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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.

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain.

A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition), but it can be none of them. They are named after Augustin Louis Cauchy.

Source: https://en.wikipedia.org/wiki/Cauchy_problem

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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 ambiguity. You can also skip or skim this section…
daidaitx
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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 entire $\mathbb{R}^2$ for any nonconstant…
TrivialCase
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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 the method of characteristics (Charpit's equations),…
glowstonetrees
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Differentiability of P.D.E With given initial conditions on Its Characteristic base curve.

I am facing a major problem in $$u_t +u u_x=1,\;\;\;\; \text{with initial condition}\;\; u \left(\frac{t^2}{4},t\right)=\frac{t}{2}$$ this P.D.E., here $x \in \mathbb{R} $ and $t>0$ Solution I tried - The auxiliry equation of this equation is…
gaurav saini
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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 $y(0)=1$ : $y(0)=\frac{1}{2}(e^{2c_1}-1)=1$ my doubt is about the fact that i don't know…
Gabriele Sortino
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Problem on the domain of the solution of a differential equation

Let $f:[0,\alpha]\to\mathbb{R}$ be a solution of the Cauchy problem: $\begin{cases} f'(t)=(f(t))^2+t \\ f(0)=0 \end{cases} $ The question is: prove that $\alpha<3$. It is clear that the problem admits a unique solution locally on some…
PozzPlot
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Attenuation estimation of the solution of the two-dimensional wave equation Cauchy problem

This is the equation given, $$\begin{array}{l} u_{tt}=a^{2}\left(u_{x x}+u_{y y}\right), \\ \left\{\begin{array}{l} \left.u\right|_{t=0}=\varphi(x, y), \\ \left.u_{t}\right|_{t=0}=\psi(x, y) . \end{array}\right. \end{array}$$ Its solution $u(x,y,t)$…
Zydragon
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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 $h \in \mathcal{C}(\partial D)$. Then there exists a unique…
Akira
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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 $x \in \mathbb{R}$ Does the limit $ \lim_{x \to \infty}…
Jesus
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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 tried substitution like $y = tx$. but it hasn't…
alphbt
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Prove that set of limits bound is open.

Consider the following problem: $\dot{x} = A(t) x$. Let $X_{A}(t,s)$ be the Cauchy operator of the following system. Let $M$ be the space of continuous functional matrices. Let's denote by: $$ \rho(A, B) = \sup_{t \in \mathbb{R}_{+}} \|A(t) -…
openspace
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$yu_x+xu_y=u$ with two conditions using method of characteristics

$\begin{cases} yu_x+xu_y=u\\u(x,0) = x^3 \\ u(0,y)=y^3\end{cases}$ $\dot X(\sigma,s)=Y \quad X(\sigma,0)=\sigma \\ \dot Y(\sigma,s)=X \quad Y(0,s)=s \\ \dot U(\sigma,s)=U \quad U(\sigma,0)=\sigma^3\quad and \quad U(0,s)=s^3$ $\frac{dU}{ds}=U\implies…
2Napasa
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How to find $\lim\limits_{t \to \infty} \int\limits_0^\infty u(x,t) dx$?

I have $u(x,t)$ - solution of Cauchy problem $$u_t=u_{xx},~~~u(x,0)=e^{-x^2},$$ where $t>0, x\in\mathbb{R}.$ Is there a way to find such a limit? I have seen several theorems which help to find value of $\lim\limits_{t \to \infty} u(x,t),$ but this…
Haldot
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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}^nf_i(x_1,\dots ,x_n)\frac{\partial w}{\partial…
KIM
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