Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

learn more… | top users | synonyms (2)

1
vote
1answer
33 views

Find wave equation initial condition such that solution consists of right-going waves only

Let $u(x,t)$ solve the wave equation $u_{tt}=c^2u_{xx}$ and let $u(x,0)=A(x)$ for some function $A(x)$. Find the function $B(x)=u_t(x,0)$ such that the solution consists of right going waves only. My ...
1
vote
1answer
37 views

An imbedding inequality in PDE.

u is a function of 3-dimension, I'm trying to prove this: $\|u\|_4^4 \leq C \|u\|^2_{H^1} \|u\|_{L^2}^2$ Anyone can shed light on this? Thanks very much.
0
votes
1answer
55 views

Inhomogeneous heat equation with source term orthogonality

This is a question on the lecture notes. Basically we have the usual heat equation: $$\frac{\partial y}{\partial t}(x,t)=k^2\frac{\partial^2 y}{\partial^2 x}(x,t)+F(x,t)$$ We also have the usual ...
0
votes
0answers
57 views

Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= (...
0
votes
1answer
53 views

Important ODE Solutions for Solving PDEs

Which ODEs pop up most often in the study of Partial Differential Equations such as the Heat Eq, Laplace Eq, Wave Eq, etc. At least in the homogeneous case. What are their solutions? I'm going to take ...
1
vote
1answer
26 views

Sturm-louville heat problem

Could I request for an example to the above question? I've read thru the regular sturm-liouville theory but have no idea how should the theory be applied to this problem. I understand that the method ...
0
votes
1answer
13 views

Clarification on matters relating the the differential operator in Sturm Liouville

My set of two notes, one of which might contain a printing error. In Sturm-Liouville problem, is the differential operator expressed as $$L(y) = \frac{d}{\text{dx}}\text{((P(x) }\frac{d}{\text{dx}}\...
10
votes
1answer
130 views

Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
0
votes
1answer
32 views

Sine Fourier series and smooth function.

I was reading through my text on PDEs and came across a theorem (or perhaps Lemma) that states: "For any smooth function $g_1(y)$ with $g_1(0) = g_1(h) = 0$, it can be expressed as a Fourier sine ...
1
vote
0answers
38 views

Converting solutions to separation constant to Cosh and Sinh

The Laplace's equation inside a rectangle is: $$u_{\text{xx}}+u_{\text{yy}}\text{=0}$$ The IC's are: $${u(0,y)=g(y)}$$ $${u(L,y)=0}$$ $${u(x,0)=0}$$ $${u(x,H)=0}$$ Via method of separation we have ...
9
votes
1answer
234 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is $C^{\infty}(\...
1
vote
1answer
33 views

Why are porous medium equations posed on connected domains? Shouldn't it be done on a domain with holes (or pores)?

The porous medium equation is supposed to model gas flow through porous media (i.e. some object with holes in it). Why then, in theory of weak solutions, do people study the equation on a sufficiently ...
1
vote
0answers
20 views

Charcteristic directions

Well in my assignment I need to find the characterstic directions of a PDE. A characteristic direction is a line along which the function behaves as an ODE. The problem PDE is: $$\frac{\partial^2 u}{...
0
votes
1answer
21 views

Motivated by Level Sets, how can I show that minimizing this functional is equivalent to this PDE?

I would like to show, that minimizing the functional $$F(g)=\alpha\int_\Omega |\nabla g(x)|^2dx+\mu \int_\Omega (g(x)-f(x))^2dx $$ is equivalent to solving the differential euqation $$-\alpha\nabla g(...
0
votes
1answer
70 views

Schwarz Reflection Principle for Harmonic Functions

Given $\Omega \subset \mathbb{R}^n$ define $\Omega^+ = \Omega \cap \{x_n>0\}$ and $\Omega^0$, $\Omega^-$ analogously let $u \in C^2(\bar{\Omega}^+)$ be harmonic and such that $\frac{\partial u}{\...
0
votes
1answer
52 views

Rewriting multivariate second order diffrential equation as system of first order

I hope someone can shed some light on the steps taken in between, as I have the answer and the problem, but no idea how to get there: Given the second order differential equation $$\frac{\partial^2 ...
5
votes
1answer
123 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with $...
0
votes
0answers
18 views

Solving equations of the form $\phi(D_{x},D_{y})u=0$

I am taking a course called partial differential equations. We are mostly learning about solving equations of the form $\phi(D_{x},D_{y})u=0$, for example $(D_{x}^{2}+D_{x}D_{y}+D_{y}^{2})u=0$ or $\...
1
vote
0answers
38 views

Dirichlet problem for Laplace equation in $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$

I'm having troubles with the proof of the following theorem: let $\Omega=\left(\overline{B\left(0,a\right)}\right)^{c}$ and $U\in \mathbb{R},U>0$, then there exists a unique function $\psi\in C^{2}\...
3
votes
1answer
43 views

References for the operator $(I-\Delta)^{\alpha /2}$

I am studying PDEs involving fractional differential operators, and I have found a few properties for the operator $(I-\Delta)^{\alpha /2}$ scattered through scientific papers. I wonder if there is a ...
1
vote
2answers
121 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial B(0,\varepsilon)}\varepsilon^{-\alpha}\,dS=\...
10
votes
3answers
1k views

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an ...
2
votes
1answer
179 views

spectral theory of Laplacian on $\mathbb R^n$ [duplicate]

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$? I am interested for which values $z \in \mathbb C$ the equation $\...
1
vote
1answer
57 views

PDE - Don't understand teacher's solution

I'm reading a solution to a problem in PDE class that the teacher gave, and I don't fully understand his solution. The problem is $\frac{dx}{1+\sqrt{z-x-y}}=dy=\frac{dz}{2}$ and what we want is to ...
0
votes
2answers
78 views

Can we show this inequality? (PDE question)

I am attempting to show that $$\int^t_0 \left[e^{-\lambda \kappa (t-s)} a(s)\right] ds \leq \int^t_0 \frac{\lambda}{2\kappa} a(s)^2 ds \tag{1}\label{1}$$ for any $t$, where $\lambda, \kappa >0$, $a(...
1
vote
1answer
29 views

How to find the cosine series when solving a PDE with Dirichlet conditions?

Suppose I have to solve $\sum_{n=0}^{\infty} A_n \cos(\frac{(n+1/2)\pi x}{L}) = x $ from $0$ to $L$. If I we want to find $A_n$ my professor uses the formula for a cosine series: $$\frac{2}{L}\int_{...
1
vote
1answer
72 views

Solve a second-order PDE with non-constant coefficients

Solve the following equation: $$\frac{\partial u}{\partial t}+ax^2(1-x)^2\frac{\partial^2u}{\partial x^2}=0$$ with the boundary condition $u(1,x)=(.5-x)^2$, $u(t,0)=u(t,1)=.25$. Domain is $t\geq0, x\...
3
votes
2answers
163 views

Partial Differential Equation with no solution - Transversality condition?

I have the following equation: $$ x u_x + y u_y = \frac{2e^u }{xy } , x>0,y>0 $$ with the initial condition (corresponding to $t=0$ ): $$ \Gamma =\{ (s,s,0) | 0<s<\infty \} $$ By using ...
0
votes
1answer
138 views

Solve laplace equation for a semi-infinite plate. Where is my mistake?

The plate is semi-infinite. 2 Of its sides have $f=0$ and the bottom part satisfies $f=cos(x)$. Its width is $\pi$. The temperature distribution $f(x,y)$ satisfies the Laplace equation $\nabla^2 f=0$. ...
0
votes
1answer
46 views

Partial Differential Equation - The Chain Rule

$\displaystyle \sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}\zeta^{2}u_{x_{j}}dx$ $\displaystyle =\sum_{i,j=1}^{n}\int_{U}a^{ij}D_{i}u\zeta^{2}D_{j}u dx$ Can someone please explain to me how we use the ...
0
votes
1answer
60 views

HELP to solve - PDE First order

I have this equation $$ u_x+uu_y=0$$ by the book "Handbook of First order Partial Differential Equations - page 290." The general solution is $F(ux-y,u)$, where $F$ is a arbitrary function. I try ...
1
vote
1answer
22 views

Transforming pde to nicer form?

I have a second order differential equation for $u$ $$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + 5u = 0$$ I am looking for a transformation $u(x,y) \rightarrow v(x,y)$ that gives $$\frac{d^2v}{dx^2} + ...
2
votes
1answer
23 views

Transformed pde but my answer doesn't match solution?

$$\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + \frac{du}{dx} + 2\frac{du}{dy} + 3u = 0$$ Let $u = ve^{ax + by}$ and find $a, b$ such that we can transform to the following equation $$\frac{d^2v}{dx^2} + \...
1
vote
0answers
33 views

Understand Dankwerts boundary conditions in plug flow pde

I'm trying to model the advection-diffusion-reaction plug flow equation in mathematica: $\frac{\partial C_a}{\partial t}$=$D_s$$\frac{\partial^2 C_a}{\partial z^2}-V\frac{\partial C_a}{\partial z}-...
1
vote
1answer
160 views

How can I solve numerically this partial differential equation?

I am reading this paper and came across a system of differential equations with 4 ODEs and 1 PDE. The PDE is given below. My question is how to solve this numerically in MATLAB , Python or Mathematica?...
1
vote
1answer
17 views

Convolution of derivatives

When transforming nonlinear PDE to its Fourier space, I encounter the following problem: Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get \begin{equation*} ...
1
vote
0answers
39 views

solubility of nonlinear overdetermined PDE for holonomic scaling of a frame

This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? . Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As ...
3
votes
2answers
69 views

How to Separate Quasi-Linear PDE

I'm attempting to solve the non-homogenous quasi-linear PDE below: $$z\frac{\partial z}{\partial x} - \alpha y\frac{\partial z}{\partial y} = \frac{\beta}{x^3y}$$ From what I've read in texts, the ...
1
vote
1answer
29 views

Equilibrium temperature in a heat equation

To find the equilibrium temperature distribution for a heat equation, $$U(x,t)$$ it is critical to note that the second partial derivatives WRT the space variables is zero. Why is this so?
2
votes
1answer
97 views

Extremizing the boundary value problem $I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$

Extremizing the boundary value problem $$I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$$ My Thought: First, we use Euler-Lagrange equation and solving we get , $y(x)=C_1x+C_2$. Then we put it in $I[y]$ ...
3
votes
2answers
209 views

Change of variables for heat equation

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of variables?...
2
votes
0answers
64 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}\sin x,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. By Duhamel's principle, ...
0
votes
1answer
143 views

Solve the Poisson equation $\Delta f = x_2$ in the unit disk

Let $D$ be the disk of radius 1 centered at (0,0). Find a formula for the solution of $\Delta u=f$ in $D$ $u=1$ on $\partial D$ In the case where $f(x)=x_2$ In polar coordinates $f(r,\theta)=rsin(...
1
vote
1answer
61 views

$\log|x|\in\text{BMO}(\mathbb R^n)$

Lemma. $A$ is fixed. For any ball $B$, there exists a constant $c_B$, which satisfies $$\frac{1}{B}\int_B|f-c_B|\leq A$$ then we have $f\in\text{BMO}(R^n)$. I want use the lemma to prove the ...
0
votes
1answer
26 views

Integral over a ball.

Let for $0<t<t_0$ $$e(t)=\int_{B(x, R(t_0-t))}v(x,t)dx$$ Given that $v(x,t)$ is a differentiable function prove that $$e'(t)=\int_{B(x, R(t_0-t))}v_t-R\int_{\partial B(x, R(t_0-t))}vdS$$ ...
1
vote
0answers
42 views

Well-posedness of semilinear elliptic equations

I am trying to understand the well-posedness of some semilinear elliptic problems, such as: $$-\Delta u +F(x,u)=0, \ \Omega$$ $$u=f \in C^{2,\alpha}, \ \partial \Omega, $$ or $$-\Delta u +F(x,\...
1
vote
1answer
56 views

The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
0
votes
0answers
25 views

Prove one property of harmonic function

Let $u(x)$ be a harmonic function defined in the square $[0,1]\times[0,1]$. Suppose that $u(x_k)=0$, where $x_k=(1/k,1/k).$ Prove that $u(x)=0$ everywhere in $[0,1]\times[0,1]$.
0
votes
1answer
31 views

$g(x,y,t) = f(x+yt,y,t)$ what is $\partial_t g$?

This is a simple question in multivariable calculus, but it confuses me. Say that $f \in C^2\left(\Bbb{R}^d \times \Bbb{R}^d \times [0,T]\right)$ and it satisfies a PDE, such as \begin{equation*} \...
1
vote
0answers
27 views

Find an explicit formula for the solution of the boundary value problem

Let $D$ be the disk of radius $1$ centered at $(0,0)$. Find an explicit formula for the solution of the boundary value problem $\Delta u=0$ in $D$ $u=x_1^2$ on $\partial D$ I just don't even know ...