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

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2
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1answer
17 views

some calculations need to be verified relates to $L^p$ norm

Hi I am having trouble verifying the equivalence of the following two inequalities. \begin{align*}\tag{1} \|u(t)\|_{L^2(\Omega)}&\le \|u_0\|_{L^2(\Omega)} \end{align*} \begin{align*}\tag{2} ...
2
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0answers
59 views

How to calculate analytical solution to fokker planck equation in finite domain?

I am looking for the solution to the following fokker planck equation. $\frac{\partial f(x,t)}{\partial t}= (k_1 x - v)\frac{\partial f(x,t)}{\partial x} + (k_2 x + D) \frac{\partial^2 ...
2
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0answers
63 views

Lax-Milgram theorem on Evans. If the mapping is injective why do we need to prove uniqueness again?

This is the theorem and its proof (From Evans L., Partial Differential Equations, p. $297-299$) If we already know that $\langle f,v \rangle = (w,v)$ = $B[u,v] = (Au, v)$ and we know that ...
2
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0answers
56 views

Solve general solution of system of first order partial differential equations

$$u_x + v_y = 0$$ $$v_x - v_y = 0$$ $$v_y - u_x = 0$$ We are instructed to solve the following system of first order equations. I have no idea where to begin. I have tried putting this into a ...
2
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1answer
38 views

Well posedness and regularity of diffusion advection with Robin BC

I have the following diffusion advection with time and space dependent coefficients with Robin BC \begin{equation} \left \{ \begin{array}{l} \partial_t u - div(B(t,x) \nabla u) + V(t,x) \nabla u = 0 ...
2
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0answers
25 views

Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
2
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0answers
88 views

Proving Compatibility of two Partial differential equation

Given two PDE(s): $F(x,y,z,p,q)=0$ and $G(x,y,z,p,q)=0$ In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of ...
2
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1answer
27 views

Solve the initial value problems for $u_t+2u_x=0$

$u_t+2u_x=0$ initial value is $u(-1,x)=\frac{x}{1+x^2}$ Using the characteristic method i find that $\zeta= x-2t$ so the solution will be $$u(t,x)=\frac{x-2t}{1+(x-2t)^2}$$ so therefore when i plug in ...
2
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0answers
29 views

Heat Equation Separation of Variables Goal

I am trying to solve a partial differential equation but I am confused on one part. The Setup: $u_t = u_{xx}$ on 0 < x < 1, t > 0. BCs: $u_x(1,t) = u(1,t)$ & $u_x(0,t) = 0$. I run into ...
2
votes
1answer
111 views

Applying a convex function to a weak solution yields a weak subsolution

PDE Evans, 2nd edition: Chapter 6, Exercise 11 Assume $u \in H^1(U)$ is a bounded weak solution of $$-\sum_{i,j=1}^n (a^{ij}u_{x_i})_{x_j} = 0 \quad \text{in }U.$$ Let $\phi : \mathbb{R} \to ...
2
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0answers
39 views

How to solve this differential equation with Fourier Transform?

Consider the differential equation $$\dfrac{\partial w}{\partial t} = -\alpha \dfrac{\partial w}{\partial x} + D \dfrac{\partial ^2 w}{\partial x^2}$$ together with the boundary conditions that ...
2
votes
1answer
79 views

Which PDE book covers these topics best?

I have an exam in January and I want to prepare ODE and PDE section first as they carry good weightage. For ODE I have S.L. Ross' book, which I like and have always referred to. But I haven't done PDE ...
2
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1answer
54 views

Solve $p^2x+qy=z$

Solve $ p^2x+qy=z$ where $p=z_x,\ q=z_y$ and $z=z(x,y)$. I have tried it by Charpit's method. Take $ f(x,y,z,p,q)=p^2x+qy-z=0$ Now $ f_x=p^2,\ f_y=q,\ f_z=-1,\ f_p=2px, f_q=y .$ So the Charpit's ...
2
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0answers
70 views

Solving PDE using Method of Characteristics or Transport Equation in N variables

I am given the following PDE to solve where $b \in \mathbb{R}^{n}$ and $a \in \mathbb{R}$ $u_t+b \cdot Du+cu=0$ in $\mathbb{R}^n\times(0,\infty)$ $u= g$ on $\mathbb{R}^n\times\{t=0\}$ I am told to ...
2
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1answer
32 views

Wave equation with nonsmooth initial data

Does this problem have a solution? $$\begin{cases} \partial_t^2u(x,t)&=\partial_x^2u(x,t) \qquad x \in[-1,1] \quad t>0 \\ u(x,0)&=1-|x| \qquad \quad x \in[-1,1] \\ \partial_tu(x,0)&=0 ...
2
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0answers
66 views

How can I prove that a PDE's solution is greater than $0$?

I have this PDE system: $\frac{d}{dt}x(r,t)=-\int_{[0,1]}J(|r-r'|)y(r',t)dr'x(r,t)$ $\frac{d}{dt}y(r,t)=\int_{[0,1]}J(|r-r'|)y(r',t)dr'x(r,t)-y(r,t)$ $x(r,0)=a(r), y(r,0)=1-a(r)$ where ...
2
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1answer
100 views

$C^\alpha$-regularity of elliptic PDE when $f$ is only continuous

Consider $\Omega\subset\mathbb{R}^n$, open bounded, $$ Lu=f\text{ in }\Omega,\quad u=0\text{ on }{\partial\Omega}, $$ with $Lu=a^{ij}D_{ij}u+b^{i}(x)D_iu+c(x)u=f(x)$, $a^{ij}=a^{ji}$, $L$: strictly ...
2
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0answers
32 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
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0answers
73 views

Mixed Dirichlet-Neumann eigenvalue problem

Let $\Omega\subset\Bbb R^2$ be a bounded $C^2$ domain. Let $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$. Does anyone know about the existence of eigenvalues and eigenfunctions for the ...
2
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1answer
52 views

$u=xf(xy)$, show that $xu_{xx}-yu_{xy} = 0$

I need to show that: $$xu_{xx}-yu_{xy} = 0$$ when $$u=xf(xy)$$ So, I did: $$u_x = xyf_x(xy)+f(xy) \implies $$ $$u_{xx} = xy^2f_{xx}(xy)+2yf_x(xy)$$ and $$u_{xy} = ...
2
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0answers
28 views

$1$st Order PDE

I was solving 1st order PDE which was \begin{equation} (x^2-y^2-z^2)p+2xyq=2xz. \tag{1} \end{equation} I had tried to solve this. Please tell me whether it is correct or not. ...
2
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0answers
31 views

How does one find the coefficients in the solution to the Laplace equation?

I'm reading this. Equation (522) gives the general solution to the Laplace equation. What I'm stuck about, is how to determine the coefficients $a_m$, $\beta_m$, and $\theta_m$ for non-trivial ...
2
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0answers
80 views

Do I have any hope with this PIDE?

$\frac{\omega(1-\omega)}{N_1} \frac{\partial^2 f}{\partial x_1^2} + \frac{\omega(1-\omega)}{N_2} \frac{\partial^2 f}{\partial x_2^2} + \cdots + \frac{\omega(1-\omega)}{N_k} \frac{\partial^2 ...
2
votes
2answers
133 views

Dirichlet Problem, Dirichlet Principle

I have some questions concerning Dirichlet Problem and it would be very nice if somebody could give me some hints or some literature tips. Actually, at the moment I am working on Dirichlet Problem ...
2
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0answers
44 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
2
votes
1answer
34 views

How to solve the following PDE

What steps should be taken in order to get a solution (that only depends on v) for the following?: $\dfrac{\partial ^2f}{\partial v^2}+\dfrac{1}{v}\dfrac{\partial f}{\partial ...
2
votes
1answer
24 views

Implicit finite differences: Sufficient conditions for non-negativity

Given the finite difference approximation for black scholes with zero interest rate, $$ \frac{V_n^{m+1}-V_n^m}{\Delta t} + \frac{1}{2}\sigma^2S^2 \frac{V_{n+1}^{m}-2V_n^m+V_{n-1}^{m}}{\Delta ...
2
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0answers
36 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
2
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0answers
25 views

A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
2
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0answers
45 views

Nonlinear Solution to PDE (sine-Gordon Equation)

So I have this nonlinear PDE, the sine-Gordon Equation, $u_{tt}-c^{2}u_{xx}+\omega_{p}^{2}\sin u=0$ whose linearized solution is given by $u_0$. ($c$ and $\omega_p$ are constant.) My reference tells ...
2
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0answers
37 views

A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq ...
2
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1answer
69 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
2
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0answers
60 views

$\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$

I have a encountered to a problem in reading article . Can someone look at the page 9 in this article and give a hint that why $\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < ...
2
votes
1answer
118 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in ...
2
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0answers
98 views

How to use Fourier transform to solve Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
2
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1answer
22 views

Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
2
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1answer
41 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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0answers
39 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
2
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0answers
85 views

Method of characteristics for a parabolic PDE

I am trying to solve a second order parabolic PDE using the method of characteristics. The PDE has the following form: $$\alpha\frac{\partial^2u}{\partial x^2}-\gamma\frac{\partial u}{\partial ...
2
votes
1answer
65 views

Riemann problem of Burgers equation with source term

How to solve $u_t+uu_x=u$ with initial condition $u(x,0)=ul$ if $x<0$ and $u(x,0)=ur$ if $x>0$ where $ul$ and $ur$ being constant.
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0answers
53 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
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0answers
179 views

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
2
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1answer
83 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
2
votes
1answer
56 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
2
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1answer
57 views

Problem understanding “formal proof” of Duhamel principle

I am now studying PDEs. My teacher referred to these notes. In particular, I'm having trouble understanding the proof on pages 20-21 of this part. This is a non-rigorous, formal proof of Duhamel's ...
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0answers
56 views

asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; ...
2
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0answers
24 views

an estimate about the Besove space?

Recently, I study a book about the Besov space and nonlinear partial differential equations, http://link.springer.com/book/10.1007%2F978-3-642-16830-7, whose authors are Hajer Bahouri, Jean-Yves ...
2
votes
1answer
32 views

Differential spherical wave equation, why is the result the same for real and imaginary parts?

I wasn't very sure whether to ask this in the physics forum or here, but the question regards mathematics much more than it does physics. The following wave function is given (spherical wave): ...
2
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0answers
32 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
2
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0answers
39 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...