The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

In this paper Heun method has been used to find numerical solution for first order nonlinear functional differential equation. Moreover, this method has been modified in order to treat system of nonlinear functional differential equations .two numerical examples are given for conciliated the results of this method.

Asmari is the main productive reservoir in Abu Ghirab oilfield in the south-east part of Iraq. It has history production extends from 1976 up to now with several close periods. Recently, the reservoir suffers some problems in production, which are abstracted as water production rising with oil production declining in most wells. The water problem type of the field and wells is identified by using Chan's diagnostic plots (water oil ratio (WOR) and derivative water oil ratio (WOR') against time). The analytical results show that water problem is caused by the channeling due to high permeability zones, high water saturation zones, and faults or fracturing. The numerical approach is also used to study the water movement inside the reser

Asmari is the main productive reservoir in Abu Ghirab oilfield in the south-east part of Iraq. It has history production extends from 1976 up to now with several close periods. Recently, the reservoir suffers some problems in production, which are abstracted as water production rising with oil production declining in most wells. The water problem type of the field and wells is identified by using Chan's diagnostic plots (water oil ratio (WOR) and derivative water oil ratio (WOR') against time). The analytical results show that water problem is caused by the channeling due to high permeability zones, high water saturation zones, and faults or fracturing. The numerical approach is also used to study the water movement insi

       In this work, the fractional damped Burger's equation (FDBE) formula    = 0,

Based on the needs of the scientific community, researchers tended to find new iterative schemes or develop previous iterative schemes that would help researchers reach the fixed point with fewer steps and with stability, will be define in this paper the multi_implicit four-step iterative (MIFSI) which is development to four-step implicit fixed point iterative, to develop the aforementioned iterative scheme, we will use a finite set of projective functions ,nonexpansive function and finite set from a new functions called generalized quasi like contractive which is an amalgamation of quasi contractive function and contractive like function , by the last function and a set of sequential organized steps, we will be able to prove the existen

Many of the dynamic processes in different sciences are described by models of differential equations. These models explain the change in the behavior of the studied process over time by linking the behavior of the process under study with its derivatives. These models often contain constant and time-varying parameters that vary according to the nature of the process under study in this We will estimate the constant and time-varying parameters in a sequential method in several stages. In the first stage, the state variables and their derivatives are estimated in the method of penalized splines(p- splines) . In the second stage we use pseudo lest square to estimate constant parameters, For the third stage, the rem

The numerical resolve nonlinear system of Volterra integral equation of the second kind (NLSVIEK2) has been considered. The exponential function is used as the base function of the collocation method to approximate the resolve of the problem. Arithmetic epitome are performed which have already been solved by weighted residual manner,  Taylor manner and block- by- block(2, 3, 5).

A particular solution of the two and three dimensional unsteady state thermal or mass diffusion equation is obtained by introducing a combination of variables of the form,
η = (x+y) / √ct , and η = (x+y+z) / √ct, for two and three dimensional equations
respectively. And the corresponding solutions are,
θ (t,x,y) = θ₀ erfc (x+y)/√8ct and θ( t,x,y,z) =θ₀ erfc (x+y+z/√12ct)