Fractional Decomposition Method Research Articles

Convergence analysis for the fractional decomposition method applied to class of nonlinear fractional Fredholm integro-differential equation

For scientists conducting research, fractional integral differential equation analysis is crucial. Therefore, in this study, we investigate analysis utilizing a novel method called the fractional decomposition method, which is applicable to fractional nonlinear fractional Fredholm integro-differential equations. Then, we apply the approach to five test problems for a general fractional derivative [Formula: see text] involving fractional Fredholm integro-differential equations. To the best of our knowledge, we are the first to ever do so because of the very complicated calculations involved when dealing with the general case [Formula: see text]. For fractional Fredholm integral-differential equations, we provide both exact and approximate solutions. Throughout this work, the fractional Caputo derivative is discussed. This technique leads us to say that the method is precise, accurate, and efficient, according to the theoretical analysis.

Convergence analysis of the fractional decomposition method with applications to time‐fractional biological population models

AbstractIn this study, we present convergence analysis along with an error estimate for time‐fractional biological population equation in terms of the Caputo derivative using a new technique called the fractional decomposition method (FDM). Further, we present exact solutions to four test problems of nonlinear time‐fractional biological population models to show the accuracy and efficiency of the FDM. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without the need of linearization, discretization and perturbations. The results prove that the FDM is very effective and simple for solving fractional partial differential equations in multi‐dimensional spaces, special cases of which we have described in this paper.

Novel higher order convolutional perfectly matched layer based on complex envelope ADI‐FDTD algorithm for truncating the unmagnetized plasma

An improved higher order convolutional perfectly matched layer (HO-CPML) algorithm is proposed, which combines the complex envelope (CE) method and the alternating direction implicit (ADI) method. This algorithm is solved by the auxiliary differential equation method in the unmagnetized plasma, so as to simulate the unconditionally stable finite-difference-time-domain (FDTD). The novel HO-CPML with fractional decomposition method different from the previous one is adopted to truncate the unmagnetized plasma, making full use of the characteristics of the HO-CPML. The numerical example in plasma simulation shows the characteristics of the algorithm directly. According to the simulation results, the algorithm is effective, and the absorption effect is better than the conventional ADI (CL-ADI) algorithm and the first order CPML (FO-CPML) algorithm.

An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method

We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.

An efficient technique to solve coupled–time fractional Boussinesq–Burger equation using fractional decomposition method

For this work, a novel numerical approach is proposed to obtain solution for the class of coupled time-fractional Boussinesq–Burger equations which is a nonlinear system. This system under consideration is endowed with Caputo time-fractional derivative. By means of the natural decomposition approach, approximate solutions of the proposed nonlinear fractional system are obtained where the exact solutions are presented in the classical case of fractional order at [Formula: see text]. Some numerical examples are given to support the theoretical framework and to point out the role and the effectiveness of the intended method. Our results clearly show the approximate analytical solutions eventually will converge quickly to the already known exact solutions. AMS Classification: 35A22, 35C05, 35C10, 35R11, 44A30.

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.

On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operators

In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equations, homogeneous or nonhomogeneous. The operators are taken in the local fractional sense. Some examples are given to demonstrate the simplicity and the efficiency of the presented methods.

Analytical Approximate Solutions of Fractional Convection-Diffusion Equation by Means of Local Fractional Derivative Operators

In this article, the local fractional decomposition method ( LFDM) is applied to obtain approximate the analytical solution of nonlinear fractional convection-dif fusion. Numerical solutions obtained by local fractional decomposition method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. A new application of loca l fractional decomposition method (LFDM) was extended to reproduce the analytical solutions to this equati on in the form of a series. It is shown that the solutions obtained by the LFDM are reliable, simple and t hat LFDM is an effective method for strongly nonlinear partial equations.

Fractal boundary value problems for integral and differential equations with local fractional operators

In the present paper we investigate the fractal boundary value problems for the Fredholm\Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results.

Local Fractional Derivative Boundary Value Problems for Tricomi Equation Arising in Fractal Transonic Flow

The local fractional decomposition method is applied to obtain the nondifferentiable numerical solutions for the local fractional Tricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions.

Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

Modified fractional decomposition method having integral w.r.t [formula omitted

In this paper, a novel analytical method is proposed for differential equations with time-fractional derivative. This method is based on the famous Adomian decomposition method and the modified Riemann-Liouville derivative. The fractional derivatives are considered in the Jumarie sense. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome this shortcoming, a fractional decomposition method is proposed with modified Riemann-Liouville derivative. This method is a more efficient approach to solve the fractional differential equations. Several illustrative examples are given to demonstrate the effectiveness of the present method.