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)

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.

The aim of this modest study was to shed some light on one of

the useful tools of complex analysis, which is the method of conformal

mapping (Also called conformal transformation). Conformal

transformations are optimal for solving various physical and

engineering problems that are difficult to solve in their original form

and in the given domain. This work starts by introducing the meaning

of a ''Conformal Mapping'', then introducing its basic Properties. In the

second part, it deals with a set of various examples that explain the

behavior of these mappings and show how they map a given domain

from its original form into a simpler one. Some of these examples

mentioned in this study showed that conformal transformations could

be used to determine harmonic functions, that is, to solve Laplace's

equation in two dimensions, which is the equation that governs a variety

of physical phenomena such as the steady-state temperature distribution

in solids, electrostatics and inviscid and irrotational flow (potential

flow). Other mathematical problems are treated. All problems that are

dealt with in this work became easier to solve after using this technique.

In addition, they showed that the harmonicity of a function is preserved

under conformal maps and the forms of the boundary conditions change

accordingly

In this paper we want to describe the algorithm to find all good starting points for

iteration of 𝑁𝑝(𝑧) to find all the roots of 𝑝(𝑧) by quick and easy way. So in this paper

we have proved that all critical points go to the roots under iteration of Newton’s

function then it will be the good starting points for iteration of 𝑁𝑝(𝑧) .

In this paper, we first recall the definition of a family of Koenig’s root-

finding algorithms known as Koenig’s algorithms (𝐾௣,௡) for polynomials. In the whole

paper p has degree 𝑑 ≥ 2 with real coefficients and real (and simple) zeros 𝑥௞ , 1 ≤

𝑘 ≤ 𝑑 .

Now we want to discuss Koenig’s algorithms in details where

𝑛 = 4, (𝐾௉,ସ(𝑧)).

In this paper we want to show that each immediate basin of fixed points of Newton’s

function for a real polynomial is simply connected.

Keywords: Newton’s function, Immediate basin, Simply connected.