On the integrals of a partial differential equation

1947 ◽  
Vol 43 (3) ◽  
pp. 348-359 ◽  
Author(s):  
F. G. Friedlander

The ordinary one-dimensional wave equationhas special integrals of the formwhich satisfy the first-order equationsrespectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight linesin an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

1980 ◽  
Vol 87 (3) ◽  
pp. 515-521
Author(s):  
Albert E. Heins

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equationcould be written asif b > rorif r > b.


1955 ◽  
Vol 7 ◽  
pp. 284-288
Author(s):  
M. H. Martin

1. Introduction. In a previous paper (3) the study of one-dimensional, unsteady flows, isentropic or anisentropic, was reduced to the integration of a Monge-Ampère partial differential equation(1)For a polytropic gas, the specific volume(2) ,takes the form,once the entropy distribution function S = S(Ψ) is specified.


2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG

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.


2015 ◽  
Vol 47 (1) ◽  
pp. 89-94
Author(s):  
C.L. Yu ◽  
D.P. Gao ◽  
S.M. Chai ◽  
Q. Liu ◽  
H. Shi ◽  
...  

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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