On a first order partial differential equation with the nonlocal boundary condition

2014 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Sueda Nur Tekalan ◽  
Abdullah Said Erdogan
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order

Conditions for Determining Areal Permeability Distribution by Calculation

1962 ◽  
Vol 2 (03) ◽  
pp. 223-224 ◽  
Author(s):  
R. William Nelson
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
General Description ◽  
Order Partial Differential Equation ◽  
Stream Tube ◽  
Partial Differential ◽  
First Order ◽  
Permeability Distribution

NELSON, R. WILLIAM, GENERAL ELECTRIC CO., RICHLAND, WASH. Introduction Methods for determining permeability presented by W. D. Krugers in a paper entitled "Determining Areal Permeability Distribution by Calculation" have considerable merit. They can be expected to contribute significantly to analysis of flow in heterogeneous systems. Some rather subtle yet important conditions must be met, however, to assure that the computed permeabilities are correct. The conditions which insure a unique permeability distribution may have been overlooked in the method as proposed and presented in the subject paper. The subtlety of the special requirements may permit good agreement between computed and measured pressures in the reservoir; but the computed permeability may still be in error. The general theory and discussion to the requirements for a unique determination of the permeability distribution are presented elsewhere in more detail. Accordingly, only major features will be presented here in the notation of Kruger. GENERAL DESCRIPTION OF REQUIREMENTS A brief mention of the route to be followed in showing the special requirements may be helpful. The equation to be solved is a quasilinear, first-order, partial differential equation in the unknown permeability. This equation can be solved through an extension of Lagrange's method by reduction to a system of subsidiary ordinary differential equations. Through consideration of the identity of one of the Lagrange subsidiary differential equations with the stream function, a special interrelationship can be shown. The interrelationship has special significance with respect to the boundary condition. If the boundary function satisfies part of the subsidiary differential equation, no unique solution for the permeability distribution exists. In the physical problem this requirement for uniqueness indicates that the permeability measurements used as a boundary condition can not be along a stream tube. The boundary condition must be along a line, or series of lines, which completely traverse the region and pass through every stream tube. DERIVATION OF SPECIAL REQUIREMENTS ON THE BOUNDARY CONDITION If the velocity components in the directions of increasing x and y are vx and vy, the equation for the streamlines is dx/vx = dy/vy ..............................(1) Eq. 1 of Kruger's paper can be expanded to give: .............................(2) khEq. 2, when is the dependent variable, is a quasilinear first-order partial differential equation. The analytical method of solution is described in several texts with Goursat's treatment as translated by Hedrick and Dunkel, being very readable and complete. Eq. 2 is reducible to the system of subsidiary differential equations. ..............................(3) Through assuming the form of the solution to be ......(4) where F is an arbitrary function of two independent integrals of the system of Eq. 3. SPEJ P. 223^

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

scholarly journals Interpretation of Frenkel’s theory of sintering considering evolution of activated pores: II. Model and reliability

2015 ◽  
Vol 47 (1) ◽  
pp. 89-94
Author(s):  
C.L. Yu ◽  
D.P. Gao ◽  
S.M. Chai ◽  
Q. Liu ◽  
H. Shi ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Liquid Phase ◽  
Liquid Phase Sintering ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Sintering Time ◽  
Sintering Mechanism ◽  
Cordierite Glass

Frenkel's liquid-phase sintering mechanism has essential influence on the sintering of materials, however, by which only the initial 10% during isothermal sintering can be well explained. To overcome this shortage, Nikolic et al. introduced a mathematical model of shrinkage vs. sintering time concerning the activated volume evolution. This article compares the model established by Nikolic et al. with that of the Frenkel's liquid-phase sintering mechanism. The model is verified reliable via training the height and diameter data of cordierite glass by Giess et al. and the first-order partial differential equation. It is verified that the higher the temperature, the more quickly the value of the first-order partial differential equation with time and the relative initial effective activated volume to that in the final equibrium state increases to zero, and the more reliable the model is.

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

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