Exact Solutions of Fractional Partial Differential Equations by Sumudu Transform Iterative Method

  • Manoj Kumar
  • Varsha Daftardar-GejjiEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


Developing analytical methods for solving fractional partial differential equations (FPDEs) is an active area of research. Especially, finding exact solutions of FPDEs is a challenging task. In the present chapter, we extend Sumudu transform iterative method to solve a variety of time and space FPDEs as well as systems of them. We demonstrate the utility of the method by finding exact solutions to a large number of FPDEs.


  1. 1.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, New York (1998)Google Scholar
  2. 2.
    Debnath, L., Bhatta, D.D.: Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics. Fract. Calc. Appl. Anal. 7(1), 21–36 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994)CrossRefGoogle Scholar
  4. 4.
    He, J.-H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3), 257–262 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Daftardar-Gejji, V., Jafari, H.: An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316(2), 753–763 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bhalekar, S., Patade, J.: An analytical solution of fishers equation using decomposition method. Am. J. Comput. Appl. Math. 6(3), 123–127 (2016)zbMATHGoogle Scholar
  7. 7.
    AL-Jawary, M.A., Radhi, G.H., Ravnik, J.: Daftardar-Jafari method for solving nonlinear thin film flow problem. Arab. J. Basic Appl. Sci. 25(1), 20–27 (2018)CrossRefGoogle Scholar
  8. 8.
    Jafari, H.: Numerical solution of time-fractional Klein-Gordon equation by using the decomposition methods. J. Comput. Nonlinear Dyn. 11(4), 041015 (2016)CrossRefGoogle Scholar
  9. 9.
    Jafari, H., Nazari, M., Baleanu, D., Khalique, C.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66(5), 838–843 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Singh, J., Devendra, S.: Homotopy perturbation sumudu transform method for nonlinear equations. Adv. Theor. Appl. Mech 4(4), 165–175 (2011)zbMATHGoogle Scholar
  11. 11.
    Kumar, D., Singh, J., Rathore, S.: Sumudu decomposition method for nonlinear equations. Int. Math. Forum 7, 515–521 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Wang, K., Liu, S.: A new sumudu transform iterative method for time-fractional cauchy reaction-diffusion equation. SpringerPlus 5(1), 865 (2016)CrossRefGoogle Scholar
  13. 13.
    Prakash, A., Kumar, M., Baleanu, D.: A new iterative technique for a fractional model of nonlinear Zakharov-Kuznetsov equations via sumudu transform. Appl. Math. Comput. 334, 30–40 (2018)MathSciNetGoogle Scholar
  14. 14.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)Google Scholar
  15. 15.
    Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)Google Scholar
  16. 16.
    Bonilla, B., Rivero, M., Rodríguez-Germá, L., Trujillo, J.J.: Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 187(1), 79–88 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Belgacem, F.B.M., Karaballi, A.A.: Sumudu transform fundamental properties investigations and applications. Int. J. Stoch. Anal. (2006)Google Scholar
  18. 18.
    Amer, Y., Mahdy, A., Youssef, E.: Solving systems of fractional nonlinear equations of Emden Fowler type by using sumudu transform method. Glob. J. Pure Appl. Math. 14(1), 91–113 (2018)Google Scholar
  19. 19.
    Bhalekar, S., Daftardar-Gejji, V.: Convergence of the new iterative method. Int. J. Differ. Equ. (2011)Google Scholar
  20. 20.
    Choudhary, S., Daftardar-Gejji, V.: Invariant subspace method: a tool for solving fractional partial differential equations. Fract. Calc. Appl. Anal. 20(2), 477–493 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Choudhary, S., Daftardar-Gejji, V.: Solving systems of multi-term fractional PDEs: Invariant subspace approach. Int. J. Model. Simul. Sci. Comput. 10(1) (2019)CrossRefGoogle Scholar
  22. 22.
    Sahadevan, R., Prakash, P.: Exact solutions and maximal dimension of invariant subspaces of time fractional coupled nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 42, 158–177 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSavitribai Phule Pune UniversityPuneIndia
  2. 2.National Defence AcademyKhadakwasala, PuneIndia

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