SpringerLink
Log in

Trustworthy Analytical Technique for Generating Multiple Solutions to Fractional Boundary Value Problems

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Using the Laplace residual power series approach, this study establishes solutions to fractional differential equations with boundary conditions. The proposed method is effective in generating multi-series solutions of the target problem that converge to exact solutions rapidly. This method is presented and illustrated in a simple algorithm. Interesting examples are solved with an illustration of the steps of the procedure. Moreover, we analyze the results and present them in tables and figures to show the efficiency and capability of the method. The accuracy of the obtained results is measured by the Laplace residual error using the Mathematica software.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

Data Availability

The data that supports the findings of this study are available in the article.

References

  1. Sintunavarat, W., Turab, A.: A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory. PLoS ONE 17(8), e0270148 (2022)

    Article  Google Scholar 

  2. Zhang, K., Fu, Z.: Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity. J. Funct. Spaces 2019, 9046472 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Tabatadze, V., Karaçuha, K., Velıyev, E., Karaçuha, E.: Diffraction of the electromagnetic plane waves by double half-plane with fractional boundary conditions. Progress Electromagnet Res M 101, 207–218 (2021)

    Article  Google Scholar 

  4. Hollkamp, J.P., Semperlotti, F.: Application of fractional order operators to the simulation of ducts with acoustic black hole terminations. J. Sound Vib. 465, 115035 (2020)

    Article  Google Scholar 

  5. Askarian, A.R., Permoon, M.R., Shakouri, M.: Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions. Int. J. Mech. Sci. 179, 105702 (2020)

    Article  Google Scholar 

  6. L’vov, P.E., Sibatov, R.T., Yavtushenko, I.O., Kitsyuk, E.P.: Time-fractional phase field model of electrochemical impedance. Fractal Fract 5(4), 191 (2021)

    Article  Google Scholar 

  7. Burqan, A., Saadeh, R., Qazza, A.: A novel numerical approach in solving fractional neutral pantograph equations via the ARA integral transform. Symmetry 14(1), 50 (2021)

    Article  Google Scholar 

  8. Jafari, H., Daftardar-Gejji, V.: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180(2), 700–706 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Daftardar-Gejji, V., Bhalekar, S.: Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Comput. Math. Appl. 59(5), 1801–1809 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Jajarmi, A., Baleanu, D.: A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Front. Phys. 8, 220 (2020)

    Article  Google Scholar 

  11. Al-Mdallal, Q.M., Syam, M.I., Anwar, M.N.: A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 15(12), 3814–3822 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ford, N., Morgado, M.: Fractional boundary value problems: analysis and numerical methods. Fract. Calculus Appl. Anal. 14(4), 554–567 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Geng, F., Cui, M.: A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett. 25(5), 818–823 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Akgül, A., Karatas Akgül, E.: A novel method for solutions of fourth-order fractional boundary value problems. Fractal Fractional 3(2), 33 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. López, B., Rocha, J., Sadarangani, K.: Lyapunov-type inequalities for a class of fractional boundary value problems with integral boundary conditions. Math. Methods Appl. Sci. 42(1), 49–58 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Al-Mdallal, Q.M., Hajji, M.A.: A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fract. Calcul. Appl. Anal. 18(6), 1423–1440 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Alkan, S., Secer, A.: Application of Sinc-Galerkin method for solving space-fractional boundary value problems. Math Problems Eng 2015, 217348 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zaky, M.A.: Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl. Numer. Math. 145, 429–457 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jong, K., Choi, H., Jang, K., Pak, S.: A new approach for solving one-dimensional fractional boundary value problems via Haar wavelet collocation method. Appl. Numer. Math. 160, 313–330 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62(5), 2364–2373 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Eriqat, T., El-Ajou, A., Moa’ath, N.O., Al-Zhour, Z., Momani, S.: A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations. Chaos, Solitons Fractals 138, 109957 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Burqan, A., El-Ajou, A., Saadeh, R., Al-Smadi, M.: A new efficient technique using Laplace transforms and smooth expansions to construct a series solution to the time-fractional Navier-Stokes equations. Alex. Eng. J. 61(2), 1069–1077 (2022)

    Article  Google Scholar 

  23. Saadeh, R., Burqan, A., El-Ajou, A.: Reliable solutions to fractional Lane-Emden equations via Laplace transform and residual error function. Alex. Eng. J. 61(12), 10551–10562 (2022)

    Article  Google Scholar 

  24. Saadeh, R., Qazza, A., Amawi, K.: A new approach using integral transform to solve cancer models. Fractal Fract. 6(9), 490 (2022)

    Article  Google Scholar 

  25. Alderremy, A.A., Shah, R., Iqbal, N., Aly, S., Nonlaopon, K.: Fractional series solution construction for nonlinear fractional reaction-diffusion Brusselator model utilizing Laplace residual power series. Symmetry 14(9), 1944 (2022)

    Article  Google Scholar 

  26. Alquran, M., Ali, M., Alsukhour, M., Jaradat, I.: Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics. Results Phys. 19, 103667 (2020)

    Article  Google Scholar 

  27. Oqielat, M.A.N., Eriqat, T., Al-Zhour, Z., Ogilat, O., El-Ajou, A., Hashim, I.: Construction of fractional series solutions to nonlinear fractional reaction–diffusion for bacteria growth model via Laplace residual power series method. Int. J. Dynm. Control 11(2), 520–527 (2022)

    Article  Google Scholar 

  28. Burqan, A., Sarhan, A., Saadeh, R.: Constructing analytical solutions of the fractional riccati differential equations using laplace residual power series method. Fractal Fractional 7(1), 14 (2022)

    Article  Google Scholar 

  29. Salah, E., Qazza, A., Saadeh, R., El-Ajou, A.: A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system. AIMS Mathematics 8(1), 1713–1736 (2023)

    Article  MathSciNet  Google Scholar 

  30. El-Ajou, A.: Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach. Eur. Phys. J. Plus 136(2), 1–22 (2021)

    Article  Google Scholar 

  31. Eriqat, T., Oqielat, M.A.N., Al-Zhour, Z., Khammash, G.S., El-Ajou, A., Alrabaiah, H.: Exact and numerical solutions of higher-order fractional partial differential equations: a new analytical method and some applications. Pramana 96(4), 1–17 (2022)

    Article  Google Scholar 

  32. Arqub, O.A., El-Ajou, A., Zhour, Z.A., Momani, S.: Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique. Entropy 16(1), 471–493 (2014)

    Article  Google Scholar 

  33. Alomari, A.K., Awawdeh, F., Tahat, N., Bani Ahmad, F., Shatanawi, W.: Multiple solutions for fractional differential equations: analytic approach. Appl. Math. Comput. 219, 8893–8903 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Abbasbandy, S., Shivanian, E.: Predictor homotopy analysis method and its application to some nonlinear problems. Commun. Nonlinear Sci. Numer. Simul. 16, 2456–2468 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sadaf, M., Akram, G., Arshed, S., Farooq, K.: A study of fractional complex Ginzburg-Landau model with three kinds of fractional operators. Chaos, Solitons Fractals 166, 112976 (2023)

    Article  MathSciNet  Google Scholar 

  36. Tariq, H., Sadaf, M., Akram, G., Rezazadeh, H., Baili, J., Lv, Y.P., Ahmad, H.: Computational study for the conformable nonlinear Schrödinger equation with cubic–quintic–septic nonlinearities. Results Phys. 30, 104839 (2021)

    Article  Google Scholar 

  37. Sadaf, M., Akram, G.: Effects of fractional order derivative on the solution of time-fractional Cahn-Hilliard equation arising in digital image inpainting. Indian J. Phys. 95, 891–899 (2021)

    Article  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, 13110, Jordan

    Aliaa Burqan, Rania Saadeh & Ahmad Qazza

  2. Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt, 19117, Jordan

    Ahmad El-Ajou

Authors
  1. Aliaa Burqan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rania Saadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmad Qazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmad El-Ajou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

AB wrote the introduction, abstract, conclusion and preliminary sections, RS prepared sections 3 and 4, AQ used Mathematica software to compute the symbolic and numerical quantities, and AEl-Ajou reviewed all the manuscript sections.

Corresponding author

Correspondence to Ahmad El-Ajou.

Ethics declarations

Conflict of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burqan, A., Saadeh, R., Qazza, A. et al. Trustworthy Analytical Technique for Generating Multiple Solutions to Fractional Boundary Value Problems. Int. J. Appl. Comput. Math 9, 89 (2023). https://doi.org/10.1007/s40819-023-01554-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-023-01554-y

Keywords

Search

Navigation