Exact Solution for Nonlinear Local Fractional Partial Differential Equations

Document Type: Research Paper


1 Laboratory of Mathematics and its Applications (LAMAP), University of Oran1, Oran, Algeria

2 Department of Mathematics, Cankaya University, Ankara, Turkey

3 Institute of Space Sciences, Magurele-Bucharest, Romania


In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this algorithm, and therefore, it can be applied to similar examples.


Main Subjects

[1] Adomian, G., Solution of physical problems by decomposition, Comp. Math. App, 27, 1994, 145-154.

[2] Adomian, G., Solutions of nonlinear P.D.E, Appl. Math. Lett., 11, 1998, 121-123.

[3] Sadeghinia, A., Kumar, P., One Solution of Multi-term Fractional Differential Equations by Adomian Decomposition Method, Int. J. Sci. Inno. Math. Res., 3(6), 2015, 14-21.

[4] Jafari, H., Daftardar-Gejji, V., Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comp. Appl. Math., 196, 2006, 644-651.

[5] Saha Ray, S., Bera, R.K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comp., 167, 2005, 561-571.

[6] Ali, U., Kamal, R., Mohyud-Din, S.T., On Nonlinear Fractional Differential Equations, Int. J. Mod. Math. Sci, 3(3), 2012, 116-124.

[7] Hamdi Cherif, M., Ziane, D., A New Numerical Technique for Solving Systems of Nonlinear fractional Partial Differential Equations, Int. J. Anal. Appl., 15(2), 2017, 188-197.

[8] Liu, Y., General Solutions of Space Fractional Fisher's Nonlinear Diffusion Equation, J. Frac. Cal. Appl., 1(2), 2011, 1-8.

[9] Gao, F., Yang, X.J., Local Fractional Euler's Method for the Steady Heat-Conduction Problem, Ther. Sci., 20(3), 2016, 735-738.

[10] Jafari, H., Jassim, H.K., Tchier, F., Baleanu, D., On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operators, Entropy, 18, 2016, 1-12.

[11] Yang, C.Y., Zhang, Y.D., Yang, X.J., Exact Solutions for the Differential Equations in Fractal Heat Transfer, Ther. Sci., 20(3), 2016, 747-750.

[12] Srivastava, H.M., Golmankhaneh, A.K., Baleanu, D., Yang, X.J., Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets, Abs. Appl. Anal., 2014, ID 176395, 1-7.

[13] Yang, X.J., Baleanu, D., Lazarevi, M.P., Caji, M.S., Fractal Boundary Value Problems for Integral and Differential Equations with Local Fractional Operators, Ther. Sci., 19(3), 2015, 959-966.

[14] Yang, X.J., Srivastava, H.M., Cattani, C., Local Fractional Homotopy Perturbation Method for Solving Fractal Partial Differential Equations Arising in Mathematical Physics, Romanian Repo. in Phys, 67(3), 2015, 752-761.

[15] Yang, X.J., Baleanu, D., Local fractional variational iteration method for Fokker-Planck equation on a Cantor set, Acta Universitaria, 23(2), 2013, 3-8.

[16] Ye, S.S., Mohyud-Din, S.T., Belgacem, F.B-M., Laplace Transform Series Expansion method for Solving the Local Fractional Heat-Transfer Equation Defined on Cantor Sets, Ther. Sci., 20(3), 2016, 867-870.

[17] Ziane, D., Baleanu, D., Belghaba, K., Hamdi Cherif, M., Local fractional Sumudu decomposition method for linear partia ldifferential equations with local fractional derivative, J. King Saud Univ-Sci, 31(1), 2019, 83-88.

[18] Ziane, D., Local Fractional Sumudu Variational Iteration Method for Solving Partial Differential Equations with Local Fractional Derivative, Int. J. Open Prob. Compt. Math., 10(3), 2017, 29-42.

[19] Ahmad, J., Mohyud-Din, S.T., Yang, X.J., Applications of Local Fractional Adomian Decomposition Method to Integral Equations, J. Sci. Arts, 1(26), 2014, 73-82.

[20] Neamah, A.A., Local Fractional Variational Iteration Method for Solving Volterra Integro-Differential Equations Within Local Fractional Operators, J. Math. Stat, 10(3), 2014, 401-407.

[21] Yang, X.J., Local Fractional Integral Equations and Their Applications, Adv. Comp. Sci. Appl, 1(4), 2012, 234-239.

[22] Jafari, H., Jassim, H.K., Mohyud-Din, S.T., Local Fractional Laplace Decomposition Method for Solving Linear Partial Differential Equations with Local Fractional Derivative, Sciendo, 1, 2015, 287-305.

[23] Yang, X.J., Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.

[24] Yang, X.J., Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.

[25] Zhu, Y., Chang, Q., Wu, S., A new algorithm for calculating Adomian polynomials, Appl. Math. Comput., 169, 2005, 402-416.

[26] Singh, J., Kumar, D., A. Kilicman., Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform, Abs. Appl. Anal., 2013, ID 934060, 1-8.

[27] Ziane, D., The combined of Homotopy analysis method with new transform for nonlinear partial differential equations, Mal. J. Math., 6(1), 2018, 34-40.

[28] Rajaraman, R., Hariharan, G., Kannan, K., Homotopy Perturbation Transform method for solving Klein-Gordan equations, Int. J. Eme. Tre. & Tech. Comp. Sci., 1(4), 2012, 150-154.