Aghdam, Y.: Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme. Comp. Appl. Math. 39, 62 (2020)

    Article  MathSciNet  Google Scholar 

  23. Mesgarani, H., Beiranvand, A., Esmaeelzade Aghdam, Y.: The impact of the Chebyshev collocation method on solutions of the time-fractional Black-Scholes. Math. Sci. (2020)

  24. Tuan, Nguyen H., Esmaeelzade Aghdam, Y., Jafari, H., Mesgarani, H.: A novel numerical manner for two-dimensional space fractional diffusion equation arising in transport phenomena. Numer. Method. Parti. Diff. Equ. 37, 1397–1406 (2021)

    Article  MathSciNet  Google Scholar 

  25. Safdari, H., Mesgarani, H., Javidi, M., Esmaeelzade Aghdam, Y.: The Chebyshev wavelet of the second kind for solving fractional delay differential equations. Math Comput. Sci. Ser. 47, 111–124 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Allahviranloo, T., Sahihi, H.: Reproducing kernel method to solve fractional delay differential equations. Appl. Math. Comput. 400, 126095 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Geng, F., Cui, M.: A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25, 818–823 (2012)

    Article  MathSciNet  Google Scholar 

  28. Li, X., Wu, B.: Approximate analytical solutions of nonlocal fractional boundary value problems. Appl. Math. Model. 39, 1717–1724 (2015)

    Article  MathSciNet  Google Scholar 

  29. Li, X., Wu, B.: A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Appl. Math. Lett. 86, 194–199 (2018)

    Article  MathSciNet  Google Scholar 

  30. Wang, Y., Chaolu, T., Jing, P.: New algorithm for second-order boundary value problems of integro-differential equation. J. Comput. Appl. Math. 229, 1–6 (2009)

    Article  MathSciNet  Google Scholar 

  31. Wang, Y., Chaolu, T., Chen, Z.: Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems. Int. J. Comput. Math. 87, 367–380 (2010)

    Article  MathSciNet  Google Scholar 

  32. Sahihi, H., Allahviranloo, T., Abbasbandy, S.: Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space. Appl. Numer. Math. 151, 27–39 (2020)

    Article  MathSciNet  Google Scholar 

  33. Sahihi, H., Abbasbandy, S., Allahviranloo, T.: Computational method based on reproducing kernel for solving singularly perturbed differential-difference equations with a delay. Appl. Math. Comput. 361, 583–598 (2019)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Abbasbandy, S., Sahihi, H., Allahviranloo, T.: Implementing reproducing kernel method to solve singularly perturbed convection-diffusion parabolic problems. Math. Model. Anal. 26, 116–134 (2021)

    Article  MathSciNet  Google Scholar 

  36. Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, Hauppauge (2009)

    MATH  Google Scholar 

  37. Li, Z.Y., Wang, Y.L., Tan, F.G., Wan, X.H., Yu, H., Duan, J.S.: Solving a class of linear nonlocal boundary value problems using the reproducing kernel. Appl. Math. Comput. 265, 1098–1105 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y., Cao, X., Li, X.: A new method for solving singular fourth-order boundary value problems with mixed boundary conditions. Appl. Math. Comput. 217, 7385–7390 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Wang, Y., Du, M., Tan, F., Li, Z., Nie, T.: Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions. Appl. Math. Comput. 219, 5918–5925 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Allahviranloo, T., Sahihi, H.: Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions. Numer. Method. Part. Diff. Equ. 36, 1758–1772 (2020)

    Article  MathSciNet  Google Scholar 

  41. Atkinson, K., Han, W.: Theoretical Numerical Analysis A Functional Analysis Framework, 3rd edn. Springer, New York (2009)

    MATH  Google Scholar 

  42. Ketabchi, R., Mokhtari, R., Babolian, E.: Some error estimates for solving Volterra integral equations by using the reproducing kernel method. J. Comput. Appl. Math. 273, 245–250 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. Li, X.Y., Wu, B.Y.: Error estimation for the reproducing kernel method to solve linear boundary value problems. J. Comput. Appl. Math. 243, 10–15 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the anonymous reviewers for helpful comments and Dr. H. Sahihi, which lead to definite improvement in the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Raziye Mohammad Hosseiny & Tofigh Allahviranloo

  2. Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, 34149-16818, Iran

    Saeid Abbasbandy

  3. Computer Science Department, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

    Esmail Babolian

Authors
  1. Raziye Mohammad Hosseiny
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tofigh Allahviranloo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saeid Abbasbandy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Esmail Babolian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saeid Abbasbandy.

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

Hosseiny, R.M., Allahviranloo, T., Abbasbandy, S. et al. Reproducing kernel method to solve non-local fractional boundary value problem. Math Sci 16, 261–268 (2022). https://doi.org/10.1007/s40096-021-00418-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-021-00418-0

Keywords

Search

Navigation