Exact Travelling Wave Solutions for Local Fractional Partial Differential Equations in Mathematical Physics

  • Xiao-Jun YangEmail author
  • Feng Gao
  • J. A. Tenreiro Machado
  • Dumitru Baleanu
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 24)


In the article, we investigate the exact travelling wave solutions for the linear and nonlinear local fractional partial differential equations. The non-differential exact solutions of the fractal diffusion, Korteweg-de Vries, and Boussinesq equations via local fractional derivative are discussed in detail. The local fractional calculus formulations are efficient in description of fractal and complex behaviors of the linear and nonlinear mathematical physics.


Exact Traveling Wave Solutions Local Fractional Derivative (LFD) Boussinesq Equations Fractional Diffusion Nonlinear Mathematical Physics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work is supported by the State Key Research Development Program of the People’s Republic of China (Grant No.2016YFC0600705), the Natural Science Foundation of China (Grant No.51323004), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD2014).


  1. 1.
    Yang, X.J.: Local Fractional Functional Analysis & Its Applications. Asian Academic Publisher Limited, Hong Kong (2011)Google Scholar
  2. 2.
    Yang, X.J.: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012)Google Scholar
  3. 3.
    Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Academic, Amsterdam (2015)zbMATHGoogle Scholar
  4. 4.
    Cattani, C., Srivastava, H.M., Yang, X.J.: Fractional Dynamics. De Gruyter Open, Berlin (2015)CrossRefGoogle Scholar
  5. 5.
    Sarikaya, M., Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145(4), 1527–1538 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Akkurt, A., Sarikaya, M.Z., Budak, H., Yildirim, H.: Generalized Ostrowski type integral inequalities involving generalized moments via local fractional integrals. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas 111(3), 797–807 (2017)Google Scholar
  7. 7.
    Choi, J., Set, E., Tomar, M.: Certain generalized Ostrowski type inequalities for local fractional integrals. Commun. Kor. Math. Soc. 32(3), 601–617 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erden, S., Sarikaya, M.Z.: Generalized Pompeiu type inequalities for local fractional integrals and its applications. Appl. Math. Comput. 274, 282–291(2016)MathSciNetGoogle Scholar
  9. 9.
    Tunç, T., Sarıkaya, M.Z., Srivastava, H.M.: Some generalized Steffensen’s inequalities via a new identity for local fractional integrals. Int. J. Anal. Appl. 13(1), 98–107 (2016)Google Scholar
  10. 10.
    Liu, Q., Sun, W.: A Hilbert-type fractal integral inequality and its applications. J. Inequal. Appl. 2017(1), 83 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mo, H., Sui, X.: Hermite–Hadamard-type inequalities for generalized s-convex functions on real linear fractal set \({\mathrm {R}}^\alpha \left ( {0<\alpha <1} \right )\). Math. Sci. 11(3), 241–246 (2017)Google Scholar
  12. 12.
    Vivas, M., Hernández, J., Merentes, N.: New Hermite-Hadamard and Jensen type inequalities for h-convex functions on fractal sets. Revista Colombiana de Matemáticas 50(2), 145–164 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Saleh, W., Kiliçman, A.: On generalized s-convex functions on fractal sets. JP J. Geom. Topol. 17(1), 63 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kiliçman, A., Saleh, W.: Notions of generalized s-convex functions on fractal sets. J. Inequal. Appl. 2015(1), 312 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kilicman, A., Saleh, W.: On some inequalities for generalized s-convex functions and applications on fractal sets. J. Nonlinear Sci. Appl. 10(2), 583–594 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yang, X.J., Machado, J.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200–206 (2017)CrossRefGoogle Scholar
  17. 17.
    Zhao, X.H., Zhang, Y., Zhao, D., Yang, X.J.: The RC circuit described by local fractional differential equations. Fundamenta Informaticae 151(1–4), 419–429 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yang, X.J., Gao, F., Srivastava, H.M.: New rheological models within local fractional derivative. Rom. Rep. Phys. 69, 113 (2017)Google Scholar
  19. 19.
    Yang, X.J., Tenreiro Machado, J.A., Baleanu, D., Cattani, C.: On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. Chaos Interdisciplinary J. Nonlinear Sci. 26(8), 084312 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Baleanu, D., Khan, H., Jafari, H., Khan, R.A.: On the exact solution of wave equations on cantor sets. Entropy 17(9), 6229–6237 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Debbouche, A., Antonov, V.: Finite-dimensional diffusion models of heat transfer in fractal mediums involving local fractional derivatives. Nonlinear Stud. 24(3), 527–535 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jafari, H., Jassim, H.K., Tchier, F., Baleanu, D.: On the approximate solutions of local fractional differential equations with local fractional operators. Entropy 18(4), 150 (2016)CrossRefGoogle Scholar
  23. 23.
    Yang, X.J., Machado, J.A., Nieto, J.J.: A new family of the local fractional PDEs. Fundamenta Informaticae 151(1–4), 63–75 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yang, X.J., Machado, J.T., Hristov, J.: Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow. Nonlinear Dyn. 84(1), 3–7 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yang, X.J., Machado, J.T., Baleanu, D.: On exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(4) 1740006 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang, X.J., Gao, F., Srivastava, H.M.: Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Comput. Math. Appl. 73(2), 203–210 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yang, X.J., Gasimov, Y.S., Gao, F., Allahverdiyeva, N.: Travelling-wave solutions for Klein-Gordon and Helmholtz equations on Cantor sets. Proc. Inst. Math. Mech. 43(1), 123–131 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yang, X.J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. (2017). zbMATHGoogle Scholar
  29. 29.
    Griffiths, G., Schiesser, W.E.: Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Methods with MATLAB and Maple. Academic, New York (2010)zbMATHGoogle Scholar
  30. 30.
    Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Springer, Berlin/Heidelberg (2010)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Xiao-Jun Yang
    • 1
    • 2
    Email author
  • Feng Gao
    • 1
    • 2
  • J. A. Tenreiro Machado
    • 3
  • Dumitru Baleanu
    • 4
  1. 1.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.School of Mechanics and Civil EngineeringChina University of Mining and TechnologyXuzhouChina
  3. 3.Instituto Superior de Engenharia do PortoPortoPortugal
  4. 4.Department of MathematicsÇankaya UniversityAnkaraTurkey

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