TRAVELING WAVE SOLUTION OF FRACTAL KdV-BURGERS–KURAMOTO EQUATION WITHIN LOCAL FRACTIONAL DIFFERENTIAL OPERATOR
About: This article is published in Fractals.The article was published on 2021-11-01. It has received 5 citations till now. The article focuses on the topics: Korteweg–de Vries equation & Fractal.
Citations
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TL;DR: In this paper , the local fractional variational iteration method is applied to a modified Fisher's equation defined on Cantor sets with the fractal conditions, and the solution process is simple and the accuracy of the approximate solution is high.
Abstract: The local fractional variational iteration method is applied to a modified
Fisher?s equation defined on Cantor sets with the fractal conditions. The
solution process is simple, and the accuracy of the approximate solution is
high. The method provides an unrivaled tool for local differential
equations. Key word: fractal Fisher?s equation, approximate analytical
solutions, local fractional variational iteration method, local fractional
derivative
2 citations
TL;DR: In this paper , the variational principles of the fractal variable coefficients and highly non-linear Schr?dinger equations are built successfully by coupling fractal semi-inverse and He?s two-scale transformation methods, which are helpful to reveal the symmetry and discover the conserved quantity.
Abstract: With the help of a new fractal derivative, a fractal model for variable
coefficients and highly non-linear Schr?dinger equations on a non-smooth
boundary are acquired. The variational principles of the fractal variable
coefficients and highly non-linear Schr?dinger equations are built
successfully by coupling fractal semi-inverse and He?s two-scale
transformation methods, which are helpful to reveal the symmetry, to
discover the conserved quantity, and the obtained variational principles
have widespread applications in numerical simulation.
1 citations
TL;DR: In this paper , a new fractal derivative, the nonlinear Boiti-Leon-Manna-Pempinelli equation (NBLMPE) with nonsmooth boundary is explored.
Abstract: With the aid of a new fractal derivative, the nonlinear Boiti–Leon–Manna–Pempinelli equation (NBLMPE) with nonsmooth boundary is explored. The variational principle of the fractal NBLMPE is successfully established by fractal wave transformation (FWT) and fractal semi-inverse method (SIM) and strong minimum condition of fractal NBLMPE is proven with the fractal Weierstrass theorem. Based on the two-scale transformation method (TSTM) and homogeneous equilibrium method (HBM), soliton-like solutions for the [Formula: see text]-dimensional (SLS [Formula: see text]D) fractal NBLMPE are acquired. A powerful means of coupling HBM and TSTM to solve fractal differential equations is proposed.
TL;DR: In this paper , a modification of the high-order long wave equation with unsmooth boundaries was considered by adopting a new fractal variational derivative, which is successfully constructed by the fractal semi-inverse method.
Abstract: In this article, we mainly consider a modification of the high-order long
water-wave equation with unsmooth boundaries by adopting a new fractal
derivative. Its fractal variational principles are successfully constructed
by the fractal semi-inverse method, the obtained principles are helpful to
study the symmetry, to discover the conserved quantity, and to have wide
applications in numerical simulation.
TL;DR: In this paper , a fractal-fractional model of the impact stress on the crusher drum is established by using He?s fractal derivative and the fluid-solid coupling vibration equation.
Abstract: In this paper, a fractal-fractional model of the impact stress on the crusher
drum is established by using He?s fractal derivative and the fluid-solid
coupling vibration equation. The two-scale transform is used to obtain its
solution, which can be used to improve the safety performance of beating
machines.
References
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TL;DR: Fractional complex transform as mentioned in this paper was proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus.
Abstract: Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.
278 citations
TL;DR: Local fractional differential equations (LDFDE) as mentioned in this paper is a new class of differential equations, which involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time.
Abstract: We propose a new class of differential equations, which we call local fractional differential equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of the Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. We solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise.
275 citations
TL;DR: In this paper, a review of the variational approach, the Hamiltonian approach, variational iteration method, the homotopy perturbation method, parameter expansion method, Yang-Laplace transform, and Yang-Fourier transform is presented.
Abstract: This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.
239 citations
TL;DR: The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.
Abstract: In this paper, a family of local fractional two-dimensional Burgers-type equations (2DBEs) is investigated. The local fractional Riccati differential equation method is proposed here for the first time. The travelling wave transformation of the non-differentiable type is presented. The non-differentiable exact travelling wave solutions for the problems are obtained. The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.
232 citations
TL;DR: In this article, a systematic perturbation method is applied to three-dimensional long waves on a viscous liquid film, and the nonlinear evolution equation incorporating the effects of dissipation and dispersion is derived.
Abstract: A systematic perturbation method is applied to three-dimensional long waves on a viscous liquid film, and the nonlinear evolution equation incorporating the effects of dissipation and dispersion is derived. It is shown that both the fourth-order derivative term as well as the three-dimensionality have stabilizing effects.
203 citations