Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cauchy-problem]

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

0
votes
2answers
100 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
330 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
153 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 ...
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=...
2
votes
1answer
73 views

Reaction-diffusion Cauchy problem

How do I solve this reaction-diffusion Cauchy problem? $$\left\{ \begin{array}{l l} u_{t} - \kappa u_{xx} +ru=0 & \quad \mbox{$x \in \mathbb{R}, t>0$,}\\ \quad u(x,0) = \phi(x), \end{array} \...
2
votes
1answer
41 views

Explaining results involving differential equations using the theorem for the existence and uniqueness of the solutions of a Cauchy Problem

Let $f(x,y) = 3(y-1)^{2/3y}$. I want to show that $y' = f(x,y)$ has two solutions for $y_0 = y(0) = 1$ and one solution for $y_0 = y(0) = 2$. I am trying to understand the solution to this problem. ...
1
vote
1answer
60 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 ...
1
vote
1answer
26 views

Conditions to apply variation of constant méthod on a Cauchy problem

I try to solve the following Cauchy Problem on the interval $[0,T[$ for a $T>0$: $ \left\{\begin{matrix}y'(t) = -\frac{4}{t} y(t) \\ y(0)=0 \end{matrix}\right. $ What I did: I found the ...
1
vote
1answer
37 views

Result of Cauchy Schwarz inequlaity (homogeneous case)

I have shown the Cauchy Schwarz inequality to be $$ \langle v_i(\vec x),v_j(\vec y)\rangle^2 \ \le \ \langle v_i(\vec x),v_i(\vec x)\rangle\langle v_j(\vec y),v_j(\vec y)\rangle.$$ How would I use ...
0
votes
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) = \...
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=...
0
votes
1answer
140 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
0
votes
1answer
88 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)=...
0
votes
1answer
65 views

Consider this Cauchy Problem

Consider the Cauchy problem $$\left\{ \begin{array}{l l} u_{t} - \kappa u_{xx}=0 & \quad \mbox{$x \in \mathbb{R}, t>0$,}\\ \quad u(x,0) = \psi(x), \end{array} \right. $$ where $\kappa > 0$ ...
0
votes
1answer
27 views

What is the solution to the following Cauchy problem?

What I have tried to do was to separate variables and integrate both sides but what I got was y+ln|y-3|=ln|t-3| + c which is undefined for y=3 and t=3. but I don't understand how to get the range for ...
0
votes
1answer
122 views

PDE Cauchy Problems

How do I find the solution to these Cauchy problems? $$ $$ 1. $$\left\{ \begin{array}{l l} yu_{x} +xu_{y}=0 & \quad \mbox{$x,y \in \mathbb{R}$}\\ u(0,y) = y^4 \end{array} \right. $$ 2. $$\left\{ \...
0
votes
1answer
41 views

How to solve Cauchy problem for $y'+2y= \exp(x)y^2 \quad y(1) = 1$?

The original equation is $$y'+2y= \exp(x)y^2 \quad y(1) = 1$$ Well, this one is Bernulli's one, so I've put $z = \frac{1}{z}$ therefore $z' = -\frac{y'}{y^2}$ and reduced it to the first-order non-...
0
votes
1answer
131 views

Solving Cauchy Problem Using Lagrange Method

The question: Solve the Cauchy Problem stated below using Lagrange Method: $2Z_x-3Z_y+(x+y)z=0$ $z(x,0)=x^2 $ I attempted solving it by writing the characteristic equation: $$\frac{dx}{2}=\frac{...
7
votes
0answers
188 views

A function such that any solution to the Cauchy problem has no entire solution in $\mathbb{R}^2$

I'm trying to figure out this problem: Find a smooth function $a(x, y)$ in $\mathbb{R}^2$ such that, for the equation of the form $$u_y + a(x, y)u_x = 0,$$ there does not exist any solution in the ...
6
votes
0answers
68 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 ...
3
votes
0answers
56 views

Extending three vectors to commuting vector fields

The following claim can be found in page 27 of Petersen's book "Riemannian Geometry": "... any three vectors can be extended to vector fields that commute." I have been unable to prove the statement,...
2
votes
0answers
20 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
votes
0answers
50 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}^...
2
votes
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 \\ \...
2
votes
0answers
101 views

Computation of a wave equation using Kirchoff's formula !!

I want to solve this wave equation in $3$-D using Kirchoff's formula but all I've done it doesn't work. In particular I'm not able to compute the integral in Kirchoff's formula. you can see the ...
2
votes
0answers
308 views

Solving a Cauchy problem

Given the equation $$ y' = e^t \sqrt[3]{y^2}$$ (a) consider the related Cauchy problem with $y(t_0)=y_0$. What $P(t_0,y_0)$ ensures the problem has a unic solution? (b) find the general integral of ...
1
vote
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 $\...
1
vote
0answers
36 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 ...
1
vote
0answers
46 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 ...
1
vote
0answers
77 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
96 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^...
1
vote
0answers
49 views

How to prove the inequality of solutions for two Cauchy problems?

Consider two continuous and Lipschitz continuous with respect to the second variable functions $g_1:[0,1]\times\mathbb{R}\to\mathbb{R}$ and $g_2:[0,1]\times\mathbb{R}\to\mathbb{R}$ such that $g_1\le ...
1
vote
0answers
26 views

Cauchy problem PDEs help please

$$U_{xx}+U_{yy}=0$$ with $U = 1$ on $x^2+y^2=1$ and $\partial U/\partial n = 1$ on $x^2+y^2=1.$ ($n$=normal). So I tried switching $U_{xx}+U_{yy}=0$ to polars and got $U_{rr}+(1/r^2)U_{\theta \theta}...
1
vote
0answers
53 views

Wave equation with real analytic data

I was wondering if the solution of a wave equation is real analytic when the coefficients, Cauchy data, and boundary data is real analytic. To be precise, consider two types of problems: \ An initial ...
1
vote
0answers
62 views

I need a hint in solving Cauchy problem for differential equation

I need a hint for the following problem. It seems that separating variables does not work. Please leave just a hint not a whole solution. Solve the Cauchy problem. $y'(x)=\frac{x^{2}}{2+\sin(x^{2})}\...
1
vote
0answers
173 views

Alternative solution to fixed point iteration, Cauchy problem

I am solving some Cauchy problems with implicit Euler and Crank Nicolson method. Obvioulsy it's necessary, with these methods, solve a non linear equation. So I choose fixed point iteration for some ...
0
votes
0answers
21 views

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 ...
0
votes
0answers
21 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: $\...
0
votes
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^\...
0
votes
0answers
87 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
0answers
23 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+...
0
votes
0answers
58 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-...
0
votes
0answers
31 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 -...
0
votes
0answers
95 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
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 ...
0
votes
0answers
83 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 ...
0
votes
0answers
172 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 ...
0
votes
0answers
130 views

Cauchy's theorem proof (sequences)

Proof thr following of cauchy's theorems: If (a$_m)m\geq$0 is a convergent sequence in $\mathbb{R}$ and $b=\lim_{m\to \infty} a_m$ then $$\lim_{m\to \infty} \frac{a_0 + a_1 + ... + a_m}{m+1}= b $$ ...
0
votes
0answers
55 views

solve cauchy problem not homogeneous with fundamental solution

some book or some method to solve cauchy problem not homogeneous with fundamental solution in distributions theory, ie, I want to solve $Lu=f(x,t) $ $u(0,x)=\phi(x) $ $u_t(0,x)=\psi(x)$ with $L=\...
0
votes
0answers
110 views

Difference between weak and strong formulation of a Cauchy problem solution

While studying the theory of differential equations (only ordinary equations in fact, I'm just getting started), the teacher told us about these two different kinds of formulation. Given a Cauchy ...