Questions tagged [cauchy-problem]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
45 views

Solving Cauchy problem PDE and drawing the solution graph

I want to solve and draw the solution of PDE such that \begin{align} \rho_t +2\rho\rho_x=0 \end{align} and with initial condition \begin{align} \rho(x,0)= \begin{cases} 1 & \text{if $x<0 $}\\ ...
JAEMTO's user avatar
  • 661
1 vote
1 answer
24 views

Logistic equation and Cauchy - Lipschitz theorem

I consider the following differential equation $$ x’(t)= rx(t)(1-\frac{x(t)}{K}) $$ Where $r$ and $K$ are constant. I consider an initial condition $x(0)=x_0\in (0,K)$. In my lecture notes, it is ...
coboy's user avatar
  • 1,195
0 votes
1 answer
37 views

Elementary proof of Cauchy-Lipschitz's theorem in a simple case

Let $a,b,c \in \mathbb{R}$, with $a \neq 0$. we consider the linear homogeneous differential equation : $(E) : ay''+by'+cy=0$. Is there an "elementary" way (i.e. without invoking "big&...
antwomorfisme's user avatar
1 vote
0 answers
29 views

Local Lipschitz continuity: existence and uniqueness of a solution to a ODE Cauchy problem

The version of the Picard–Lindelöf that I know, which is possibly one of the most common statements of the theorem states that, given a closed rectangle $D\subset \mathbb{R}\times\mathbb{R}^n$ and let ...
Mr. Feynman's user avatar
1 vote
0 answers
15 views

Qualitative study of the solution of the Cauchy problem: $x' = \log x$, $x(0) = a$ where $a > 0$

I have the following exercise: Consider the Cauchy problem: $$\begin{cases}x' = \log x \\ x(0) = a \end{cases} \quad \text{where } a > 0 .$$ Perform a qualitative study of the solution of the ...
user665110's user avatar
0 votes
0 answers
23 views

Unique existence of a weak solution for inhomogeneous parabolic equations in a broader settng than $L^2$? - request for references

For notations, let $p \in (1,\infty)$ and $p':=\frac{p}{p-1}$. For $n \in \mathbb{N}$, define $\mathbb{T}^n:= (\mathbb{R} / \mathbb{Z})^n$. Let us consider the following cauchy problem on $[0,T] \...
Keith's user avatar
  • 7,411
0 votes
0 answers
30 views

Extending solutions of a differential equation

I have a following exercise: Prove that every solution of an equation $x' = \sin(x^2+t^2) + 3|x|$ can be extended to all $t$ in real numbers. How do I go about proving this? I know theorems stating ...
de_michael's user avatar
0 votes
0 answers
6 views

Hypothesis on a parameter to ensure the unique solvability of a modified Laplace equation

Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following ...
SAKLY's user avatar
  • 465
0 votes
0 answers
29 views

Show supremum norm of solution to ode is less than $k$ i.e. $||e^{At}|| \leq k$.

So I'm trying to make an exercise in ode's. Let $A \in M_5(\mathbb{R})$ be a matrix with 3 eigen-values $\lambda_1 = -1$ with multiplicity $3$ and $\dim(\text{Ker}(A+I))=1$, $\lambda_{2,3} = \pm 2i$. ...
Dorelanië's user avatar
1 vote
0 answers
27 views

Help with the solution to the Cauchy problem of the firdst order partial differential equation (quasi linear case)

I am looking for the help with the solution to the following PDE (along with IC): $$ uu_x + u_t = u, u(x, 0) = 2x $$ Solution (without parameterization): Applying the method of characteristics one can ...
Катерина Ковальова's user avatar
-2 votes
1 answer
33 views

How do you solve Cauchy problem to the first order PDE [closed]

How do you solve Cauchy problem to the first order PDE of the form: $y^{-1}u_x+u_y=u^2, u(x,1)=x^2$
Катерина Ковальова's user avatar
0 votes
0 answers
25 views

Cauchy conversion Criteria Integral

I have the following question: I want to show, that the improper integral exists: $$\int_{0}^{b}\tfrac{\sqrt{1+y'(x)^2}}{\sqrt{y(x)}}$$ I stated: For $$[t,b] \subset(0,b]$$, all t must be in the ...
Bastian Sommerfeld's user avatar
1 vote
0 answers
37 views

A particular Cauchy problem for first-order PDE

Is the following result true? Suppose $M$ is a smooth manifold, $W\subseteq J^1M\cong M\times T^\ast M$ is an open subset, $F\colon W\to \mathbb R$ is a smooth function, and $(x_0,u_0,p_0)\in W$. Let ...
Parco Macelli's user avatar
1 vote
1 answer
45 views

Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian

I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...
kumquat's user avatar
  • 137
3 votes
1 answer
42 views

About differentiable dependence in a Cauchy problem

Let $\epsilon >0$, consider the Cauchy problem: $$\epsilon x' = x^2 + (1-\epsilon)t \quad , x(0)=1$$ If $x(t;\epsilon)$ denotes the solution (defined on the maximum interval) of the problem, I'm ...
Yauset Cabrera's user avatar
2 votes
1 answer
136 views

On number of solutions of ODE $y’=-y^a, y(0)=0$.

How many solutions the ODE of the type $y’=-y^a, y(0)=0, 0<a<1$ have? I am able to prove that the IVP $y’=y^a, y(0)=0$, where $0<a<1$, has infinite number of solutions by finding it’s ...
neelkanth's user avatar
  • 6,038
0 votes
0 answers
35 views

Cauchy problem for parabolic equation

Let $\mathcal{L}=a \partial_{xx}+b\partial_{x}+c$, where $a>0,c\in\mathbb{R},b\in \mathbb{R}$, I want to know if the following parabolic equation gets a smooth and unique solution \begin{align} &...
yxyt's user avatar
  • 73
1 vote
0 answers
24 views

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 ...
MicahW's user avatar
  • 11
1 vote
0 answers
34 views

Does this Cauchy problem $x'=tx^{2/3}$, $x(0)=0$ have unique solution? [duplicate]

Consider the following Cauchy problem: $$ \left\{ \begin{array}{lcc} x'=tx^{2/3} \\ \\ x(0)=0 \\ \end{array} \right. $$ I proceed as usual. First we notice that $x(t)=0 \ \forall t \in \mathbb{R}$ ...
José's user avatar
  • 1,020
0 votes
0 answers
13 views

Understanding the meaningful of the vanishing data on the boundary of a half space. (Hormander)

Hi, I am reading about the concept of a characteristic Cauchy problem in Hormander's books, but not understanding the following: Question. What is the meaningful of vanishing data on the boundary of ...
eraldcoil's user avatar
  • 3,458
-1 votes
1 answer
54 views

Exercise1.1 of Tao's book <Nonlinear Dispersive equation> which is about Cauchy Kowalevski Theorem

In the book of Tao's nonlinear dispersive equations Exercise1.1, the author want us to prove $$\|\partial^{m}_{t}u(0)\|_{\mathcal{D}}\leq K^{m+1}m!$$ use the equation $\partial_{t}u=F(u(t))$ where $F:\...
monotone operator's user avatar
4 votes
0 answers
191 views

How to prove Picard's existence and uniqueness theorem by Tonelli sequence instead of Picard sequence? For O.D.E./ODE.

$\qquad$First of all, this question for O.D.E. comes from an end-of-book exercise with no answer. $\qquad$Secondly, allow me to give the definitions of the relevant contents in the question to avoid ...
daidaitx's user avatar
  • 137
3 votes
0 answers
109 views

Problem with the differential equation $2u_{xx}-3u_{xy}+u_{yy}+u_x-u_y=1$

Specify the largest domain in which the given Cauchy problem has a single solution, and find this solution $$2u_{xx}-3u_{xy}+u_{yy}+u_x-u_y=1, \; u\Bigg|_{x=0,y>0}=-2y, \; u_x\Bigg|_{x=0,y>0}=-1$...
Dmitry's user avatar
  • 1,218
4 votes
1 answer
76 views

A relation between Dirichlet problem and Brownian motion

I'm reading about Dirichlet problem and Brownian motion in these notes, i.e., Fact. Let $D$ be an open and bounded domain in $\mathbb{R}^n$ and $\partial D$ be its (smooth) boundary. Let $h \in \...
Akira's user avatar
  • 17.2k
0 votes
1 answer
395 views

Solve the Cauchy problem by the method of characteristic $pz+q=1$ with initial data $y=x,z=x/2$ . Indicate the region where the solution is valid.

Solve the Cauchy problem by the method of characteristic $pz+q=1$ with initial data $y=x, z=x/2$. Indicate the region where the solution is valid. How to solve this problem. Lagrange's auxiliary ...
math131's user avatar
  • 165
1 vote
0 answers
46 views

solution of EDO defined on a specific interval

How can I prove that$$ \left\{\begin{array}{l} y^{\prime}(t)=1-(t+1) e^{y(t)} \\ y(0)=y_0 \end{array}\right. $$ the solution is defined in $[0,+\infty)?$. My teacher said first to prove that $e^{y(t)} ...
Dsrksidemath's user avatar
2 votes
0 answers
48 views

Can you solve $y'(x) = y^2(x)\;\land\;y(x_0) = y_0 $ on $]-1, 1[$ using the Banach contraction theorem?

I would like to solve the following Cauchy problem: $$ \begin{cases} y'(x) = y^2(x)\\ y(x_0) = y_0 \end{cases}\tag 1 $$ In my opinion, using the Banach contraction theorem it can only be solved in $I:=...
user avatar
-2 votes
1 answer
70 views

Application of Cauchy's integral formula [duplicate]

Let $f(z)$ be analytic in a neighbourhood of $z_0$, where $f'(z_0)$ does not equal $0$. Show that $$\int_C\frac{\mathrm{d}z}{f(z)-f(z_0)} = \frac{2\pi i}{f'(z_0)}$$ where $C$ is a small (as small as ...
Bond's user avatar
  • 53
2 votes
1 answer
109 views

Unqueness of the weak solutions of transport equation

Let $a\in \mathbb{R}$ and consider the IVP \begin{eqnarray} u_t+au_x&=&0 \quad \quad (x,t) \in \mathbb{R} \times \mathbb{R}^+\\ u(x,0)&=&u_0(x) \quad \quad x \in \mathbb{R}. \end{...
Rosy's user avatar
  • 995
1 vote
1 answer
79 views

Find initial condition so that ODE has multiple solutions

Given Cauchy problem $$y'=\dfrac{x^2}{y(1+x^3)}, \;y(x_0)=y_0.$$ Is it possible to find $(x_0,y_0)$ such that given Cauhcy problem has multiple solutions? It is obvious that equation is with separable ...
Mark's user avatar
  • 354
2 votes
1 answer
117 views

Convergence of solution of non linear problem using heat equation

I am considering the following non linear problem: $u_{t}(t,x)-u_{xx}(t,x)+(u_{x}(t,x))^2=f(t,x)$ for $t>0, x \in (0,1)$ $u(0,x)=u_{0}(x)$ for $x \in [0,1]$ $u(t,0)=u(t,1)=0$ for $t>0$ where f ...
Aron's user avatar
  • 99
1 vote
1 answer
112 views

How can I find the solution of the Hopf equation $u_t + u u_x = 0$ with the initial condition $u(0,x) = \rho(x)$?

Here, $\rho$ is equal to: \begin{align}{\rho(x) = } \begin{cases} a & x<0 \\ \frac{a}{L} (L-x) & 0 \le x \le L \\ \frac{a}{L} (x-L) & L < x < 2L \\ a & x \ge 2L \end{cases} ...
Student's user avatar
  • 317
0 votes
2 answers
48 views

PDE question that hard to see second first integral

Solve the IVP. $x\dfrac{\partial z}{\partial x}+y\dfrac{\partial z}{\partial y} = z- x^2-y^2, \; z\vert_{y=-2} = x-x^2 $ $\underline{\text{My Attempt:}}$ I first wrote Characteristic Equation for the ...
beingmathematician's user avatar
4 votes
1 answer
187 views

Does the solution of $y' = (x^2 + y^2) e^{-(x^2+y^2)}$ have a limit for $x \to \infty$?

An old exam problem I am trying to solve is as follows: Given the cauchy problem $y' = (x^2 + y^2) e^{-(x^2+y^2)}, y(x_0) = y_0$, do the following: Show that there is a unique solution for all $x \in ...
Jesus's user avatar
  • 1,788
3 votes
0 answers
58 views

Solve the Initial-Value Problem $\quad x\frac{\partial u}{\partial x}+ u\frac{\partial u}{\partial y}\,= u + 2x^2$

Solve the initial value problem: $$\,$$ $$ x\;\frac{\delta u}{\delta x}\,+\, u\;\frac{\delta u}{\delta y}\,=\, u + 2x^2,\qquad \mathrm{ with\;initial\, conditions}:\,u\,(x,\,\frac{1}{4}-x^2)=x$$ $$\,$$...
MarkoVoNeumann's user avatar
2 votes
1 answer
73 views

Cauchy problem for the wave equation.

I have the system $$u_{tt} = u_{xx}$$ for $x \in \mathbb{R}$ and $t>0$ with initial conditions $$u(x,0) = \phi(x) = 0, \hspace{5pt} u_{t}(x,0) = \psi(x) = \left\{ \begin{array}{ll} 1 & ...
Humberto M. Peña's user avatar
1 vote
0 answers
29 views

Proof full conditionals are normal

I am confused on how to approach this problem. How can I show the conditional in question a?
MathMan 99's user avatar
0 votes
1 answer
34 views

Can someone help me fill in the gaps of a Cauchy PDE problem?

I was given the following problem to solve: Ux + Uy = U^2 x=0, U = e^(-y^2) This is how I was taught to solve it: ...
Sam Moldenha's user avatar
-1 votes
1 answer
364 views

How to find the angle of a projectile launch knowing only the initial velocity and the coordinates?

I need to find the angle [α] of a projectile launch knowing only the initial velocity and the coordinates of a target that projectile needs to hit [$x_*$, $y_*$]. Coordinates of the start point are ...
Eddy's user avatar
  • 1
1 vote
1 answer
64 views

Has variable shuffling in linear maps been studied? [duplicate]

Specifically, say I have a linear map $\mathbb{R}^2\rightarrow\mathbb{R}^2$. I want to construct from it a map between a regrouping of the vector spaces. Concretely, I start with \begin{align} \left[...
Sean Lake's user avatar
  • 1,737
0 votes
2 answers
55 views

Cauchy integral formula with a singularity inside the set

Im struggling a bit with a Cauchy integral formula exercise. Compute $\int_{C(0,4)} \frac{e^z}{z(z-3)} dz$ In past exercises i've been able to manipulate the integral to not have any singularities for ...
uoiu's user avatar
  • 591
0 votes
0 answers
53 views

Cauchy's Problems

While reading the Cauchy's problem I have some following doubts: The number of the solutions of Cauchy's problem can be finite? What is the necessary condition for the unique solution?
Mathemusica's user avatar
1 vote
1 answer
102 views

Cauchy problem solution

Consider the Cauchy problem defined by $$ \left\{ \begin{array}{c} y' = e^{-x^2}\sin(y) \\ y(0) = a \end{array} \right. $$ with $ a \in \mathbb R $. If $ y_a(x) $ is the solution, for which values ...
GN00Fu's user avatar
  • 59
2 votes
0 answers
117 views

Linear reaction-diffusion PDE with variable reaction coefficient

I am considering a Cauchy problem for linear reaction-diffusion PDE of the form: $$\dfrac{\partial u}{\partial t} = a\Delta u + b(x)u, \quad u(t=0,x) = u_0(x), \quad x\in\mathbb{R}^{n},$$ where $\...
Abhishek Halder's user avatar
1 vote
1 answer
81 views

Numerical solution to second order ODE with an infinity at r = 0 (Auxiliary Cauchy Equation)

I am interested in numerically solving the ODE below for the function $\theta(r)$: $$-\theta''=\frac{1}{r}\theta'-\frac{1}{r^2}\sin(\theta)\cos(\theta)-\frac{D}{Ar}\sin^2(\theta)+\frac{K}{A}\sin(\...
B-G's user avatar
  • 11
4 votes
2 answers
299 views

Solving the first-order non-linear differential equation $y' = y^2 - 2 x$

I am trying to solve this Cauchy's problem: $$ y' = y^2 - 2x $$ with condition $y(0) = 2$ It's very similar to Bernoulli equation $$ y' + a(x)y = b(x)y^2$$ however doesn't contain $a(x)y$. I also ...
alphbt's user avatar
  • 73
2 votes
0 answers
30 views

Exististence result for semilinear parabolic PDE

I am considering the following semilinear parabolic PDE: $$ \partial_tu+\theta(x-y)\partial_yu+\frac12x^2\partial_{xx}u+\frac12y^2\partial_{yy}u + y^{-3/2}u^2+a=0$$ with terminal condition $$u(T,x,y) =...
Zwei's user avatar
  • 65
2 votes
1 answer
149 views

existence of a solution of a differential equation

we have a continuous function $\omega:\mathbb{R}_+\rightarrow\mathbb{R}_+ $ such that $$ p'(t)=\omega(p(t))\\ p(0)=0$$ has only the trivial solution $p=0$ on $J=[0,b]$. Now the author ...
Skimann's user avatar
  • 45
2 votes
1 answer
62 views

Confusion about Finite Differences Method Notation

I would need a confirmation that I am understanding one of the steps in the lecture notes about numerical methods right. We have an ODE with $y\in C^1(t)$: \begin{align} y'(t) &= f(t, y(t))\\ y(...
adal's user avatar
  • 21
1 vote
1 answer
97 views

Existence of a unique solution to Cauchy problem

Let $X$ Hilbert space. Let $\mathcal{A}:D(\mathcal A)\subset X\rightarrow X$ nonlinear and $f:[0,+\infty)\rightarrow X$ $$\begin{cases}U_{t}(t)=\mathcal{A} U(t)+f(t),& \forall t>0, \\ U(0)=U_{0}...
user1072831's user avatar

1
2 3 4 5 6