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Questions tagged [cauchy-problem]

Use this tag for questions about partial differential equations that satisfy certain conditions given on a hypersurface in the domain.

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain.

A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition), but it can be none of them. They are named after Augustin Louis Cauchy.

Source: https://en.wikipedia.org/wiki/Cauchy_problem

303 questions
2
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Solution to Cauchy Problem

I am trying to solve the following Cauchy Problem: $y'(t) = A(t)y(t), A=\begin{pmatrix} t &-1 \\ 1 &t \end{pmatrix}, y(0)=y_0$ What I did: I know that $ \forall\ t, s \in\ \mathbb{R}: A(s)A(t)=A(t)A(s)$ So the solution would be: $y(t)= e^{…
Conjecture
  • 3,088
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When can we prescribe the speed of a normal flow from a hypersurface?

Suppose that $X_0:M^n \rightarrow \mathbb{R}^{n+1}$ is a smooth embedding of a compact hypersurface. I'm given a scalar function $\eta \in C^{\infty}\left( M \times [0, T) \right)$ and I'd like to solve the problem $$ \frac{dX}{dt}(x,t) =…
MicahW
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0
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Why "we set $\xi^{2}=A, \quad \xi^{2}=B \quad \Longrightarrow \quad A=\xi^{2}, \quad B=\xi^{2}$?"

I fail to understand why we assume that $$\xi^{2}=A, \quad \xi^{2}=B \quad \Longrightarrow \quad A=\xi^{2}, \quad B=\xi^{2}$$ in the process of obtaining a solution to a PDE. Example 4 Find the solution of the Cauchy problem governed by the linear…
user817548
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2 answers

How to solve the Cauchy problem?

I have a task that looks like $$ \frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial x} + u + 3e^{t},\, u\left(x, 0\right)=\sin{x}$$ Please help, thanks :)
Pavel Naberezhnev
  • 13