Questions tagged [cauchy-problem]
The cauchy-problem tag has no usage guidance.
2
votes
0answers
47 views
Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)
Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...
3
votes
0answers
47 views
How to solve this linear Cauchy Problem
within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...
1
vote
2answers
156 views
Method of characteristics for 2x2 systems
In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE ...
3
votes
0answers
151 views
Reference request on connection between PDE problems
I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
1
vote
1answer
192 views
Infinitesimal generator of a semigroup with parameter
When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
...
1
vote
0answers
47 views
Supnorm problem involving kernel of Cauchy problem
Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem
$$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
0
votes
0answers
168 views
Domain of dependence for Hyperbolic system of PDES
In Leveque's "Numerical Methods for Conservation Laws", Ch. 3.1.1., he says that given a system of Hyperbolic PDE's and a point $(\bar{t},\bar{x} )$, its domain of dependence $D(\bar{t},\bar{x} )$ is ...
2
votes
0answers
129 views
An question about Cauchy Problem in General Relativity [closed]
Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question:
Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there ...
3
votes
2answers
130 views
An English version Borok's work on finite-infinite systems of ordinary differential equations
I am looking for the English translation of the paper by V. M. Borok (originally in Russian)
The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...
1
vote
2answers
192 views
Integral representation of the Cauchy problem solution for the heat equation
Consider the Cauchy problem for the heat equation
$u_t=\Delta u$, $u|_{t=0}=\varphi$.
S. Täcklind showed its solution $u$ is unique in the class $|u|\le e^{|x|h(|x|)}$, $|x|>1$, iff $\int_1^\infty \...
