Local fractional Laplace homotopy analysis method for solving non-differentiable wave equations on Cantor sets

  • Shehu MaitamaEmail author
  • Weidong Zhao


In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy analysis method (LFLHAM) for solving wave equations with local fractional derivatives. The LFLHAM is based on the homotopy analysis method and the local fractional Laplace transform method, respectively. The proposed analytical method was a modification of the homotopy analysis method and converged rapidly within a few iterations. The nonzero convergence-control parameter was used to adjust the convergence of the series solutions. Three examples of non-differentiable wave equations were provided to demonstrate the efficiency and the high accuracy of the proposed technique. The results obtained were completely in agreement with the results in the existing methods and their qualitative and quantitative comparison of the results.


Local fractional Laplace homotopy analysis method Local fractional wave equations Local fractional Laplace transform Homotopy analysis method Numeric and symbolic computations 

Mathematics Subject Classification

34K50 34A12 34A30 45A05 44A05 44A20 



Funding was provided by China Scholarship Council (2017GXZ025381), National Natural Science Foundation of China (11571206).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Algahtani OJJ (2016) Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fract 89:552–559MathSciNetCrossRefzbMATHGoogle Scholar
  2. Atangana A (2016) On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation. Appl Math Comput 273:948–956MathSciNetzbMATHGoogle Scholar
  3. Atangana A, Baleanu D (2016) New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm Sci 20(2):763–769CrossRefGoogle Scholar
  4. Atangana A, Gómez-Aguilar JF (2017) Numerical approximation of Riemann–Liouville definition of fractional derivative: from Riemann–Liouville to Atangana–Baleanu. Numer Methods Partial Differ Equ. CrossRefzbMATHGoogle Scholar
  5. Atangana A, Koca I (2016) Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos Solitons Fract 89:447–454MathSciNetCrossRefzbMATHGoogle Scholar
  6. Baleanu D, Güvenc ZB, Tenreiro, Machado JA (2010) New trends in nanotechnology and fractional calculuc applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 2:731–85Google Scholar
  8. Chen Y, Yan Y, Zhang K (2010) On the local fractional derivative. J Math Anal Appl 362:17–33MathSciNetCrossRefzbMATHGoogle Scholar
  9. Golmankhaneh AK, Yang XJ, Baleanu D (2015) Einsten field equations within local fractional calculus. Rom J Phys 60:22–31Google Scholar
  10. Hemeda AA, Eladdad EE, Lairje IA (2018) Local fractional analytical methods for solving wave equations with local fractional derivative. Math Methods Appl Sci. MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hu M-S, Ravi PA, Yang XJ (2012) Local fractional Fourier series with applications to wave equation in fractal vibrating string. Abstr Appl Anal 2012:1–15 (article ID: 567401) MathSciNetGoogle Scholar
  12. Jafari H, Kamil HJ (2015) Local fractional variational iteration method for solving nonlinear partial differential equations within local fractional operators. Appl Appl Math Int J 10(2):1055–1065MathSciNetzbMATHGoogle Scholar
  13. Jafari H, Tajadodi H, Johnston SJ (2015a) A decomposition method for solving diffusion equationa via local fractional time derivative. Therm Sci 19(1):S123–S129CrossRefGoogle Scholar
  14. Jafari H, Ünlü C, Moshoa SP, Khalique CM (2015b) Local fractional Laplace variational iteration method for solving diffusion and wave equations on Cantor sets within local fractional operators. Entropy 2015:1–9 (article ID: 309870) Google Scholar
  15. Jassim HK (2015) Local fractional Laplace decomposition method for nonhomogeneous heat equation arising in fractal heat flow with local fractional derivative. Int J Adv Appl Math Mech 2:1–7MathSciNetzbMATHGoogle Scholar
  16. Jumarie G (2001) Fractional master equation: non-standard analysis and Liouville–Riemann derivative. Chaos Solitons Fract 12:2577–2587MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jumarie G (2005a) On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. Appl Math Lett 18:817–826MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jumarie G (2005b) On the representation of fractional Brownian motion as an integral with respect to \((dt)^a\). Appl Math Lett 18:739–748MathSciNetCrossRefzbMATHGoogle Scholar
  19. Jumarie G (2009) Laplace’s transform of fractional order via Mittag–Leffler function and modified Riemann–Liouville derivative. Appl Math Lett 22:1659–1664MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kolwankar KM, Gangal AD (1996) Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 6:505–513MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kolwankar KM, Gangal AD (1997) Hölder exponents of irregular signals and local fractional derivatives. Pramana J Phys 48:49–68CrossRefGoogle Scholar
  22. Kolwankar KM, Gangal AD (1998) Local fractional Fokker–Planck equation. Phys Rev Lett 80:214–217MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kumar D, Singh J, Mehmet HB, Bulut H (2017a) An effictive computational approach to local fractional telegraph equations. Nonlinear Sci Lett A 8(2):200–206Google Scholar
  24. Kumar D, Singh J, Baleanu D (2017b) A hybrid computational approach for Klein–Gordon equations on Canto sets. Nonlinnear Dyn 87:511–517CrossRefzbMATHGoogle Scholar
  25. Liao SJ (1995) An approximate solution technique not depending on small parameters: a special example. Int J Non-Linear Mech 30:371–380MathSciNetCrossRefzbMATHGoogle Scholar
  26. Liao SJ (2003) Beyond perturbation: introduction to the homotopy analysis method. CRC Press, Boca RatonCrossRefGoogle Scholar
  27. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 169(2):1186–1194MathSciNetzbMATHGoogle Scholar
  28. Liao SJ (2010) An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul 362:2003–2016MathSciNetCrossRefzbMATHGoogle Scholar
  29. Liu K, Hu RJ, Cattani C, Xie GN, Yang XJ, Zhao Y (2014) Local fractional Z-transforms with applications to signals on Cantor sets. Abstr Appl Anal 2013:1–6 (article ID: 638648) MathSciNetzbMATHGoogle Scholar
  30. Losada J, Nieto JJ (2015) Properties of the new fractional derivative without singular kernel. Progr Fract Differ Appl 2(1):87–92Google Scholar
  31. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422–437MathSciNetCrossRefzbMATHGoogle Scholar
  32. Oldham KB, Spanier J (1974) The fractional calculus. Acadamic Press, New YorkzbMATHGoogle Scholar
  33. Raja MAZ, Manzar MA, Samar R (2015) An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP. Appl Math Model 39(10–11):3075–3093MathSciNetCrossRefGoogle Scholar
  34. Raja MAZ, Samar R, Manzar MA, Shah SM (2016) Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley–Torvik equation. Math Compute Simul. CrossRefGoogle Scholar
  35. Singh J, Kumar D, Nieto JJ (2016) A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow. Entropy 18:1–8MathSciNetCrossRefGoogle Scholar
  36. Srivastava HM, Golmankhaneh AK, Baleanu D, Yang XJ (2014) Local fractional Sumudu transform with application to IVPs on Cantor sets. Abstr Appl Anal 2014:1–7 (article ID: 176395) MathSciNetzbMATHGoogle Scholar
  37. Wang SQ, Yang YJ, Kamil HJ (2014) Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative. Abstr Appl Anal 2014(2014):1–7MathSciNetzbMATHGoogle Scholar
  38. Yan SP, Jafari H, Jassim HK (2014) Local fractional Adomian decomposition and function decomposition methods for Laplace equation within local fractional operators. Adv Math Phys 2014(2014):1–8MathSciNetCrossRefzbMATHGoogle Scholar
  39. Yang XJ (2011a) Local fractional fucntional analysis and its applications. Asian Academic, Hong KongGoogle Scholar
  40. Yang XJ (2011b) Local fractional Laplace transform based on the local fractional calculus. In: Shen G, Huang X (eds) Advanced Research on computer science and information engineering (communications in computer and information science, vol 153. Springer, BerlinGoogle Scholar
  41. Yang XJ (2012) Advance local fractional calculus and its applications. World Science Publisher, New YorkGoogle Scholar
  42. Yang XJ, Kang Z, Liu C (2010) Local fractional Fourier’s transform based on local fractional calculus. In: The 2010 ICECE 2010. IEEE Computer Society, pp 1242–1245Google Scholar
  43. Yang XJ, Baleanu D, Zhong WP (2013a) Approximate solutions for diffusion equations on Cantor space-time. Proc Rom Acad Ser A 14:127–133MathSciNetGoogle Scholar
  44. Yang XJ, Srivastava HM, He JH, Baleanu D (2013b) Cantor-type cylindrical-coordinate fractional derivatives. Proc Rom Acad Ser A 14:127–133Google Scholar
  45. Yang XJ, Baleanu D, Yang XJ (2013c) A local fractional variational iteration method for Laplace equation within local fractional operators. Abstr Appl Anal 2013:1–6 (article ID: 202650) MathSciNetzbMATHGoogle Scholar
  46. Yang AM, Zhang YZ, Cattani C, Xie GN, Rashidi MM, Zhou YJ, Yang XJ (2014a) Application of local fractional series expansion method to solve Klein–Gordon equations on Cantor sets. Abstr Appl Anal 2014:1–7MathSciNetzbMATHGoogle Scholar
  47. Yang XJ, Hristov J, Srivastava HM, Ahmad B (2014b) Modelling fractal waves on shallow water surfaces via local fractional Korteweg–de Vries equation. Abstr Appl Anal 2013:1–10 (article ID: 278672) MathSciNetzbMATHGoogle Scholar
  48. Yang AM, Li J, Srivastava HM, Xie GN, Yang XJ (2014c) Local fractional variational iteration method for solving linear partial differential equation with local fractional derivative. Discrete Dyn Nat Soc 2014:1–8 (article ID: 365981) MathSciNetzbMATHGoogle Scholar
  49. Yang XJ, Srivastava HM, Cattani C (2015a) Local fractional homotopy perturbation method for solving fractional partial differential equations arising in mathematical physics. Rom Rep Phys 67:752–761Google Scholar
  50. Yang XJ, Baleanu D, Srivastava HM (2015b) Local fractional integral transforms and their applications. Academic Press, New YorkzbMATHGoogle Scholar
  51. Yang XJ, Tenreiro JAM, Baleanu D, Gao F (2016a) A new numerical technique for local fractional diffusion equation in fractal heat transfer. J Nonlinear Sci Appl 9:5621–5628MathSciNetCrossRefzbMATHGoogle Scholar
  52. Yang XJ, Machado JT, Baleanu D, Cattani C (2016b) On exact traveling-wave solutions for local fractional Korteweg–de Vries equation. Chaos Interdiscip J Nonlinear Sci 26(8):084312MathSciNetCrossRefzbMATHGoogle Scholar
  53. Yang XJ, Machado JA, Hristov J (2016c) Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow. Nonlinear Dyn 84(1):3–7MathSciNetCrossRefzbMATHGoogle Scholar
  54. Yang XJ, Machado JT, Cattani C, Gao F (2017a) On a fractal LC-electric circuit modeled by local fractional calculus. Commun Nonlinear Sci Numer Simul 47:200–206CrossRefGoogle Scholar
  55. Yang XJ, Machado JAT, Baleanu D (2017b) Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(4):1740006, 1-7MathSciNetGoogle Scholar
  56. Yang XJ, Gao F, Srivastava HM (2017c) Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Comput Math Appl 73(2):203–210MathSciNetCrossRefzbMATHGoogle Scholar
  57. Yang XJ, Machado JA, Nieto JJ (2017d) A new family of the local fractional PDEs. Fundam Inform 151(1–4):63–75MathSciNetCrossRefzbMATHGoogle Scholar
  58. Yang XJ, Gao F, Srivastava HM (2018) A new computational approach for solving nonlinear local fractional PDEs. J Comput Appli Math 339:285–296MathSciNetCrossRefzbMATHGoogle Scholar
  59. Zhang Y, Cattani C, Yang XJ (2015) Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains. Entropy 17:6753–6764MathSciNetCrossRefGoogle Scholar
  60. Zhao D, Singh J, Kumar D, Rathore S, Yang XJ (2017) An efficient computational technique for local fractional heat conduction equation in fractal media. J Nonlinear Sci Appl 10:1478–1486MathSciNetCrossRefzbMATHGoogle Scholar
  61. Ziane D, Baleanu D, Belghaba K, Cherif M (2017) Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative. J King Saud Univ Sci. CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanPeople’s Republic of China

Personalised recommendations