Abstract: The exact solution of a nonlinear fractional Volterra-Fredholm integro-differential equation is found in this paper through the successful application of the Abaoub Shkheam decomposition method. These techniques have a wider range of applications due to their dependability and decreased computational effort. 

ABSTRACT: This paper applies the Abaoub – Shkheam Decomposition Method (QDM) to obtaining solutions of linear fractional diffusion equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method. 

Abstract: In this paper, a nonlinear fractional diffusion and wave equations are solved using the Abaoub-Shkheam Decomposition Method (QDM). The Caputo sense is used to characterise the fractional derivative. An example is provided to demonstrate the method's efficiency and practicality.

Abstract: The Abaoub-Shkheam Decomposition Method (QDM) is employed in this paper to solve nonlinear initial value problems of second order. Adomian polynomial is used to decompose nonlinear functions that exist in a given equation. We are using this method (QDM) to find the exact solution of different types of non-linear ordinary differential equations, which is based on the Abaoub Shkheam transform method (QTM) and the Adomian Decomposition Method (ADM). An example is provided to demonstrate the efficacy of this approach. 

Abstract: In this study, we introduce a novel approach to the solution of a nonlinear Volterra -Fredholm integral equations by applying the Adomian decomposition method under the effect of the Abaoub- Shkheam transform. We demonstrate the existence and uniqueness of the solution in Banach space and illustrate this idea with an example.

 عن الدوال التوافقية، لذلك قدمنا أوال أهه النظزيات املتعلقة بالدوال التوافقية،

وطزح فكزة املزافق التوافقي. وقدمنا أيضا مفووو جزين ومسألة ديزيشليى. وخلصنا إىل أن الدوال التوافقية

هلا العديد من التطبيقات ميكن استدداموا يف دراسة القزص وفوه مسألة ديزيشليى.

In this paper, we apply the auxiliary equation method with the aid of symbolic

computer algebra system (CAS) such as Maple or Mathematica to construct

many new exact traveling wave solutions of a singularly perturbed generalized

Gardner equation with nonlinear terms of any order. Using a simple

transformation, this equation can be reduced to a nonlinear ordinary differential

equation (ODE). A comparison between our results and the well-known results

is given. Further, plotting 2D and 3D graphics of the exact solutions are shown.

The main problem of this paper is to conjugate the cubic equation to quadratic

equation by using Mӧbius transformation, so the work to finding the roots of

polynomials with Complex variable and double roots using Newton’s method will

be easier. Also we make some correction to the linear fractional transformation (or

Mobius transformation)

Abstract: We employ Abaoub - Shkheam transformation to solve linear Volterra integro-differential equation of the first kind, we considered the kernel of that equation is a deference type kernel. Moreover, we prove the existence and uniqueness of solutions of the equation under some conditions in the Banach space and fixed-point theory. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique. 

In this article, we propose a new method to construct many new exact solutions with

parameters for the generalized KdV–mKdV (GKdV–mKdV) equation with higher-

order nonlinear terms. The proposed method is a generalization of the well-known( G′

G , 1

G

)

-expansion method in the case of the hyperbolic function solutions. Also, we

use a direct algebraic method based on the generalized Liénard equation to find other

diffrent new exact solutions of the above GKdV–mKdV equation. Soliton solutions,

periodic solutions, rational functions solutions, hyperbolic functions solutions and

symmetrical hyperbolic Lucas functions solutions are obtained. Comparing our new

results obtained in this article with the well-known results are given. The generalized( G′

G , 1

G

)

-expansion method presented in this article is straightforward, concise and it

can also be applied to other nonlinear partial differential equations in mathematical

physics.