scholarly journals On solving non-homogeneous partial differential equations with right-hand side defined on the grid

2021 ◽  
Vol 31 (3) ◽  
pp. 443-457
Author(s):  
L.I. Rubina ◽  
O.N. Ul'yanov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Grid Point ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Ode System ◽  
Right Hand ◽  
The Right

An algorithm is proposed for obtaining solutions of partial differential equations with right-hand side defined on the grid $\{ x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}\},\ (\mu=1,2,\ldots,N)\colon f_{\mu}=f(x_{1}^{\mu}, x_{2}^{\mu}, \ldots, x_{n}^{\mu}).$ Here $n$ is the number of independent variables in the original partial differential equation, $N$ is the number of rows in the grid for the right-hand side, $f=f( x_{1}, x_{2}, \ldots, x_{n})$ is the right-hand of the original equation. The algorithm implements a reduction of the original equation to a system of ordinary differential equations (ODE system) with initial conditions at each grid point and includes the following sequence of actions. We seek a solution to the original equation, depending on one independent variable. The original equation is associated with a certain system of relations containing arbitrary functions and including the partial differential equation of the first order. For an equation of the first order, an extended system of equations of characteristics is written. Adding to it the remaining relations containing arbitrary functions, and demanding that these relations be the first integrals of the extended system of equations of characteristics, we arrive at the desired ODE system, completing the reduction. The proposed algorithm allows at each grid point to find a solution of the original partial differential equation that satisfies the given initial and boundary conditions. The algorithm is used to obtain solutions of the Poisson equation and the equation of unsteady axisymmetric filtering at the points of the grid on which the right-hand sides of the corresponding equations are given.

III. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

1863 ◽  
Vol 12 ◽  
pp. 420-424
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamical Problem ◽  
General Character ◽  
Partial Differential ◽  
Central Function ◽  
First Order ◽  
Linear Partial Differential Equations

Jacobi in a posthumous memoir, which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the differential equations of dynamics which was established by Sir W. R. Hamilton in the 'Philosophical Transactions’ for 1834-35. The knowledge, indeed, that the solution of the equation of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results.

scholarly journals Invariant measures for the flow of a first order partial differential equation

1985 ◽  
Vol 5 (3) ◽  
pp. 437-443 ◽  
Author(s):  
R. Rudnicki
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Invariant Measures ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order

AbstractWe prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

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^

scholarly journals XXII. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

1863 ◽  
Vol 153 ◽  
pp. 485-501
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Auxiliary System ◽  
Systems Of Equations ◽  
Dynamical Equations ◽  
Partial Differential ◽  
First Order ◽  
Linear Partial Differential Equations

Jacobi, in a posthumous memoir* which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the Differential Equations of Dynamics which was esta­blished by Sir W. B. Hamilton in the Philosophical Transactions for 1834‒35. The knowledge, indeed, that the solution of the equations of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results. But in the order of those speculative truths which enable us to perceive unity where it was unperceived before, its place is a high and enduring one. Given a system of dynamical equations, it is possible, as Jacobi had shown, to con­struct a partial differential equation such that from any complete primitive of that equation, i. e . from any solution of it involving a number of constants equal to the number of the independent variables, all the integrals of the dynamical equations can be deduced by processes of differentiation. Hitherto, however, the discovery of the com­plete primitive of a partial differential equation has been supposed to require a previous knowledge of the integrals of a certain auxiliary system of ordinary differential equa­tions; and in the case under consideration that auxiliary system consisted of the dynamical equations themselves. Jacobi’s new methods do not require the preliminary integration of the auxiliary system. They require, instead of this, the solution of certain systems of simultaneous linear partial differential equations. To this object therefore the method developed in my recent paper on Simultaneous Differential Equa­tions might be applied. But the systems of equations in question are of a peculiar form. They admit, in consequence of this, of a peculiar analysis. And Jacobi’s methods of solving them are in fact different from the one given by me, though connected with it by remarkable relations. He does indeed refer to the general problem of the solution of simultaneous partial differential equations, and this in language which does not even suppose the condition of linearity. He says, “Non ego hic immorabor qusestioni generali quando et quomodo duabus compluribusve æquationibus differentialibus partialibus una eademque functione Satisfied possit, sed ad casum propositum investigationem restringam. Quippe quo præclaris uti licet artificiis ad integrationem expediendam commodis. ” But he does not, as far as I have been able to discover, discuss any systems of equations more general than those which arise in the immediate problem before him.

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.

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