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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.

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19 views

Where is the solution uniquely determined by the data?

Question: Solve the PDE $$y\frac{\partial u^2}{\partial x^2} + (y-x)\frac{\partial u^2}{\partial x \partial y} -x \frac{\partial u^2}{\partial y^2} = \frac{y-x}{y+x}\bigg(\frac{\partial u}{\partial ...
2
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1answer
29 views

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 ...
0
votes
0answers
19 views

PDE and Implicit Function Theorem

Consider the PDE: $\partial_tu+u\partial_xu=u$ With boundary conditions: $u(0,x)=u_0(x)=-\tanh(x)$ Let $X(t,y,z)$ and $U(t,y,z)$ be the solutions to ODE Cauchy Problem: $\...
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2answers
40 views

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 ...
2
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0answers
35 views

Uniqueness of solution for linear first-order partial differential equations

In Polyanin's Handbook of First Order Partial Differential Equations (2002), in Section 10.1.2, it is stated that the non-homogeneous linear, first-order partial differential equation: $$\sum_{i=1}^...
0
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1answer
35 views

Uniqueness of Cauchy problem

I have the following problem: Let $\Omega \subset \mathbb{R}^n$ be an open and bounded subset with piecewise smooth boundary $\partial\Omega$. $a:\Omega\to]0,\infty[$ is a smooth function. $f:\...
0
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1answer
36 views

Cauchy problem with specific value of starting point, find limit

I have the following Cauchy problem: $$ y' = \vert y \vert - \arctan{e^x} $$ $$ y(0) = y_0 $$ I want to prove that there exists a specific value of $y_0$ such that: $$ \lim_{x\to\infty} y(x) = \...
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0answers
40 views

Studying the following Cauchy problem

I have the following Cauchy problem: $$ y' = y^2 - (\arctan{x})^2$$ $$ y(1) = 0 $$ I want to draw a the plot of the solution. This is what I have so far: $f(x,y) = y^2 - (\arctan{x})^2 $ is $C^\...
1
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0answers
24 views

Existence and uniqueness of solution for first order Cauchy problems

Suppose we have two functions $f=f(x,a):\mathbb{R}^n\times A\to \mathbb{R}^n$, where $A\subset \mathbb{R}^m$ is compact, and also $\alpha:[0,+\infty)\to A$. Under the hypothesis that A')$f$ and $\...
2
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2answers
49 views

Solving a Cauchy problem, differential equation

I have the following Cauchy problem \begin{cases} y'(x) + \frac{1}{x^2-1}y(x) = \sqrt{x+1} \\ y(0) = 0 \end{cases} I proceed by finding $e^{A(x)} $ where $A(x)$ is the primitive of $a(x)= \frac{1}{...
3
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1answer
86 views

Continuity of mild solution

This is a question on the proof of Theorem 6.1.2, Pazy's book Semigroups of Linear Operators and Applications to Partial Differential Equations. Also the title might not be too accurate, as my ...
0
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1answer
24 views

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 ...
4
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1answer
46 views

Demonstrate that solution of differential equation is bounded

Let $b(t) \in C^1([0,+\infty))$. I have to find a formula for the solution of this Cauchy's problem: $$ \left\{\begin{aligned} x''(t)+x(t)&=b(t) \\ x(0)&=x_0\\ x'(0)&=x_1 \end{aligned}\...
4
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1answer
41 views

Limit of solution of Cauchy problem

I have the following Cauchy problem: \begin{align} y'(t) = \arctan(t^3(y-1)) \\ y(0) = \alpha \end{align} I want to study the limit of the solution on the boundary. This is what I have done so far:...
1
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2answers
100 views

If $\vert F(t,x)\vert \leq \alpha (t)\vert x\vert + \beta(t)$ the maximal solutions are global for an ODE

Let $F: \mathbb{R} \times\mathbb{R}^2 \rightarrow \mathbb{R}$ localy lipschitz in its second variable. Let the Cauchy problem be: $$x'=F(t,x),\\ x(0) = x_0 $$ Let $\alpha : \mathbb{R} \rightarrow \...
0
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1answer
34 views

How do I study the domain of a Cauchy's problem without solving it?

I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example, $\begin{cases} y'=(y-\sin x)^2+1+\cos x\\ y(0)=0 \end{...
1
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0answers
35 views

existence of solutions for Cauchy problems

Consider the equation $$(1-\cos x)u_{tt} - u_{tx} - u_{xx} = 0$$ with Cauchy data $$u(x,0) = f(x), u_t(x,0) = g(x),\text{ for } f,g\in\mathcal{C}^2$$ What compatibility condition do $f$ and $g$ have ...
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0answers
43 views

$u$ is a spherical wave $\iff$ $f$ and $g$ are radial

Consider the Cauchy problem for the wave equation : \begin{cases} u_{tt}= \Delta u \\u(x,0)=f(x) \\ u_t(x,0)= g(x)\end{cases} with $t>0$ and $x\in\mathbb{R^d}$ This is an exercise from Stein's ...
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0answers
68 views

Mixed Cauchy and Dirichlet and unspecified boundary conditions for Laplace equation on $I^2$

I am looking for a reference where the following problem is discussed: $u \in C^{\infty}(I^2)$ so that $\Delta u = 0$ $u(0,y) = f(y)$, $u(1,y) = g(y)$, $u(x,0) = h(x)$ $\nabla u(x,0) \cdot \hat{n} (...
0
votes
1answer
40 views

Solving Cauchy's problem with a discontinuous function

I have the Cauchy problem: $$\begin{cases} f'=g(t)+2(f-5) \\ f(0)=2\end{cases}$$ Now $g(t)$ is a periodic function: $$g(t)=\begin{cases} 0,t\in(24k,24k+8)\\ 2,t\notin(24k,24k+8) \end{cases}$$ for $k=...
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0answers
22 views

Continuity of a piecewise function (rarefaction wave for quasilinear pde's) - question on domain

In studying elementary theory of PDE's from Salsa, the author talks about weak solution for conservation laws (quasilinear first order PDE). In particular, the Cauchy problem $$\begin{cases} \rho_t+...
6
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0answers
62 views

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 ...
2
votes
1answer
149 views

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 ...
0
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1answer
130 views

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
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0answers
55 views

Definition of (classical) solution for a boundary-value-problem

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-...
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29 views

Bayesian analysis: what is the kernel of Cauchy distribution

Here is the Cauchy model by putting 𝑛 = 1, 𝜇 = 𝜃, 𝜎 = 1 in the student's t-model: $f(y|\theta)=\frac{1}{\pi\cdot(1 + (y - \theta)^2)}$ Is the kernel (the parts that contain y) $\frac{1}{(1 + (y -...
1
vote
2answers
52 views

Solve for $u$ the PDE $(x − y − 1)u_x + (y − x − u + 1)u_y = u$ if $u=1$ on $x^2+(y+1)^2=1.$

Solve the Cauchy problem $$(x − y − 1)u_x + (y − x − u + 1)u_y = u,$$ if $u=1$ on $x^2+(y+1)^2=1.$ Attempt. $$\frac{dx}{x-y-1}=\frac{dy}{y-x-z+1}=\frac{dz}{z}$$ so $$\frac{dx+dy}{(x-y-1)+(y-x-z+1)}...
0
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2answers
91 views

How to solve the Cauchy problem $y'+xy=1+x$; $y(3/2)=0$

Given \begin{equation} y'+xy=1+x; \text{ } y(3/2)=0 \end{equation} I am able to solve the non homogeneous linear differential equation to find: \begin{equation} y=e^{-\frac{x^{2}}{2}}(\int e^{\...
3
votes
1answer
305 views

Solving a Cauchy Problem using Method of Characteristics

I came across a partial differential equation (IVP PDE) that I would like to solve: $$\{u_y+cos(ky)u_x=ax^2 | u(x,0)=0\}$$ This should be a quasilinear PDE, and is in the format of a Cauchy ...
2
votes
1answer
49 views

Show that problem is well defined for each time

We have the Cauchy problem of the equation $u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$ with some given smooth ($C^1$) function $g$ as initial value. I want to check if the problem is well ...
1
vote
2answers
72 views

Studying a solution of a Cauchy problem

Consider the Cauchy problem $y' = \frac{t+2}{t^2+y^2}$, $y(0)=1$, study the behavior of its solutions, if it possible, when $t\to +\infty$. By Cauchy-Lipschitz there exists a unique local solution; ...
0
votes
1answer
411 views

How to solve $y^{\prime\prime}=y^2-y, y^{\prime}=0$ when $y(x_0)=1, y^{\prime}(x_0)=2$ where $x_0=\sqrt{2}$

I use $v=y^{\prime}$ and get a first order non-linear ODE, then how to solve $(y^{\prime})^2=\frac{2}{3}y^3-2y^2+\frac{4}{3}$?
2
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0answers
34 views

PDE Cauchy Problem w/ Two Boundaries

I'm trying to solve this Cauchy problem, which appears to be rather basic, but it seems like I'm missing something. The problem is as follows: $ u_x+u_t+u=0 \\ 0<\alpha*t<x<\infty \\ \...
0
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0answers
91 views

Picard-Lindelöf theorem application with absolute value

I have to verify the applicability of Picard-Lindelöf theorem for the Cauchy problems associated with the ODE $$x'(t) = |\sin{x(t)}|(t-e^{x(t)}).$$ In order to answer I have to verify that: $f(x,t)=|...
0
votes
2answers
60 views

Cauchy problem for diffusion equation and condition $x^2$ if $x\in [0,1]$

Solve the Cauchy problem for diffusion equation $$u_t=u_{xx}, t\ge0,x\in\mathbb R,\\ u(x,0)=\begin{cases}x^2, x\in [0,1]\\0,x\not\in [0,1] \end{cases}$$ Give the solution in terms of $erf(x)=\sqrt{2/\...
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0answers
13 views

Cauchy problem of a density probability

I can't find the solution for this Cauchy problem: $$\frac{\partial \rho(q,p,t)}{\partial t} = - \frac{p}{m} \frac{\partial \rho(q,p,t)}{\partial q}$$ $$\rho(q,p,0) = \delta(q)g(p)$$ Probably the ...
2
votes
1answer
52 views

Solve $\dot X = AX$ and find the time at which the area doubles.

We have a linear operator $X(t): \mathbb{E}^2 \to \mathbb{E}^2$ such that $$\dot X = AX\quad X(0)=\mathbf{1}$$ To be clear $X$ is a $2\times 2$-matrix. An ink spot is contained in $\mathbb{E}^2$ at $t=...
1
vote
1answer
58 views

Coincidence of the Characteristic projection and the Data Curve

While studying first order Semi-linear PDE and the method of characteristics, I found this part in the note that I just cannot comprehend. From what I’ve understood, the solution surface is swept ...
3
votes
1answer
70 views

Uniqueness of solutions of Cauchy problem with nonzero function

Consider the initial value problem $$y'(x)=f(x,y(x)), \\ y(x_0)=y_0,$$ where the function $f \colon D \to \mathbb R$ is defined and continuous on some open set $D \subseteq \mathbb R \times \mathbb R$ ...
0
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0answers
74 views

Cauchy problem, boundary vs initial problem

I'm trying to understand the notion of initial value problem and boundary value problem. I think I understand that, to have a well-posed problem we need to define a surface over a which we can define ...
-2
votes
1answer
72 views

Cauchy Problem - Why no unique solution? [closed]

Why does the Cauchy problem $$ \begin{cases} y'= y^{3/4},\\ y(0)=0 \end{cases} $$ not admit a unique solution? And please how we justify that $(y^{3/4})'$ is not continuous?
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1answer
83 views

Convergence of the Picard sequence

Consider the Cauchy problem $$ \begin{cases} y'= \cos(y)=f(x,y)\\ y(0)=0 \end{cases} $$ The question is: Does the Picard sequence converge? My attempt: We have that $y_0=0$, $y_1(x)= x$, $y_2(x)=...
1
vote
1answer
58 views

Find exact solution of Cauchy Problem using the convergence of a sequence defined by Fixed Point Theorem

I have the following Cauchy problem $$ \begin{cases} y'= y^2 - x^2= f(x,y)\\ y(0)=1 \end{cases} $$ in the rectangle $R=\{(x,y), 0 \leq x \leq 1, |y-1| \leq 1\}$. The question is to prove the ...
1
vote
2answers
40 views

Solve the Cauchy problem for the linear PDE

I have the linear PDE $$yu_x - xu_y = 0 \qquad x^2 + y^2 < a^2 \\ u(0,y) = (a^2-y^2)^{\frac{1}{2}} \qquad y \in(-a,a)$$ where $a > 0$ is a constant. So what I have done is to say that $$a_1 ...
5
votes
2answers
305 views

Cauchy differential equation

I'm trying to resolve this cauchy problem: $ y'=2y+1$ such as $y(0)=1$ the general integral for the differential equation is $\frac{1}{2}(e^{2x+2c_1}-1)$ for $y(0)=1$ : $y(0)=\frac{1}{2}(e^{2c_1}-...
2
votes
1answer
49 views

Exisence and unicity for Cauchy Problem

i have the following theorem: Let the Cauchy problem $$ y'=f(x,y),~~ y(x_0)= y_0 $$ in $$ R=\{(x,y) \in \mathbb{R}^2: |x-x_0| \leq a, |y-y_0| \leq b\} $$ If $f$ is continuous and bounded ...
0
votes
0answers
158 views

Subsidiary Differential Equation

I was presented with the equation: $\begin{equation} \begin{split} \frac{dx}{x(x+y)} = \frac{dy}{y(x+y)} = \frac{dz}{(x-y)(2x+2y+z)} \end{split} \end{equation}$ So, we immediately know that the ...
2
votes
2answers
157 views

Explanation to PDE $u_{tt}-c^2u_{xx}=0$ solution

Exercise: Solve the Cauchy problem $u_{tt}-c^2u_{xx}=0$ with conditions $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$, where $g(x)=0,\ h(x)=\begin{cases} 0,\ x<0 \\ 1,\ x\ge0 \end{cases}$. Please ignore this ...
1
vote
0answers
70 views

Using Green's function in Cauchy problem

Green's function is a well known tool for solving differential equations with boundary conditions by turning it into integral equations. I wonder if Green's function can be applied somehow in a Cauchy ...
1
vote
0answers
87 views

Use an energy argument for to show that the global Cauchy problem for the three-dimensional wave equation has a unique solution

I am trying to use an energy argument for to show that the global Cauchy problem for the three-dimensional wave equation has a unique solution. The wave equation is $$\partial^2u/\partial t^2=\nabla^...