Wafaa Hadi Hanoon Alshmerty

The goal of this paper is to evaluate numerically a double integral of partial Derivatives Using RTRT Method. For Trapezoidal method (one of Newton-Cotes formula) which will be based on two dimensions x and y. In addition to that Romberg acceleration rule will be used to get more accurate results together with less time (faster convergence) and number of subintervals which are involved. We shall refer to this method by RTRT, where R stands for Romberg acceleration and T for Trapezoidal rule.

In this paper we find the solution for some kinds of nonlinear third order partial differential equations of homogeneous degree with three independent variables where A, B, C, D, E, … ,S and T are linear functions of dependent variable Z and partial derivatives of dependent variable with respect to the independent variables x , y and t, and that by using some of the assumptions .

In this paper ,we used search method for calculating the approximate values of double integrals , when its integrands continuous and we choose rules (Trapezoidal and mid-point)because the approximate values resulting from the use of Trapezoidal rule be faster closer to the real value of the integrals compared to another formulas of Newton-Cotes formulas ,but for the rule mid-point we can use it in the calculation of continuous integrand or non-continuous integrands in the region of integration because it is not use the values of function of integration in both ends of the integration period and the symbol of this rule is .We choose ( the number of subintervals of )is equal to (number of subintervals of )and we improved the results by applying Aitken accelerate with and denote to the method by , where the symbol A indicates to the Aitken’s acceleration method and (indicates to the mid-point rule on the external dimension and Trapezoidal rule on the internal dimension ).

The main aim of this search is to find the complete solution for special kinds of nonlinear second order partial differential equations with three independent variables where A , B , C , D , E , F , G , H , I and J are functions of . And in these functions are of first degree and not multiplied with together.

The main aim of this search is to solve kind of partial differential equations of second order with variable coefficients. Where some of and are functions of or both . For this purpose, we willuse one kind of the partial differential equation we found the substitution transformsthe above kind to the first order ordinary differential equation with two independent functions . We found the general form of its complete solution and we appliedthis form for solving the heat equation as physical application insphericalcoordinates

— in this study, the connection between the fuzzy semi-T-ABSO module and the fuzzy socle is compared. To address this query, we provide an idea known a the fuzzy socle semi-TABSO modules. Several characteristics were found in the research that supported the new theory. Also, using simple algebraic methods, the connection between the fuzzy socle-semi T-ABSO module and the fuzzy socle-T-ABSO module was found. Also, we looked into the structure of the fuzzy socle semi-TABSO module for the fuzzy direct sum. The connections between the fuzzy socle semi-TABSO module and other fuzzy module types were also discussed as a divisible module and a comultiplication module. The definition of the new fuzzy system semi-TABS concepts will benefit from the research's conclusions.

Let Vbe aring with identity and W be a unitary left-module. TheSoc-L-small-T-ABSO submodule, a generalization of the small-T-ABSO, is a new class of submodules that we introduce in this study, and we also outline some of its key char-acteristics and examples that are achieved and not achieved in it and its relationship to other submodules

Let R be a commutative ring with identity, and Y be a fuzzy module of an R- module M where M is a left untial module. In this paper we investigate a class of F. submodules called weakly essential fuzzy submodules that lie between essential fuzzy submodules and semi-essential fuzzy submodules. Also, we present the concept of weakly uniform fuzzy modules as a generalization of uniform fuzzy modules. This concept lies between Uniform fuzzy modules and semi-uniform fuzzy modules. Some basic properties, characteristics and examples of weakly essential fuzzy submodules and weakly uniform fuzzy modules are investigated, and some relationships among weakly essential fuzzy submodules, weakly uniform fuzzy modules and other related fuzzy submodules (fuzzy modules) are given.

— in this study, the connection between the fuzzy semi-T-ABSO module and the fuzzy socle is compared. To address this query, we provide an idea known a the fuzzy socle semi-TABSO modules. Several characteristics were found in the research that supported the new theory. Also, using simple algebraic methods, the connection between the fuzzy socle-semi T-ABSO module and the fuzzy socle-T-ABSO module was found. Also, we looked into the structure of the fuzzy socle semi-TABSO module for the fuzzy direct sum. The connections between the fuzzy socle semi-TABSO module and other fuzzy module types were also discussed as a divisible module and a comultiplication module. The definition of the new fuzzy system semi-TABS concepts will benefit from the research's conclusions.