An accurate method for nonlinear local fractional Wave-Like equations with variable coefficients

We discuss the line partial differential equations arising in fractal vehicular traffic flow. The nondifferentiable approximate solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. The obtained results show the efficiency and accuracy of implements of the present method.

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

In this article, an analytical solution procedure is described for solving one dimensional second order hyperbolic telegraph equation using a reliable semi-analytic method so called the local fractional variational iteration method (LFVIM) subject to the appropriate initial conditions. Various illustrate examples are carried out to check the accuracy, efficiency, and convergence of the described method.

Recently the local fractional operators have started to be considered a useful tool to deal with fractal functions defined on Cantor sets. In this paper, we consider the Fokker-Planck equation on a Cantor set derived from the fractional complex transform method. Additionally, the solution obtained is considered by using the local fractional variational iteration method.

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

In this paper, we apply a new technique, namely local fractional variational iteration transform method on homogeneous/non-homogeneous non-linear gas dynamic and coupled KdV equations to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative and integral operators. This method is the combination of the local fractional Laplace transform and variational iteration method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Entropy plays an important role in the simulation of traffic flow distribution. This paper studies the entropy condition for the Lighthill–Whitham–Richards model of the fractal traffic flows described by local fractional calculus. We also discuss the solutions of non‐differentiability with graphs by using the local fractional variational iteration method. Copyright © 2015 John Wiley & Sons, Ltd.

A novel approach to solving nonlinear fractional wave-like differential equations

This paper uses the differential transform method (DTM) to obtain analytical solutions of fractional heat- and wave-like equations with variable coefficients. The time fractional heat-like and wave-like equations with variable coefficients were obtained by replacing a first-order and a second-order time derivative by a fractional derivative of order . The approach mainly rests on the DTM which is one of the approximate methods. The method can easily be applied to many problems and is capable of reducing the size of computational work. Some examples are presented to show the efficiency and simplicity of the method.

In this paper, the local fractional homotopy perturbation method and the local fractional Sumudu transform are used to study diffusion and wave equations defined on Cantor sets with the fractal conditions with local fractional derivatives. The LFSHPM analytical method minimizes the computational size and may be applied directly to fractional differential equations without any linearization, discretization of variables, transformation, or restrictive assumptions. It provides series solutions that converge quickly in a few iterations. The proposed analytical method is successfully applied to diffusion and wave equations defined on cantor sets with fractal conditions, and proved to be highly efficient and computational accurate.
 Abstract: To solve the diffusion and wave equations on the Cantor set, the local fractional Sumudu homotopy perturbation method (LFSHPM) is used. In the local sense, the operators are used. The local fractional Sumudu homotopy perturbation method, which is the coupling method of the local fractional homotopy perturbation method and the Sumudu transform, is used to obtain non-differentiable approximate solutions. Illustrative examples are provided to show the new algorithm's excellent accuracy and fast convergence.

Variational iteration method for fractional heat- and wave-like equations

In this paper, we propose a copulation of the local fractional variational iteration technique and local fractional natural transform (LFNT) for solving the dissipative and damped wave equations with local fractional derivatives. The convergence analysis for the solution obtained through this newly suggested method is also discussed. The numerical simulations for computed solutions are presented on the Cantor set. The computational procedure shows that the newly proposed local fractional method pertaining to the LFNT is very easy and effective to generate solutions for given local fractional wave equations (LFWEs). Moreover, the generated solutions are also in nice correspondence with the previous results acquired by other existing techniques.

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

Yang-Laplace transforms is an alternative approach to nonlinear fractional equations with local fractional derivative and local fractional integral. This paper presents a new wave equation with local fractional derivative. Finally, by using the Yang-Laplace transforms, its solution to nonlinear fractional wave equation is investigated in detail.

<p>The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.</p>