Aboodh Transform Research Articles

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar–Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar–Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilizing ATDM.

Solution of Fishers equation using homotopy perturbation Aboodh transform method

In this study, a nonlinear one-dimensional reaction Diffusion Equation called Fisher’s equation is solved using Aboodh Transform coupled with HPM. This method is applied to the general one-dimensional Fishers equation. This technique is demonstrated with three unique examples with different initial conditions solved with the values of β=1 and a=0. This simplicity of Aboodh transform makes the rapid rate of convergence from approximate to exact solution answers makes it easy and dependable.

A comparative analytical investigation for some linear and nonlinear time-fractional partial differential equations in the framework of the Aboodh transformation

This article discusses two simple, complication-free, and effective methods for solving fractional-order linear and nonlinear partial differential equations analytically: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). The Caputo operator is utilized to define fractional order derivatives. In these methods, the analytical approximations are derived in series form. We calculate the first terms of the series and then estimate the absolute error resulting from leaving out the remaining terms to ensure the accuracy of the derived approximations and determine the accuracy and efficiency of the suggested methods. The derived approximations are discussed numerically using some values for the relevant parameters to the subject of the study. Useful examples are thought to illustrate the practical application of current approaches. We also examine the fractional order results that converge to the integer order solutions to ensure the accuracy of the derived approximations. Many researchers, particularly those in plasma physics, are anticipated to gain from modeling evolution equations describing nonlinear events in plasma systems.

Study of One Dimensional Groundwater Recharge Through Porous Media by Aboodh Transform Homotopy Perturbation Method

Study of One Dimensional Groundwater Recharge Through Porous Media by Aboodh Transform Homotopy Perturbation Method

The Aboodh Transform Techniques to Ulam Type Stability of Linear Delay Differential Equation

The Aboodh Transform Techniques to Ulam Type Stability of Linear Delay Differential Equation

Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach

In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The AT converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.

AN INTEGRAL TRANSFORM TOGETHER WITH TAYLOR SERIES AND DECOMPOSITION METHOD FOR THE SOLUTIONONLINEAR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER

This work aims to determine the approximate solutions of nonlinear boundary value problems of higher order obtained through the Aboodh Transform Series Decomposition Method (ATSDM), a method designed to find the integral and the inverse transform of the problems, expand the exponential function, and simultaneously decompose the nonlinear terms. The results obtained demonstrate that ATSDM is an excellent and trusted approximate method that can be employed to obtain accurate results for any problem similar to the one presented in this work.

An Integral Transform-Weighted Residual Method for Solving Second Order Linear Boundary Value Differential Equations with Semi-Infinite Domain

Solving boundary value problem with semi-infinite domain in a conventional way or using some of the available approximate methods poses a lot of challenges or better still seems almost impossible over the years, and some of the alternative ways for crossing this hurdle is by fixing value for infinity that is in the domain. Here in this work, we present an Integral Transform (Aboodh Transform) - Weighted Residual Based Method (AT-WRM) to address the afore-mentioned challenge. Aboodh Transform was used to transform and at the same time used to find the inverse of the given differential equations while Weighted Residual via Collocation Method was used in order to avoid fixing value for infinity as usual by introducing e-ix in trial function to decay the infinity that was part of the boundary condition. The accuracy of the presented method was authenticated by solving three different problems. The excellent results obtained from the three solved problems validate the accuracy and effectiveness of the method

Using Fuzzy Aboodh Transform to Solve A System of First Order Fuzzy Linear Differential Equations

In this paper, fuzzy Aboodh transform used to solve system of fuzzy linear differential equations of first order in dimension two. Moreover, support work with an applied example in one of the institution.

Local Fractional Aboodh Transform and its Applications to Solve Linear Local Fractional Differential Equations

In this work we focus on presenting a method for solving local fractional differential equations. This method based on the combination of the Aboodh transform with the local fractional derivative (we can call it local fractional Aboodh transform), where we have provided some important results and properties. We concluded this work by providing illustrative examples, through which we focused on solving some linear local fractional differential equations in order to obtain nondifferential analytical solutions.

General Solution of Telegraph Equation Using Aboodh Transform

This paper describes using the Aboodh integral transform in solving one-dimensional second-order hyperbolic telegraph equations with satisfying initial conditions by solving some telegraph equations with specific initial conditions and then generalizing the solution to the telegraph equation using the Aboodh integral transform.

Solution of Space‐Time Fractional Differential Equations Using Aboodh Transform Iterative Method

A relatively new and efficient approach based on a new iterative method and the Aboodh transform called the Aboodh transform iterative method is proposed to solve space‐time fractional differential equations, the fractional order is considered in the Caputo sense. This method is a combination of the Aboodh transform and the new iterative method and gives the solution in series form with easily computable components. The nonlinear term is easily handled by the new iterative method, to affirm the simplicity and performance of the proposed method, five examples were considered, and the solution plots were presented to show the effect of the fractional order. The outcome reveals that the approach is accurate and easy to implement.

Aboodh and Mahgoub Transform in Boundary Value Problems of System of Ordinary Differential Equations

Aboodh and Mahgoub Transform in Boundary Value Problems of System of Ordinary Differential Equations

Aboodh Transform Iterative Method for Solving Fractional Partial Differential Equation with Mittag–Leffler Kernel

The major aim of this paper is the presentation of Aboodh transform of the Atangana–Baleanu fractional differential operator both in Caputo and Riemann–Liouville sense by using the connection between the Laplace transform and the Aboodh transform. Moreover, we aim to obtain the approximate series solutions for the time-fractional differential equations with an Atangana–Baleanu fractional differential operator in the Caputo sense using the Aboodh transform iterative method, which is the modification of the Aboodh transform by combining it with the new iterative method. The relation between the Laplace transform and the Aboodh transform is symmetrical. Some graphical illustrations are presented to describe the effect of the fractional order. The outcome reveals that Aboodh transform iterative method is easy to implement and adequately captures the behavior and the fractional effect of the fractional differential equation.

Efficiency Algorithm for Solving Some Models of Nonlinear Problems

In this paper, a numerical algorithm proposed by composing the Aboodh transform (AT) and Adomian decomposition method (ADM) and it has been named (ATADM). This algorithm is tested for solving some models of nonlinear problems. The modification gives robust tool for large size of enumerations. The obtained results of comparing approximate solution with exact solution for this scheme appeared high accuracy and efficiency for solving nonlinear problems.

Aboodh Transform Iterative Method for Spatial Diffusion of a Biological Population with Fractional-Order

In this paper, a new approximate analytical method is proposed for solving the fractional biological population model, the fractional derivative is described in the Caputo sense. This method is based upon the Aboodh transform method and the new iterative method, the Aboodh transform is a modification of the Laplace transform. Illustrative cases are considered and the comparison between exact solutions and numerical solutions are considered for different values of alpha. Furthermore, the surface plots are provided in order to understand the effect of the fractional order. The advantage of this method is that it is efficient, precise, and easy to implement with less computational effort.

Application of the Aboodh Transform for Solving Fractional Delay Differential Equations

In this article, we extend the concept of the Aboodh transform to the solution of partial differential equations of fractional order using Caputo's fractional derivative. The transformation concerned is applicable to solve many classes of partial differential equations with derivatives of order and integrals. Consequently, the fractional Delay Differential Equations (DDEs) which we are going to study in this work. The resulting numerical proofs show that the method converges favorably towards the analytical solution.

The Analytical Solution of Telegraph Equation of Space-Fractional Order Derivative by the Aboodh Transform Method

In this article, an analytical solution based on the series expansion method is proposed to solve the telegraph equation of space - fractional order (TESFO), namely the Aboodh transformation method (ATM) subjected to the appropriate initial condition. Using ATM, it is possible to find exact solution or a closed approximate solution of a differential equation. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.

Connections between Aboodh Transform and Some Effective Integral Transforms

Integral transforms are the most useful techniques of the mathematics which are used to finding the solution of heat transfer problems, mixing problems, electrical networks, bending of beams, signal processing problems, which generally appears in the various disciplines of engineering and sciences. In this research paper, connections between Aboodh transform and some effective integral transforms (Laplace transform, Kamal transform, Elzaki transform, Sumudu transform, Mahgoub transform, Mohand transform and Sawi transform) are discussed and integral transforms of some typical functions are given in table form in application section to signify the fruitfulness of connections between Aboodh transform and some effective mention integral transforms.

Q-Homotopy Analysis Aboodh Transform Method based solution of proportional delay time-fractional partial differential equations

In this manuscript, a hybrid method namely q-Homotopy Analysis Aboodh Transform Method (q-HAATM) has been applied for obtaining the solutions of fractional partial differential equations (PDEs) with proportional delay (PD) described in the Caputo sense. q-HAATM is the amalgamation of the Homotopy Analysis Method (HAM) and Aboodh Transform Method (ATM). Three example problems are solved, and the results are obtained in series form. It is noticed from the obtained results that the present method is easy to apply and provides a proper way of choosing parameters ħ and n which give a better approximate solution.