Questions tagged [cauchy-problem]

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

112 questions
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Positive domain of the Cauchy problem

If I consider the standard Cauchy Problem in one-dimension of the form: \begin{cases} a(x,t) \frac{\partial^2u}{\partial^2x} + b(x,t)\frac{\partial u}{\partial x} + c(x,t) u - \frac{...
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Solutions of a PDE problem given in the Riemann invariant diagonal form

First, let's say we have a Cauchy problem: $$(1) \hspace{0.5cm} u_t (x,t)+ div f(u(x,t))=0,$$ where the initial condition is given with $u(x,0)=u_0(x)$, $x \in A \subseteq \mathbb{R}^d, d \geq 1$,...
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find 2-3 terms of power expansion of parameter $\mu$

I have the equation: $$xy' = \mu x^2 + \ln y \quad y(1)=1$$ I am given the polynomial: $$y(x,\mu) = y_0(x)+\mu y_1(x) + \mu^2 y_2(x) \ldots$$ I honestly do not know how to approach this topic at ...
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Triple-valued solution to Riemann problem - Profile of the bulge

I am studying conservation laws and hyperbolic systems, particularly, Burgers' equation and shocks, and have a doubt at pages 48/49 of the book Numerical Methods for Conservation Laws by R.J. LeVeque (...
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Infinitely many solutions for a first order Cauchy problem.

Is this correct that the following Cauchy problem has infinitely many solutions? ‎\begin{cases}‎ ‎xu_t+u_x=0 \\‎ ‎u(x,0)=\cos x‎ ‎\end{cases}‎ Using the method of characteristics it is obvious ...
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Solution of a Cauchy ODE

Let $y(t)$ be a solution of a Cauchy Problem: $$\dot{y}=\ln\left(\sqrt{1+y^2}\right)$$ with the initial condition $y(0)=y_0$. Prove that if $y_0>0$, then $y$ is a strictly increasing ...
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Can the unique, stationary solution of a Cauchy problem be prolonged on all $\Bbb R$ even if the differential equation is undefined at one point?

Consider the Cauchy problem$$(*)\begin{cases}y'=\frac{y^2-1}{x^2} \\y(1)=1.\end{cases}$$ We note that $y'=h(x)k(y)$ with $h(x)$ continuous on $\Bbb R\setminus\{0\}$ and $k(y)$ continuously ...
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Non-unique solution of first order PDE

Question: $$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}=1 \qquad \qquad u=0 \; \text{ when } \; x+y=1$$ Find all possible solutions and state where each one exists. Attempt: Using ...
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Local uniqueness of solution for quasi linear PDE

I've a little troubles in proving local uniqueness of solution for Cauchy problems concerning quasilinear PDE's. It's a little bit boring, but I tried to be as clear as possible. Suppose $\Omega$ is ...
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Cauchy simulation in R

How do I simulate Cauchy distribution from Uniform distribution in (-pi/2, pi/2) in R? Not allowed to used any functions that already exist in R that generate Cauchy
Let $k\geq 1$, $n\geq 2$ and let $U\subseteq \mathbb{R}^n$ be open and connected. Let $F:U\times \mathbb{R}\times\mathbb{R}^n\times \cdots \times \mathbb{R}^{n^k}\to \mathbb{R}$ a given function. A k-...