Local fractional Laplace homotopy analysis method for solving non-differentiable wave equations on Cantor sets
- PDF / 2,084,450 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 106 Downloads / 218 Views
(2019) 38:65
Local fractional Laplace homotopy analysis method for solving non-differentiable wave equations on Cantor sets Shehu Maitama1
· Weidong Zhao1
Received: 10 October 2018 / Revised: 14 February 2019 / Accepted: 23 February 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract 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. Keywords 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
1 Introduction Historically, more than two hundred years many problems in mathematical biology, plasma physics, analytical chemistry, finance, quantum mechanics, and many other applications in science and engineering were formulated using the fractional calculus (Losada and Nieto 2015; Caputo and Fabrizio 2015; Atangana and Baleanu 2016; Atangana and Koca 2016; Algahtani 2016; Raja et al. 2015, 2016; Atangana 2016; Mandelbrot and Van Ness 1968;
Communicated by José Tenreiro Machado.
B
Shehu Maitama [email protected] Weidong Zhao [email protected]
1
School of Mathematics, Shandong University, Jinan 250100, Shandong, People’s Republic of China
123
65
Page 2 of 22
S. Maitama, W. Zhao
Atangana and Gómez-Aguilar 2017; Jumarie 2001, 2005a, b, 2009; Baleanu et al. 2010) The origin of fractional calculus was dated back to the work of the German mathematician Gottfried Wilhelm Leibniz in 1695 (see Oldham and Spanier 1974). The theory of fractional calculus can easily be used to study the memory effects of dynamic systems, and have the embedded efficiency to described these systems in the best way. Unfortunately, the concept of classical fractional calculus cannot be used to study some continuous dynamical systems with highly irregular surfaces and curves (Kolwankar and Gangal 1998). These dynamics systems are continuous but nowhere differentiable and arise naturally in many fields of physical science and engineering (Kolwankar and Gangal 1997). To overcome the limitations of classical fractional calculus, the concept of local fractional calculus was introduced by Kolwankar and Gangal (1996). The l
Data Loading...
