Abstract
A mathematical model of fractal waves on shallow water surfaces is developed by using the concepts of local fractional calculus. The derivations of linear and nonlinear local fractional versions of the Korteweg-de Vries equation describing fractal waves on shallow water surfaces are obtained.
Highlights
The mathematical model of shallow water waves, conceived by Boussinesq [1], was rediscovered by Korteweg and de Vries [2]. It is commonly known as Korteweg-de Vries equation (KdV) [2, 3] and is given by
Several versions of the KdV equations found in the literature are listed below
There are certain nondifferentiable physical quantities describing the physical parameters locally, where the concept of differentiable functions is not applicable. In such cases the local fractional calculus (LFC) concept allows to obtain solutions adequate to such nondifferentiable problems [17,18,19,20,21,22,23,24,25] such as local fractional Helmholtz and diffusion equations [19], local fractional Navier-Stokes equations in fractal domain [21], local fractional Poisson and Laplace equations arising in the electrostatics in fractal domain [23], fractional models in forest gap [24], inhomogeneous local fractional wave equations [25], local fractional heat conduction equation [26], and other results [26,27,28,29,30]
Summary
The mathematical model of shallow water waves, conceived by Boussinesq [1], was rediscovered by Korteweg and de Vries [2]. There are certain nondifferentiable physical quantities describing the physical parameters locally, where the concept of differentiable functions is not applicable In such cases the local fractional calculus (LFC) concept allows to obtain solutions adequate to such nondifferentiable problems [17,18,19,20,21,22,23,24,25] such as local fractional Helmholtz and diffusion equations [19], local fractional Navier-Stokes equations in fractal domain [21], local fractional Poisson and Laplace equations arising in the electrostatics in fractal domain [23], fractional models in forest gap [24], inhomogeneous local fractional wave equations [25], local fractional heat conduction equation [26], and other results [26,27,28,29,30]. We focus on the derivation of the linear and the nonlinear local fractional versions of the Korteweg-de Vries equation describing fractal waves on shallow water surfaces.
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