A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform (original) (raw)

Novel Investigation of Fractional-Order Cauchy-Reaction Diffusion Equation Involving Caputo-Fabrizio Operator

Journal of Function Spaces, 2022

In this article, the new iterative transform technique and homotopy perturbation transform method are applied to calculate the fractional-order Cauchy-reaction diffusion equation solution. Yang transformation is mixed with the new iteration method and homotopy perturbation method in these methods. The fractional derivative is considered in the sense of Caputo-Fabrizio operator. The convection-diffusion models arise in physical phenomena in which energy, particles, or other physical properties are transferred within a physical process via two processes: diffusion and convection. Four problems are evaluated to demonstrate, show, and verify the present methods’ efficiency. The analytically obtained results by the present method suggest that the method is accurate and simple to implement.

New homotopy analysis transform method for solving multidimensional fractional diffusion equations

Arab Journal of Basic and Applied Sciences, 2020

In this paper, we introduce a new semi-analytical method called the homotopy analysis Shehu transform method (HASTM) for solving multidimensional fractional diffusion equations. The proposed technique is a combination of the homotopy analysis method and the Laplace-type integral transform called the Shehu transform which is a generalization of the Laplace and the Sumudu integral transforms. Shehu transform is user-friendly, and its visual-ization is easier than the Sumudu, and the natural transforms. The convergence analysis of the method is proved, and we provide some applications of the fractional diffusion equations to validate the efficiency and the high accuracy of the technique. The results obtained using the HASTM are in complete agreement with the results of the existing techniques.

ANALYTICAL SOLUTIONS FOR TIME-FRACTIONAL CAUCHY REACTION-DIFFUSION EQUATIONS USING ITERATIVE LAPLACE TRANSFORM METHOD

In the present work, the iterative Laplace transform method (ILTM) is implemented to derive approximate analytical solutions for the time-fractional Cauchy reaction-diffusion equations (CRDEs) within the Caputo fractional derivative. The proposed technique is an elegant amalgam of the Iterative method and the Laplace transform method. The ILTM produces the solution in a rapid convergent series which may lead to the solution in a closed form. The obtained analytical outcomes with the help of the proposed technique are examined graphically.

An Effective New Iterative Method to Solve Conformable Cauchy Reaction-Diffusion Equation via the Shehu Transform

Journal of Mathematics

For the first time, we establish a new procedure by using the conformable Shehu transform (CST) and an iteration method for solving fractional-order Cauchy reaction-diffusion equations (CRDEs) in the sense of conformable derivative (CD). We call this recommended method the conformable Shehu transform iterative method (CSTIM). To evaluate the efficacy and consistency of CSTIM for conformable partial differential equations (PDEs), the absolute errors of four CRDEs are reviewed graphically and numerically. Furthermore, graphical significances are correspondingly predicted for several values of fractional-order derivatives. The results and examples establish that our new method is unpretentious, accurate, valid, and capable. CSTIM does not necessarily use He’s polynomials and Adomian polynomials when solving nonlinear problems, so it has a strong advantage over the homotopy analysis and Adomian decomposition methods. The convergence and absolute error analysis of the series solutions is...

HOMOTOPY PERTURBATION SHEHU TRANSFORM METHOD FOR SOLVING FRACTIONAL MODELS ARISING IN APPLIED SCIENCES

Journal of Applied Mathematics and Computational Mechanics, 2021

Using the recently proposed homotopy perturbation Shehu transform method (HPSTM), we successfully construct reliable solutions of some important fractional models arising in applied physical sciences. The nonlinear terms are decomposed using He's poly-nomials, and the fractional derivative is calculated in the Caputo sense. Using the analytical method, we obtained the exact solution of the fractional diffusion equation, fractional wave equation and the nonlinear fractional gas dynamic equation. MSC 2010: 34K50, 34A12, 34A30, 45A05, 44A05, 44A20

A fractional model of the diffusion equation and its analytical solution using Laplace transform

SCIENTIA IRANICA, 2012

In this study, the homotopy perturbation transform method (HPTM) is performed to give analytical solutions of the time fractional diffusion equation. The HPTM is a combined form of the Laplace transform and homotopy perturbation methods. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is very effective and simple in performing a solution to the fractional partial differential equation. A solution has been plotted for different values of α., and some numerical illustrations are given.

A Novel Analytical Approach for Solving Time-Fractional Diffusion Equations

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH

This article investigates the approximate analytical solutions of the time-fractional diffusion equations using a novel analytical approach, namely the Sumudu transform iterative method. The time-fractional derivatives are considered in the Caputo sense. The analytical solutions are found in closed form, in terms of Mittag-Leffler functions. Furthermore, the findings are shown graphically, and the solution graphs demonstrate a strong relationship between the approximate and exact solutions.

A New Analytical Method for Solving Linear and Nonlinear Fractional Partial Differential Equations

In this paper, a new analytical method called the Natural Homotopy Perturbation Method (NHPM) for solving linear and the nonlinear fractional partial differential equation is introduced. The proposed analytical method is an elegant combination of a well-known method, Homotopy Perturbation Method (HPM) and the Natural Transform Method (NTM). In this new analytical method, the fractional derivative is computed in Caputo sense and the nonlinear terms are calculated using He's polynomials. Exact solution of linear and nonlinear fractional partial differential equations are successfully obtained using the new analytical method, and the result is compared with the result of the existing methods.

Numerical treatment of fractional order Cauchy reaction diffusion equations

Chaos, Solitons & Fractals, 2017

In this manuscript, an approximate method for the numerical solutions of fractional order Cauchy reaction diffusion equations is considered. The concerned method is known as optimal homotopy asymptotic method (OHAM). With the help of the mentioned method, we handle approximate solutions to the aforesaid equation. Some test problems are provided at which the adapted technique has been applied. The comparison between absolute and exact solution are also provided which reveals that the adapted method is highly accurate. For tabulation and plotting, we use matlab software.