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Adaptation of reproducing kernel method in solving Atangana–Baleanu fractional Bratu model

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Abstract

In this analysis, a reliable computational technique is implemented for the solution of fractional-order initial and boundary value problems of Bratu-type equations, called reproducing kernel method. The model under consideration is endowed with Atangana–Baleanu fractional derivative. The proposed technique is developed to create an exact solution in a convergent series formula in terms of the Atangana–Baleanu operator with easily calculable components. By means of reproducing kernel theory, an iterative operational algorithm is built for dealing with the fractional Bratu-type equations. Furthermore, the convergence and stability analysis of the method is discussed. To these aims, meaningful numerical examples are included to demonstrate the feasibility and reliability of this technique and to test the qualitative effect of using the Atangana–Baleanu operator on the quality of the acquired accounts. Also, a numerical comparison is made with some common numerical methods. From a numerical viewpoint, the gained results clearly show a high level of accuracy and great technical skills of the proposed method of dealing with such types of nonlinear fractional equations.

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References

  1. Zaslavsky GM (2005) Hamiltonian chaos and fractional dynamics. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Qureshi S, Yusuf A (2019) Modeling chickenpox disease with fractional derivatives: from Caputo to Atangana–Baleanu. Chaos Solitons Fractals 122:111–118

    MATH  Google Scholar 

  3. Qureshi S, Memon ZUN (2020) Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan. Chaos Solitons Fractals 131:109478

    MATH  Google Scholar 

  4. Yusuf A, Qureshi S, Inc M, Aliyu AI, Baleanu D, Shaikh AA (2018) Two-strain epidemic model involving fractional derivative with Mittag–Leffler kernel. Chaos Interdiscip J Nonlinear Sci 28(12):123121

    MATH  Google Scholar 

  5. Qureshi S, Yusuf A (2019) Fractional derivatives applied to MSEIR problems: comparative study with real world data. Eur Phys J Plus 134(4):171

    Google Scholar 

  6. Khan H, Gómez-Aguilar JF, Abdeljawad T, Khan A (2020) Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation. Fractals 28(8):2040048

    MATH  Google Scholar 

  7. Abdeljawad T, Atangana A, Gómez-Aguilar JF, Jarad F (2019) On a more general fractional integration by parts formulae and applications. Phys A Stat Mech Appl 536:122494

    MATH  Google Scholar 

  8. Al-Smadi M, Abu Arqub O, Momani S (2020) Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense. Phys Scr 95(7):075218

    Google Scholar 

  9. Al-Smadi M, Freihat A, Khalil H, Momani S, Khan RA (2017) Numerical multistep approach for solving fractional partial differential equations. Int J Comput Methods 14(3):1750029

    MATH  Google Scholar 

  10. El-Ajou A (2020) A modification to the conformable fractional calculus with some applications. Alex Eng J 59(4):2239–2249

    Google Scholar 

  11. El-Ajou A, Al-Zhour Z (2021) A vector series solution for a class of hyperbolic system of Caputo time-fractional partial differential equations with variable coefficients. Front Phys 9:525250

    Google Scholar 

  12. Atangana A, Baleanu D (2016) New fractional derivatives with non-local and non-sin- gular kernel: theory and application to heat transfer model. Thermal Sci 20:763–9

    Google Scholar 

  13. Khan A, Khan H, Gómez-Aguilar JF, Abdeljawad T (2019) Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 127:422–427

    MATH  Google Scholar 

  14. Atangana A, Gómez-Aguilar JF (2018) Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114:516–35

    MATH  Google Scholar 

  15. Bedi P, Kumar A, Abdeljawad T, Khan A, Gómez-Aguilar JF (2021) Mild solutions of coupled hybrid fractional order system with Caputo–Hadamard derivatives. Fractals 29(6):2150158

    MATH  Google Scholar 

  16. Khan H, Abdeljawad T, Gómez-Aguilar JF, Tajadodi H, Khan A (2021) Fractional order Volterra integro-differential equation with Mittag–Leffler kernel. Fractals 29(6):2150154

    MATH  Google Scholar 

  17. Martínez-Fuentes O, Meléndez-Vázquez F, Fernández-Anaya G, Gómez-Aguilar JF (2021) Analysis of fractional-order nonlinear dynamic systems with general analytic kernels: Lyapunov stability and inequalities. Mathematics 9(17):2084

    Google Scholar 

  18. Asma Gómez-Aguilar JF, Rahman G, Javed M (2021) Stability analysis for fractional order implicit Y-Hilfer differential equations. Math Methods Appl Sci 45(5):2701–2712

    Google Scholar 

  19. Abro KA, Laghari MH, Gómez-Aguilar JF (2020) Application of Atangana–Baleanu fractional derivative to carbon nanotubes based non-Newtonian nanofluid: applications in nanotechnology. J Appl Comput Mech 6:1260–1269

    Google Scholar 

  20. Dubey VP, Kumar R, Kumar D (2019) Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration. Physica A 521:762–772

    MATH  Google Scholar 

  21. Singh J, Kumar D, Swroop R, Kumar S (2018) An efficient computational approach for time-fractional Rosenau–Hyman equation. Neural Comput Appl 30:3063–3070

    Google Scholar 

  22. Ascher UM, Matheij R, Russell RD (1995) Numerical solution of boundary value problems for ordinary differential equations. SIAM, Philadelphia

    Google Scholar 

  23. Chandrasekhar S (1967) Introduction to the study of stellar structure. Dover, New York

    MATH  Google Scholar 

  24. Wan YQ, Guo Q, Pan N (2004) Thermo-electro-hydrodynamic model for electrospinning process. Int J Nonlinear Sci Numer Simul 5:5–8

    Google Scholar 

  25. Caglara H, Caglarb N, Özerc M, Valarıstosd A, Anagnostopoulose AN (2010) B-spline method for solving Bratu’s problem. Int J Comput Math 87:1885–1891

    Google Scholar 

  26. Sakar MG, Saldır O, Akgül A (2018) Numerical solution of fractional Bratu type equations with legendre reproducing kernel method. Int J Appl Comput Math 4:126

    MATH  Google Scholar 

  27. Singh H, Ghassabzadeh FA, Tohidi E, Cattani C (2020) Legendre spectral method for the fractional Bratu problem. Math Meth Appl Sci 43(9):5941–5952

    MATH  Google Scholar 

  28. Abbasbandy S, Hashemi MS, Liu CS (2011) The Lie-group shooting method for solving the Bratu equation. Commun Nonlinear Sci Numer Simul 16:4238–4249

    MATH  Google Scholar 

  29. Deeba E, Khuri SA, Xie S (2000) An algorithm for solving boundary value problems. J Comput Phys 159:125–138

    MATH  Google Scholar 

  30. Khuri SA (2004) A new approach to Bratu’s problem. Appl Math Comput 147:131–136

    MATH  Google Scholar 

  31. Pirmohabbati P, Sheikhani AHR, Ziabari AA (2020) Numerical solution of nonlinear fractional Bratu equation with hybrid method. Int J Appl Comput Math 6:162

    MATH  Google Scholar 

  32. Aksoy Y, Pakdemirli M (2010) New perturbation-iteration solutions for Bratu-type equations. Comput Math Appl 59:2802–2808

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  34. Bergman S (1950) The kernel function and conformal mapping. American Mathematical Society, New York

    MATH  Google Scholar 

  35. Cui M, Lin Y (2009) Nonlinear numerical analysis in the reproducing kernel space. Nova Science Publishers Inc., New York

    MATH  Google Scholar 

  36. Al-Smadi M, Abu Arqub O, 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

    MATH  Google Scholar 

  37. Momani S, Djeddi N, Al-Smadi M, Al-Omari S (2021) Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method. Appl Numer Math 170:418–434

    MATH  Google Scholar 

  38. Hasan S, El-Ajou A, Hadid S, Al-Smadi M, Momani S (2020) Atangana–Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. Chaos Solitons Fractals 133:109624

    MATH  Google Scholar 

  39. Djeddi N, Hasan S, Al-Smadi M, Momani S (2020) Modified analytical approach for generalized quadratic and cubic logistic models with Caputo–Fabrizio fractional derivative. Alex Eng J 59(6):5111–5122

    Google Scholar 

  40. Hasan S, Al-Smadi M, Freihet A, Momani S (2019) Two computational approaches for solving a fractional obstacle system in Hilbert space. Adv Differ Equ 2019:55

    MATH  Google Scholar 

  41. Al-Smadi M, Abu Arqub O, Gaith M (2021) Numerical simulation of telegraph and Cattaneo fractional-type models using adaptive reproducing kernel framework. Math Methods Appl Sci 44(10):8472–8489

    MATH  Google Scholar 

  42. Al-Smadi M, Djeddi N, Momani S, Al-Omari S, Araci S (2021) An attractive numerical algorithm for solving nonlinear Caputo-Fabrizio fractional Abel differential equation in a Hilbert space. Adv Differ Equ 2021:271

    MATH  Google Scholar 

  43. Al-Smadi M (2018) Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation. Ain Shams Eng J 9(4):2517–2525

    Google Scholar 

  44. Hasan S, Al-Smadi M, Dutta H, Momani S, Hadid S (2022) Multi-step reproducing kernel algorithm for solving Caputo-Fabrizio fractional stiff models arising in electric circuits. Soft Comput 26:3713–3727

    Google Scholar 

  45. Al-Smadi M, Abu Arqub O, Momani S (2013) A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations. Math Probl Eng 2013:832074

    MATH  Google Scholar 

  46. Al-Smadi M, Abu Arqub O (2019) Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates. Appl Math Comput 342:280–294

    MATH  Google Scholar 

  47. Hasan S, Al-Smadi M, El-Ajou A, Momani S, Hadid S, Al-Zhour Z (2021) Numerical approach in the Hilbert space to solve a fuzzy Atangana–Baleanu fractional hybrid system. Chaos Solitons Fractals 143:110506

    Google Scholar 

  48. Wu BY, Li XY (2010) Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl Math Lett 23:1301–1304

    MATH  Google Scholar 

  49. Akgül A, Inc M, Karatas E (2015) Reproducing kernel functions for difference equations. Discrete Contin Dyn Syst Ser-S 8:1055–64

    MATH  Google Scholar 

  50. Vahidi RA, Hasanzade M (2012) Restarted Adomian’s decomposition method for the Bratu-type problem. Appl Math Sci 6(10):479–486

    MATH  Google Scholar 

  51. Ghomanjani F, Shateyi S (2017) Numerical solution for fractional Bratu’s initial value problem. Open Phys 15(1):1045–1048

    Google Scholar 

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Author information

Authors and Affiliations

  1. College of Commerce and Business, Lusail University, Lusail, Qatar

    Mohammed Al-Smadi

  2. Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

    Shaher Momani & Nadir Djeddi

  3. Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan

    Shaher Momani & Nadir Djeddi

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

    Ahmad El-Ajou

  5. Department of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam, 31441, Saudi Arabia

    Zeyad Al-Zhour

Authors
  1. Mohammed Al-Smadi
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  2. Shaher Momani
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  3. Nadir Djeddi
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  4. Ahmad El-Ajou
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  5. Zeyad Al-Zhour
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Correspondence to Zeyad Al-Zhour.

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Al-Smadi, M., Momani, S., Djeddi, N. et al. Adaptation of reproducing kernel method in solving Atangana–Baleanu fractional Bratu model. Int. J. Dynam. Control 11, 136–148 (2023). https://doi.org/10.1007/s40435-022-00961-1

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  • DOI: https://doi.org/10.1007/s40435-022-00961-1

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