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

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Analytical solution of Partial Differential Equation

How can I solve the following partial differential equation analytically ? $\dfrac{\partial T}{\partial t}+u\dfrac{\partial T}{\partial x}=\alpha\dfrac{\partial^2T}{\partial x^2}$ where $u$ and ...
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1answer
152 views

Proof of an elliptic equation.

I'd like to see a proof of the theorem Theorem:Let $u \in H^1(B_1)$ a weak solution of \begin{equation} - \operatorname{div}(a_{ij}(x)\nabla u(x)) = 0 \quad \text{in} \quad B_1 \end{equation} ...
4
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1answer
152 views

Elliptic equation on riemannian manifolds

Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth ...
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1answer
178 views

Weak solution and Rankine-Hugoniot condition

I know from PDE lectures that in case of the following Cauchy problem: $$ \left\{ \begin{array}{l} u_{t}+uu_{x}=0\\ u(x,0)=g(x)\\ \end{array} \right. $$ if a piecewise $C^{1}$ function ...
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1answer
68 views

meaning of standard ways of saying in the context of PDEs

I am studying a family of PDEs depending on a real parameter $\alpha$, say $\alpha \in [0,1]$. What does it mean that $\alpha = 0$ is a singular limit for the equation? In my case the equation is ...
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1answer
98 views

$L^2(U)$ compact embedded in $H^{-1}(U)$?

Let $U$ be an open subset of $R^d$. We already knew that $L^2(U)$ is a subset of $H^{-1}(U)$. Question: is this a compact embedding?
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1answer
465 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
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2answers
2k views

Solving partial differential equation using laplace transform with time and space variation

I have a equation like this: $\dfrac{\partial y}{\partial t} = -A\dfrac{\partial y}{\partial x}+ B \dfrac{\partial^2y}{\partial x^2}$ with the following I.C $y(x,0)=0$ and boundary conditions ...
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1answer
87 views

Partial Differential Equation Conversion

Convert the partial differential equation $u_{x}-3u_{y}=2x$ from $u(x,y)$ to $u(\varepsilon, \eta)$ given $\varepsilon = x$ and $\eta = 3x + y$. Edit: Convert the partial differential equation ...
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2answers
90 views

$\dfrac{\partial u}{\partial x}\dfrac{\partial u}{\partial y}=1$

Can you please help me to solve this question? Find 2 solutions of equation: $\dfrac{\partial u}{\partial x}\dfrac{\partial u}{\partial y}=1$ $u(2x,2x)=5x$ Thank you.
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1answer
71 views

Show $f(x^2 + y^2 , \ln(\frac{x}{y}))$ is a solution to a partial differential equation?

Given function is $z = f(x^2 + y^2, \ln(x/y))$ Let $z = f(u,v)$, let $u = x^2 + y^2, v = \ln(x/y)$, show that $z$ satisfies the equation $$x \frac{\partial z}{\partial x} + y\frac{\partial ...
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1answer
144 views

Use chain rule to express $f_x$ and $f_y$ in terms of $f_\xi$ and $f_\eta$?

Consider partial differential equation: $f_x + 2 f_y = 1$ Trick is to introduce new variables $\xi = x$ and $\eta = 2x - y$ Using chain rule to express $f_x$ and $f_y$ in terms of $f_\xi$ and ...
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0answers
141 views

Fourier Series Expansion of the Partial Differential Equation

Partial differential equation: ...
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1answer
69 views

$x^2\dfrac{\partial u}{\partial x}+y^2\dfrac{\partial u}{\partial y}=u^2$

Help me please to solve the following PDE equation: $x^2\dfrac{\partial u}{\partial x}+y^2\dfrac{\partial u}{\partial y}=u^2,\; \: u(x,2x)=1$ Thanks a lot!
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1answer
151 views

General quasilinear PDE - derivation of characteristic equation

A general inhomogeneous quasilinear PDE is given as $a(x,t,u)u_t + b(x,t,u)u_x = c(x,t,u)$. In the derivation of the characteristic equations it says one can consider the solution to this PDE as the ...
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1answer
52 views

Generalization of zero-diagonal square matrices to linear operators

Which linear operators in Banach or Hilbert spaces (e.g., partial differential operators or some other operators in functional spaces) are generalizations of square matrices $A=(a_{ij})$ such that ...
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0answers
35 views

Weak Derivative is $0$ [duplicate]

If the weak derivative of some function $u$ on a open connected set is 0. Does it mean that $u$ is constant almost everywhere on the given set?
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72 views

Numerical integration of a derivative dataset

I have an experimentally measured derivative data ($\frac {dy}{dx}$) at a range of times i.e $\frac {dy}{dx}$ for $0 \le t \le tf$. Integrating for $t=0$ is fine, since $y(x,0)$ is known. However, ...
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0answers
109 views

Eigenfunctions of elliptic operator form an orthonormal basis for $L_2$?

Theorem 6.5.1 of Evans PDE is a standard result that says given a symmetric elliptic operator, there exists an orthonormal basis consisting of the Dirichlet eigenfunctions of the operator. But this ...
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1answer
79 views

Is this function harmonic? [G-T] page 121

On page 121 of Gilbarg-Trudinger's book (Elliptic PDE of second order) they have the following Green's function in $\mathbb{R}^n (n\geq 3)$: \begin{equation} G(x, ...
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2answers
118 views

Decomposition of functionals on sobolev spaces

It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on ...
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0answers
55 views

Boundary value problem domain, norm?

I'm starting some research on a boundary value problem for a memoire for the 2nd semester of a masters, and my mentoring professor asked me to find the answer to the following question: For $u \in ...
2
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1answer
671 views

2D Partial Integration

I have a (probably very simple problem): I try to find the variational form of a PDE, at one time we have to partially integrate: $\int_{\Omega_j} v \frac{\partial}{\partial x}E d(x,y)$ where v is ...
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1answer
270 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
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1answer
159 views

Solve with the use of Duhamel’s Principle: $U_t-4U_{xx}=e^t\sin\dfrac{x}{2}-\sin t$?

Solve with the use of Duhamel’s Principle $$ U_t-4U_{xx}=e^t\sin\frac{x}{2}-\sin t,\quad 0\leq x\leq \pi, t\geq0 $$ $$ U(0,t)=\cos t,\quad U_x(\pi,t)=0 $$ $$ U(x,0)=f(x)=1 $$ I know that the ...
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0answers
57 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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2answers
283 views

Cannot understand method of characteristics

$u_t+uu_x =0$ $u(x,0)\equiv u_0(x)=\begin{cases} 0 & x<0 \\ 1 & x>0 \end{cases}$ I want to parametrise $u(x(s),t(s))$. This is the first thing that is conceptually quite difficult to ...
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1answer
100 views

How can I modify the Eikonal equation to have smooth iso-contours?

I am solving the Eikonal Equation in 2D: $ | \nabla T(x,y)|=1/V(x,y) $ for the traveltime, T(x,y), from a starting point: $ T(x_0,y_0) = 0$. The curves $ T(x,y)=C$ forms closed contours around the ...
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2answers
1k views

Fourier transform of Laplace equation with boundary conditions

The function $u(x,y)$ satisfies the partial differential equation $$\nabla^{2}u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\text{ in }0<y<a, ...
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1answer
126 views

How to derive this existence result from Rabinowitz's book?

Rabinowitz puts forth the following example from his book "Minimax Methods in Critical Point Theory with Applications to Differential Equations" p.25 I will copy below the statement exactly as ...
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2answers
122 views

A Poisson Equation: how to solve it?

I wish to solve the Poisson Equation:$$u_{xx}+u_{yy}=1,$$with $u(x,y)$vanishing on $r=a$. I know that the final solution should be the sum of the homogenous part and the non-homogeneous part, which ...
2
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1answer
92 views

Show that the characteristic that passes through the point $(x,y)$ is given by $y(x)=\frac{1}{2}(x^{-2}-x_{0}^{-2})$

The function $u(x,y)$ satisfies the partial differential eqaution $x^{3}\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}=0$ Show that the characteristic that passes through the point ...
2
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1answer
233 views

Solve the Poisson Equation on a Riemannian Manifold

Imagine that I have a field that obeys to the Poisson equation. To solve the equation, in my bag of tools I only have the divergence theorem or the Fourier/Laplace transform. They usually are enough ...
2
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1answer
817 views

PDE Question in Evans

The question has been posted here previously, however, I cannot quite put all the information together from the responses there. Hopefully you can help me now. The problem is as follows: Let $U$ ...
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1answer
262 views

Does holomorphic a.e. and continuous imply holomorphic everywhere?

Suppose $D$ is a domain in $\mathbb{C}$, $f:D\rightarrow \mathbb{C}$ is a continuous function. Suppose $f$ is holomorphic outside the zero set $f^{-1}(0)$, and $f^{-1}(0)$ has Lebesgue measure zero. ...
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1answer
57 views

Characteristic definition for higher order PDE

Given PDE $u_{x_1 x_2}-4u_{x_3 x_3}+u_{x_1}=0$ in the space where $x_1 \cos (\theta) + x_2 \sin (\theta)\ge0$. Where values of $u$ and its normal derivative are given on plane $x_1 \cos (\theta) + x_2 ...
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1answer
91 views

Paradoxical argument when applying Lax-Milgram theorem to a piecewisely defined elliptic problem

$\newcommand{\v}{\boldsymbol}$ To make the problem easier, simply consider a smooth simply-connected domain $\Omega\subset \mathbb{R}^2$. $\overline{\Omega} = ...
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2answers
216 views

A question concerning measurability of a function

Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on ...
2
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1answer
284 views

Elliptic operators on compact space are Fredholm

I have come across this fact in a reading of mine, but I cannot seem to prove it, and I cannot seem to find a proof of it. Mostly, I am confused why the range of an elliptic operator between ...
2
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1answer
59 views

Showing PDE uniqueness in box

Given PDE $\Delta u= -1$ for $|x|<1,|y|<1$. With boundary conditions $u=0$ for $|x|=1$ and $\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}=0$ for $|y|=1$. Show that there is at most ...
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2answers
2k views

Discretization of Inhomogeneous Dirichlet Boundary Conditions for 2D Poisson's Equations

Let our problem be $$ \begin{align*} -\Delta u &= f(x,y), \quad (x,y) \in [0,5]\times [0,5]\\ u(x,y) &= g(x,y), \quad (x,y) \in \partial\{[0,5] \times [0,5]\} \end{align*} $$ Suppose you ...
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1answer
93 views

The characterization of Sobolev space

If $\Omega$ is a bounded open set in $\mathbb R^n$ and $u$ is a distribution with supp$u \subset \subset \Omega$. For any $s \in \mathbb R$, if ${(I - \Delta )^{\frac{s}{2}}}u \in L_{loc}^2(\Omega ...
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1answer
78 views

Problem Cauchy and the definition of a “well-put problem”

Reading a bit about the Laplace equation in some lecture notes, appeared the following questions: (1) What would a Cauchy problem for the Laplace equation? (2) What does it mean to be a problem well ...
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0answers
96 views

Is this pde Elliptic?

I have seen that there is a classification of elliptic pde's. It says the pde $$au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y+f =0$$ is elliptic if $b^2 - ac < 0$. There is another definition that is ...
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0answers
104 views

Hölder norm estimates

How do you prove the following estimate for composition of functions: If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that $$ \|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
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1answer
294 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
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147 views

Existence of the degenerate elliptic PDE coefficient condition

This is a question related to the theory presented in a book on degenerate elliptic PDEs. The book builds a theory for equations of the form: $$\sum a_{ij}u_{x_ix_j}+\sum b_{i}u_{x_i}+cu=0$$ with ...
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0answers
140 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
2
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2answers
79 views

What function satisfies these conditions?

In order to solve the PDE I am working with, I need to determine a function $u(x,t)$ that satisfies both of these conditions $$u(0,t)=\sin(t)$$ $$u(1,t)=\cos(t)$$ I know it is just trial and error ...
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1answer
88 views

Behavior of the pointwise norm of the gradient w.r.t. to boundary conditions in elliptic PDEs

Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$. If I know that \begin{align} &\bullet\quad ...