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.

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. 

ABSTRACT In this work, we show the theorem of the convolution of the Abaoub-Shkheam transform and employed it for solving one dimensional linear Volterra integral equations of the second kind.  

Abstract-In this paper a new integral transform namely Abaoub-Shkheam transform was introduced. Fundamental properties of this transform were derived and presented such as linearity, change of scale, and first translation or shifting. It is proven and tested to covering equation for temperature distributions in a semi-infinite bar. This transform may solve some different kind of integral and differential equations and it competes with other known transforms like Sumudu and Yang Transform.

Abstract: In this paper, we will use the combination of Regularization method and Direct computation method, or shortly, Regularization-Direct method for solve two dimension- al linear Fredholm integral equations of first kind, by converting the first kind of equation to the second kind by applying the obtain a solution. A few examples are provided to prove the validity and applicability of this approach. regularization method. Then the Direct compotation method is applying to getting the resulting second kind of equation to 

Abstract: In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference. 

 Abstract. We investigate harmonic Bergman spaces bp = bp(Ω), 0 < p < ∞, where Ω = Rn \Zn and prove that bq ⊂ bp for n/(k + 1) ≤ q < p < n/k. In the planar case we prove that bp is non empty for all 0 < p < ∞. Further, for each 0 < p < ∞ there is a non-trivial f ∈ bp tending to zero at infinity at any prescribed rate.

Abstract. We prove that ωu(δ) ≤ Cωf(δ), where u: Ω → Rn is the harmonic extension of a continuous map f: ∂Ω → Rn, if u is a K-quasiregular map and Ω is bounded in Rn with C2 boundary. Here C is a constant depending only on n, ωf and K and ωh denotes the modulus of continuity of h. We also prove a version of this result for Λω-extension domains with c-uniformly perfect boundary and quasiconformal mappings.