Questions tagged [cauchy-problem]

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

1
vote
1answer
133 views

Unique solution of Cauchy problem in a neighbourhood of given set

Consider the Cauchy problem $yu_x-xu_y=0$ where $u=g$ on $S=${$(x,y): x+y=1, x>1$} Prove that the Cauchy problem has a unique solution in a neighbourhood of $S$ for every differentiable function $...
2
votes
2answers
389 views

Cauchy-Riemann equations, $f(z) = \bar{z} \sin z$

Find all $z \in \mathbb{C}$ such that the function $f(z) = {\bar z}\sin(z)$ satisfies the Cauchy-Riemann equations. Own work: Let $z = x+iy$ where $x$ and $y$ are real numbers Thus, ${\bar z} = x-...
0
votes
2answers
44 views

Laplace transform of this problem's solution

Hi have to determinate the Laplace Transform of u where u is the solution to this problem: I really have no idea on how to manage this mess.
1
vote
1answer
90 views

Laplace transform of a Cauchy's problem

I have to determine the Laplace transform of $u$, where $u$ is the solution to this Cauchy's problem $$ u'(t) + \int_0^t e^{t-s} \left( \int_0^s u(r) ~ dr \right) ~ ds = 0, t > 0, u(0) = 1$$ $L[u](...
1
vote
1answer
3k views

Find limit by using Cauchy's second limit theorem

I have to find the limit of this problem. I have found that with problems like this, I always get wrong answers when I try to use Cauchy's second limit theorem. I do not understand how to solve this ...
1
vote
0answers
175 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 ...
2
votes
0answers
309 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 ...
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 ...
1
vote
1answer
93 views

A boundary value problem on the unit disk

I'm working through some problems in PDE by myself and came across one that I can't seem to figure out, I'm betting it's an easy observation I'm just not making. Problem Let $B$ be the unit disc in $...
1
vote
1answer
64 views

Cauchy problem has no global solution (Lee Smooth Manifolds 9.23.c)

$\def \pux{ {\partial u \over \partial x}} \def \puy{ {\partial u \over \partial y}} \def \px{ {\partial \over \partial x}} \def \pz{ {\partial \over \partial z}} \def \py{{ \partial \over \partial y}}...
2
votes
1answer
98 views

Ordinary Differential Equation (1°ord.), positive or negative constant when solving Cauchy problem?

I have this easy ODE: EDIT: $$y'(x)=2\sqrt{y(x)}, \\y(0)=1$$ (OLD: $y' = (x - 2)/2;$ $y(0) = 1$) The general integral/solution is $\sqrt{y} = x + c$, so $y= (x + c)^2$ Substituting the initial ...
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
133 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{...
3
votes
1answer
428 views

Find the solution of the Cauchy problem $u_t-uu_x=0$.

For the Cauchy problem $u_t-uu_x=0, ~x\in \mathbb{R}, t>0$ with $u(x,0)=x,~x\in \mathbb{R} $, which of the following statements is true? the solution $u$ exists for all $t>0$. the ...