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The Laplace Optimized Decomposition Method for Solving Systems of Partial Differential Equations of Fractional Order

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Abstract

In this paper, a new hybrid technique Laplace optimized decomposition method (LODM) had been proposed to construct the approximate solution of the nonlinear system of fractional partial differential equations (FPDEs) for the fractional derivative in the Caputo sense. LODM is a combination of the Laplace transform and optimized decomposition method. This technique is based on the linear approximation of the nonlinear system of FPDEs. Numerical examples are presented to show the accuracy and reliability of LODM on a class of nonlinear problems. Moreover, the results illustrate a strong agreement between the approximate and the exact solutions.

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References

  1. Abdou, M.A.: An analytical method for space-time fractional nonlinear differential equations arising in plasma physics. J. Ocean Eng. Sci. 2(4), 288–292 (2017)

    Article  Google Scholar 

  2. Goswami, A., Singh, J., Kumar, D., Gupta, S.: An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma. J. Ocean Eng. Sci. 4(2), 85–99 (2019)

    Article  Google Scholar 

  3. Tamboli, V.K., Tandel, P.V.: Solution of the time-fractional generalized Burger–Fisher equation using the fractional reduced differential transform method. J. Ocean Eng. Sci. 6, 66 (2021)

    Google Scholar 

  4. Islam, M.T., Akbar, M.A., Gómez-Aguilar, J.F., Bonyah, E., Fernandez-Anaya, G.: Assorted soliton structures of solutions for fractional nonlinear Schrodinger types evolution equations. J. Ocean Eng. Sci. 6, 66 (2021)

    Google Scholar 

  5. Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T., Bates, J.H.T.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)

    Article  MathSciNet  Google Scholar 

  6. Zubair, T., Usman, M., Ali, U., Mohyud-Din, S.T.: Homotopy analysis method for system of partial differential equations. Int. J. Mod. Eng. Sci. 1(2), 67–79 (2012)

    Google Scholar 

  7. Elsaid, A.: Homotopy analysis method for solving a class of fractional partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(9), 3655–3664 (2011)

    Article  MathSciNet  Google Scholar 

  8. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys. Lett. A 361(6), 478–483 (2007)

    Article  Google Scholar 

  9. Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Approximate analytical solutions of systems of PDEs by homotopy analysis method. Comput. Math. Appl. 55(12), 2913–2923 (2008)

    Article  MathSciNet  Google Scholar 

  10. Vahidi, J.: The combined Laplace-homotopy analysis method for partial differential equations. J. Math. Comput. Sci. 16, 88–102 (2016)

    Article  Google Scholar 

  11. Morales-Delgado, V.F., Gómez-Aguilar, J.F., Yépez-Martínez, H., Baleanu, D., Eskobar-Jimenez, R.F., Olivares-Peregrino, V.H.: Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv. Differ. Equ. 2016(1), 1–17 (2016)

    Article  MathSciNet  Google Scholar 

  12. El-Sayed, A.M., Elsaid, A., El-Kalla, I.L., Hammad, D.: A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains. Appl. Math. Comput. 218(17), 8329–8340 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365(5–6), 345–350 (2007)

    Article  MathSciNet  Google Scholar 

  14. Wang, Q.: Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Solitons Fract. 35(5), 843–850 (2008)

    Article  MathSciNet  Google Scholar 

  15. Jafari, H., Nazari, M., Baleanu, D., Khalique, C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66(5), 838–843 (2013)

    Article  MathSciNet  Google Scholar 

  16. Odibat, Z., Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 32(1), 28–39 (2008)

    Article  Google Scholar 

  17. Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7(1), 27–34 (2006)

    Article  MathSciNet  Google Scholar 

  18. Inc, M.: The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345(1), 476–484 (2008)

    Article  MathSciNet  Google Scholar 

  19. Abassy, T.A., El-Tawil, M.A., El-Zoheiry, H.: Modified variational iteration method for Boussinesq equation. Comput. Math. Appl. 54(7–8), 955–965 (2007)

    Article  MathSciNet  Google Scholar 

  20. Haq, I.U., Ullah, Z.: Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations. Results Nonlinear Anal. 3(1), 35–44 (2020)

    Google Scholar 

  21. Rawashdeh, M.S., Maitama, S.: Solving coupled system of nonlinear PDE’s using the natural decomposition method. Int. J. Pure Appl. Math. 92(5), 757–776 (2014)

    Article  Google Scholar 

  22. Momani, S., Arqub, O.A., Maayah, B.: Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana–Baleanu–Caputo model of the Lienard’s equation. Fractals (2020). https://doi.org/10.1142/S0218348X20400071

  23. Maayah, B., Yousef, F., Arqub, O.A., Momani, S., Alsaedi, A.: Computing bifurcations behavior of mixed type singular time-fractional partial integrodifferential equations of Dirichlet functions types in Hilbert space with error analysis. Filomat 33(12), 3845–3853 (2019)

    Article  MathSciNet  Google Scholar 

  24. Arqub, O.A., Hayat, T., Alhodaly, M.: Reproducing kernel Hilbert pointwise numerical solvability of fractional Sine–Gordon model in time-dependent variable with Dirichlet condition. Phys. Scr. (2021). https://doi.org/10.1088/1402-4896/ac0c58

    Article  Google Scholar 

  25. Arqub, O.A.: 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 (2017)

    Article  MathSciNet  Google Scholar 

  26. Arqub, O.A., Shawagfeh, N.: Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media. J. Porous Media 22(4), 66 (2019)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Arqub, O.A.: The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math. Methods Appl. Sci. 39(15), 4549–4562 (2016)

    Article  MathSciNet  Google Scholar 

  29. Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations. Comput. Math. Appl. 21(5), 101–127 (1991)

    Article  MathSciNet  Google Scholar 

  30. Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135(2), 501–544 (1988)

    Article  MathSciNet  Google Scholar 

  31. Wang, Q.: Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Appl. Math. Comput. 182(2), 1048–1055 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Khan, H., Shah, R., Kumam, P., Baleanu, D., Arif, M.: Laplace decomposition for solving nonlinear system of fractional order partial differential equations. Adv. Differ. Equ. 2020(1), 1–18 (2020)

    Article  MathSciNet  Google Scholar 

  33. Shah, R., Khan, H., Arif, M., Kumam, P.: Application of Laplace–Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy 21(4), 335 (2019)

    Article  MathSciNet  Google Scholar 

  34. Jafari, H., Khalique, C.M., Nazari, M.: Application of the Laplace decomposition method for solving linear and nonlinear fractional diffussion-wave equations. Appl. Math. Lett. 24(11), 1799–1805 (2011)

    Article  MathSciNet  Google Scholar 

  35. Mahmood, S., Shah, R., Khan, H., Arif, M.: Laplace Adomian decomposition method for multi dimensional time fractional model of Navier–Stokes equation. Symmetry 11(2), 149 (2019)

    Article  Google Scholar 

  36. Mohammed, O.H., Salim, H.A.: Computational methods based Laplace decomposition for solving nonlinear system of fractional order differential equations. Alex. Eng. J. 57(4), 3549–3557 (2018)

    Article  Google Scholar 

  37. Mohamed, M.Z., Elzaki, T.M.: Comparison between the Laplace decomposition method and Adomian decomposition in time-space fractional nonlinear fractional differential equations. Appl. Math. 9(04), 448–458 (2019)

    Article  Google Scholar 

  38. Odibat, Z.: An optimized decomposition method for nonlinear ordinary and partial differentail equations. Phys. A Stat. Mech. Appl. 541, 66 (2020). https://doi.org/10.1016/j.physa.2019.123323

    Article  Google Scholar 

  39. Odibat, Z.: The optimized decomposition method for a reliable treatment of IVPs for second order differential equations. Phys. Scr. 6, 66 (2021)

    Google Scholar 

  40. Shah, R., Khan, H., Kumam, P., Arif, M.: An analytical technique to solve the system of nonlinear fractional partial differential equations. Mathematics 7(6), 505 (2019)

    Article  Google Scholar 

  41. Veeresha, P., Prakasha, D.G., Qurashi, M.A., Baleanu, D.: A reliable technique for fractional modified Boussinesq and approximate long wave equations. Adv. Differ. Equ. 2019(1), 1–23 (2019)

    Article  MathSciNet  Google Scholar 

  42. Wang, L., Chen, X.: Approximate analytial solutions of time fractional Whitham–Broer–Kaup equations by a residual power series method. Entropy 17(9), 6519–6533 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Wahiba Beghami & Banan Maayah

  2. Department of Basic Sciences, Princess Sumaya University for Technology, 11941, Amman, Jordan

    Samia Bushnaq

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

    Omar Abu Arqub

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  1. Wahiba Beghami
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  2. Banan Maayah
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  3. Samia Bushnaq
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  4. Omar Abu Arqub
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All authors equally contributed this paper and approved the final version.

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Correspondence to Samia Bushnaq.

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Beghami, W., Maayah, B., Bushnaq, S. et al. The Laplace Optimized Decomposition Method for Solving Systems of Partial Differential Equations of Fractional Order. Int. J. Appl. Comput. Math 8, 52 (2022). https://doi.org/10.1007/s40819-022-01256-x

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