We propose a new definition of a fractional-order Sumudu transform for fractional differentiable functions. In the development of the definition we use fractional analysis based on the modified Riemann-Liouville derivative that we name the fractional Sumudu transform. We also established a relationship between fractional Laplace and Sumudu duality with complex inversion formula for fractional Sumudu transform and apply new definition to solve fractional differential equations.
In this paper, we introduce the Maxwell equations of time-fractional order in lossy media. We derive the solution of these equations by using Sumudu transform techniques.
In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.
In the present paper, an efficient approach based on homotopy perturbation method by using natural transform is adopted to solve some linear and non-linear space-time fractional Fokker-Planck Equations (FPE) in closed form. The space and time fractional derivatives are considered in Caputo sense. The homotopy perturbation natural transform method is a combined form of natural transform, homotopy perturbation method and He's polynomials. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical result shows the effectiveness and good accuracy of the method.
The differential form method is applied to study the wave equation with dissipation and extended to determine potential symmetries. The symmetry group are given and group invariant solutions associated to the symmetries are obtained.
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