Local fractional Laplace homotopy analysis method for solving non-differentiable wave equations on Cantor sets
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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.
KeywordsLocal fractional Laplace homotopy analysis method Local fractional wave equations Local fractional Laplace transform Homotopy analysis method Numeric and symbolic computations
Mathematics Subject Classification34K50 34A12 34A30 45A05 44A05 44A20
Funding was provided by China Scholarship Council (2017GXZ025381), National Natural Science Foundation of China (11571206).
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Conflict of interest
The authors declare that they have no conflict of interest.
- Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Progr Fract Differ Appl 2:731–85Google Scholar
- Golmankhaneh AK, Yang XJ, Baleanu D (2015) Einsten field equations within local fractional calculus. Rom J Phys 60:22–31Google Scholar
- 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
- 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
- Losada J, Nieto JJ (2015) Properties of the new fractional derivative without singular kernel. Progr Fract Differ Appl 2(1):87–92Google Scholar
- Yang XJ (2011a) Local fractional fucntional analysis and its applications. Asian Academic, Hong KongGoogle Scholar
- 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
- Yang XJ (2012) Advance local fractional calculus and its applications. World Science Publisher, New YorkGoogle Scholar
- 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
- Yang XJ, Srivastava HM, He JH, Baleanu D (2013b) Cantor-type cylindrical-coordinate fractional derivatives. Proc Rom Acad Ser A 14:127–133Google Scholar
- 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