Fractional Sumudu Research Articles

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

Conformable Triple Sumudu Transform with Applications

One of the important topics in applied mathematics is the topic of integral transformations, due to their importance in electrical engineering applications, including communications in particular, and other sciences. In this work, one of the most important transformations in its three dimensions was presented, which is the triple Sumudu transform, including solving some real-life applications of physics, some of which have not been solved using such an integral transform before. In this work, we extend the Sumudu transform formula to the conformable fractional order, as well as other interesting and significant rules. The general analytical solution of a singular and nonlinear conformable fractional differential equation based on the conformable fractional Sumudu transform is also presented in this paper. The general solutions of several linear and nonhomogeneous conformable fractional differential equations can be obtained using the method we’ve proposed. As a result, our results reveal that our proposed method is an efficient one that can be used for solving all conformable fractional differential equations. The relationship between the Sumudu integral transform and other important and recently proposed integral transforms are also discussed. Finally, the triple Sumudu transform is used to solve boundary value problems, such as the heat equation with boundary values. The triple Sumudu integral transform is also used to solve linear partial integro-differential equations. The transform capability to handle such equations has been proven via its utilization in three applications.

LOCAL FRACTIONAL SUMUDU DECOMPOSITION METHOD TO SOLVE FRACTAL PDEs ARISING IN MATHEMATICAL PHYSICS

In this paper, we investigate solutions of telegraph, Laplace and wave equations within the local fractional derivative operator by using local fractional Sumudu decomposition method. This method is coupled by the Sumudu transform and decomposition 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 presented method.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Homotopy Analysis Fractional Sumudu Transform Method for Semi-analytic Solution of 2-D Time-Fractional Black–Scholes Equations

Homotopy Analysis Fractional Sumudu Transform Method for Semi-analytic Solution of 2-D Time-Fractional Black–Scholes Equations

Exact solution for local fractional Diffusion and Wave Equations on Cantor Sets

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.

Time-Fractional Differential Equations with an Approximate Solution

This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with the Caputo fractional operator (CF) to solve time fractional linear and nonlinear partial differential equations. The Sumudu transform (ST) and the homotopy perturbation technique (HP) are combined in this approach. In the Caputo definition, the fractional derivative is defined. In general, the method is straightforward to execute and yields good results. There are some examples offered to demonstrate the technique's validity and use.

Computational Study of a Local Fractional Tricomi Equation Occurring in Fractal Transonic Flow

Abstract In this paper, we present the application of local fractional methods in combination with the local fractional Sumudu transform (LFST) for a local fractional Tricomi equation (LFTE). The numerical simulations for obtained results are presented for the local fractional Tricomi equation with different initial conditions on the Cantor set. The computational approach shows that the implemented methods are very impressive to derive solutions for a local fractional Tricomi equation. Moreover, the solutions obtained by using these schemes are in quite good agreement with already computed solutions in the literature.

An efficient analytical scheme with convergence analysis for computational study of local fractional Schrödinger equations

In this paper, we present a newly proposed local fractional method pertaining to the local fractional Sumudu transform (LFST) for computational study of local fractional Schrödinger’s equations (LFSEs). The error analysis for the present method is also discussed here. The uniqueness and convergence analyses for the solution obtained by using the proposed scheme are also established. The numerical simulations for achieved results have been performed for different orders of a local fractional derivative. The results depict that the proposed method efficiently provides the solution for given equations in a smooth manner.

A comparative analysis of two computational schemes for solving local fractional Laplace equations

In this paper, we implement the semi‐analytical schemes, namely, local fractional homotopy perturbation Sumudu transform method (LFHPSTM) and local fractional homotopy analysis Sumudu transform method (LFHASTM), for finding the approximate analytical solutions of local fractional Laplace equations under different initial conditions on Cantor sets. The novelty of the work lies in the application of these suggested schemes which were never used to solve the local fractional Laplace equation. The Laplace equation mainly contributes to electrostatistics, mechanical engineering, and theoretical physics. This equation also helps in formulation of the electrostatic problem of a rod under a torsion load. Moreover, it helps in handling potential field problems also. These aspects enhance the significance of getting a solution of the local fractional Laplace equation in fractal media. The suggested methods are copulation of local fractional homotopy perturbation method and local fractional homotopy analysis method with the local fractional Sumudu transform, respectively. The computational process indicates that the schemes are very efficient to obtain nondifferentiable solutions for given equations in a smooth way. Moreover, the solution analysis depicts that both the local fractional hybrid homotopy schemes are in a good agreement with each other. These methods depict the accuracy and efficiency of the implemented methods in view of correspondence with solutions obtained with other methods in previous works. The numerical simulations for obtained results are discussed for various values of order of a local fractional derivative. The 3D graphs on the Cantor set are constructed with the help of Matlab software.

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

The basic motivation of the present study is to apply the local fractional Sumudu variational iteration method (LFSVIM) for solving nonlinear wave-like equations with variable coefficients and within local fractional derivatives. The derivatives operators are taken in the local fractional sense. The results show that the LFSVIM is an appropriate method to find non-differentiable solutions for similar problems. The results of the solved examples showed the flexibility of applying this method and its ability to reach accurate results even with these new differential equations.

A Modification Fractional Homotopy Analysis Method for Solving Partial Differential Equations Arising in Mathematical Physics

In this paper, we apply a new technique, namely fractional Sumudu homotopy analysis method (FSHAM) on fractional partial differential equations to obtain the analytical approximate solutions. The fractional derivative is described in the Caputo sense. This method is the combination of the Sumudu transform (ST) and homotopy analysis method (HAM). 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.

One dimensional fractional frequency Sumudu transform by inverse α−difference operator

In this paper, we define fractional frequency Sumudu transform by inverse α−difference operator. Here we present certain new results on Sumudu transform of polynomial factorial,trigonometric and geometric functions using shift value. Finally, we provide the relation between convolution product and fractional Sumudu transform of polynomial and exponential function.Numerical results are verified and analysed the outcomes by graphs.

Exact Solution for Nonlinear Local Fractional Partial Differential Equations

In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this algorithm, and therefore, it can be applied to similar examples.

Time-Fractional Nonlinear Dispersive Type of the Zakharov–Kuznetsov Equation via HAFSTM

In this paper, a semi-analytic solution of time-fractional nonlinear dispersive type of the Zakharov–Kuznetsov (ZK) equations is obtained by the means of homotopy analysis fractional Sumudu transform method. Several examples are evaluated and compared with exact solutions to show the effectiveness of arbitrary order of ZK equations. The results demonstrate that the method is highly reliable and convenient to apply and evaluate without restrictions like finite domain, finite discretization or perturbation of small and large physical parameters.

Fractional Sumudu Decomposition Method for Solving PDEs of Fractional Order

. In this paper, the fractional Sumudu decomposition method (FSDM) is employed to handle the time-fractional PDEs and system of time-fractional PDEs. The fractional derivative is described in the Caputo sense. The approximate solutions are obtained by using FSDM, which is the coupling method of fractional decomposition method and Sumudu transform. The method, in general, is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

New Results on the Conformable Fractional Sumudu Transform: Theories and Applications

In this paper, we generalize the formula of Sumudu transform to the conformable fractional order and some interesting and important rules of this transform and conformable fractional Laplace transform are derived and discussed. Moreover, we present the general analytical solution of a singular and nonlinear conformable fractional Poisson- Boltzmann differential equation based on the conformable fractional Sumudu transform. Also, our proposed method is applied successfully for obtaining the general solutions of some linear and nonhomogeneous conformable fractional differential equations. Finally, the results show that our proposed method is an efficient and can be applied for finding the general solutions of the all cases realted to the conformable fractional differential equations.

An efficient hybrid computational technique for solving nonlinear local fractional partial differential equations arising in fractal media

AbstractIn present work, nonlinear fractional partial differential equations namely transport equation and Fokker-Planck equation involving local fractional differential operators, are investigated by means of the local fractional homotopy perturbation Sumudu transform method. The proposed method is a coupling of homotopy perturbation method with local fractional Sumudu transform and is used to describe the non-differentiable problems. Numerical simulation results are projected to show the efficiency of the proposed technique.

Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs

In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives.

Local Fractional Sumudu Variational Iteration Method for Solving Partial Differential Equations with Local Fractional Derivative

Local Fractional Sumudu Variational Iteration Method for Solving Partial Differential Equations with Local Fractional Derivative