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

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Derivation of method of characteristics for first order quasilinear equations

I am trying to follow the method of characteristics for a first order scalar quasilinear equation $$ a(x, y, u(x, y))u_x + b(x, y, u(x, y))u_y = c(x, y, u(x, y). $$ So far, I understand why we look ...
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Solve for u $ \frac{\partial^{2}u}{\partial y^{2}} = \frac{1}{\mu} \frac{\partial p}{\partial x} $

Solve for u $ \frac{\partial^{2}u}{\partial y^{2}} = \frac{1}{\mu} \frac{\partial p}{\partial x} $ my teacher solve this equation and gets $ u = \frac{1}{2\mu} \frac{ \partial p }{\partial x} ...
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Exterior Robin Boundry Condition

Exterior Robin boundary is expressed as the following in the book $\partial{u}/ \partial{v}-\lambda u=f$ on the boundary and $v$ is normal. Also u satisfies Laplace Equation in exterior domain in ...
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Good books on numerically solving nonlinear PDEs

I had one course in PDEs and we weren't taught numerical methods in this course, and from the books I've read on the topic it seems very hard to impossible to solve with methods like F.D or F.E to the ...
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A posteriori measures of numerical dissipation and dispersion

In PDEs, it is typical to find out how dissipative or dispersive a numerical method is by writing down the modified PDE corresponding to the numerical method, and seeing if that modified PDE contains ...
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1answer
38 views

How this series is produced?

All derivatives can be expressed exactly in terms of infinite series of forward, backward, or central-differences. For example $$\frac{\partial^2 U}{\partial x^2}=\frac{1}{h^2} ( \delta_x^2 U - ...
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23 views

Vocabulary : Sturm-Liouville operator

I came across this paper : http://www.pjaeckel.webspace.virginmedia.com/FiniteDifferencingSchemesAsPad%C3%A9Approximants.pdf where the author considers PDE's of the form $$\frac{\partial ...
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1answer
54 views

Given an exact differential; How do you find the function that satisfies the differential?

Suppose we have a differential: $$\mathrm{d}u=y\mathrm{d}x + (x+2y)\mathrm{d}y\tag{1}$$ and a general differential: $$\mathrm{d}u=\underbrace{P(x,y)}_{\color{#F80}{\dfrac{\partial u}{\partial ...
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46 views

Which initial functions can be solved by separation of variables

consider the wave equation on $(0,\pi)$, i.e. let $f,g\in L^2(0,\pi)$ be two fixed functions and consider the following problem: $$\partial_{tt}u=\partial_{xx}u, \quad x\in(0,\pi), t\in (0,\infty) \\ ...
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constrained heat equation

Consider the following constrained minimum energy problem for 1-D heat equation for $x\in[0,1],t\in[0,\infty)$: $$u(x,t)=\underset{{0\leq u(x,t)\leq1}}{argmin}~\frac{\partial}{\partial ...
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39 views

What kind of differential equation is this one and how to solve it?

On working on a physical problem that is of interest to me, I arrived at a differential equation that I desperately need to solve$$\frac{dy}{dt}+c y=\frac{\partial y}{\partial t}$$ where c is constant ...
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1answer
58 views

Is there a solution to this unidirectional wave equation, with initial value $v=f(x)$ and $x=t^2$

unidirectional wace equation: $$\frac{du}{dt}+c\frac{du}{dx}=0$$ The initial value $u=f(x)$ is given on the parabola $x=t^2$. Is there a solution to this problem, discuss why the solution is unique ...
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60 views

How To Prove The following equation?

The equation arised in the paper:Exact and asympototic representations of the sound field in a stratified ocean.That is the equation(3.12) for solving the problem $$\Delta ...
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Precise analysis of physical interpretations of elliptic

When I read Evans' PDE, I find the below content. I am curious about it. Where I can get the precise analysis ?
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Elliptic operators for a Laplacian transform

I need to show that the operator: $L[u]=(1-x^2)\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x \partial y}+(1-y^2)\frac{\partial^2 u}{\partial y^2}$ Find the transformation of ...
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15 views

Galerkin and Ritz method

I know Galerkin method gives us the best approximation. But i couldn't understand the difference between this method and Ritz method for approximation. Do not give us the same results? What is the ...
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60 views

Whittakers general solution to Laplace and its relation to separable variables

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates $u=x+iy$ and $v=x-iy$. This can be extended to n dimensions as long as the complex ...
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1answer
12 views

Potential such that the time-independent Schroedinger equation has an explicit solution

Consider the time-independent Schroedinger equation $\phi'' (x) +V(x)\phi(x)=\lambda \phi(x)$, $\quad x\in\mathbb{R}$. For testing my numerics, I would like to now: For which choices of $V$ are ...
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42 views

Traveling delta function $\delta(x - ct)$ as a distributional solution of the wave equation

I'm trying to show that a delta function $\delta(x - ct)$ is a distributional solution of the PDE $$ D_{(0,2)}u(x, t) = c^2 D_{(2,0)}u(x,t). $$ Here $D_{(i,j)}$ means $i$-th partial differentiation on ...
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Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial ...
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Does argsup function satisfy a property of the supremum

Let $X$ be a compact set of $d\times d$ matrices, and let $f\in C(\overline{\Omega})$, and $u\in C^2(\overline{\Omega})$. Define $A(x)=\operatorname{argsup}_{W\in ...
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Finite difference discretisation of the heat equation

Here is the equation to be discretised: $$ k\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = \dot{q} $$ Using the following discretisation scheme: $$ ...
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Problem 5 of chapter 6 of Evans PDE 1st.

I don't know how to start it. I try to compute the $Lv$ , but nothing I get.Maybe, I think some hint is suitable for me . Thanks.
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1answer
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How was the expression relating two general solutions to $x\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}- 2u = 0$ obtained?

This was the question: Find the general solution of $$x\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}- 2u = 0$$ ...
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Connection of expectation and PDE; help understanding step in proof

The following is taken from a set of lecture notes available online by Nizar Touzi. Some points before reading the proposition: $\mathcal{A}$ is the generator of a stochastic process $X_s^{t,x}$. The ...
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1answer
31 views

diffusion equation with boundary $xe^x$

I am preparing for the preliminary examination, and I have a question: Problem B.4 in http://www.math.mcmaster.ca/images/uploads/AM-prelim-201505.pdf In part (a), doing the direct calculations I ...
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1answer
57 views

Partial differential equality

Let $f \colon \mathbb{R}^2 \setminus \{0,0\} \to$ $\mathbb{R}$ be a smooth function such that $$ x\frac {\partial f(x,y)}{\partial y} + y\frac {\partial f(x,y)}{\partial x} =f(x,y)$$ ...
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What can we say about the number of linearly independent solutions of a PDE

What can we say about the number of linearly independent solutions to a PDE? Firstly, in what function space are we talking about linearity? Secondly, if there is not a general conclusion, what if ...
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36 views

Solution to the Schrödinger equation

Let $$\begin{cases} i\frac{\partial}{\partial t} \Psi(x,t) = \Delta \Psi(x,t);\\ \Psi(x,0) = \varphi(x) \end{cases}$$ Why do physicists seek a solution of this equation in the form: $$ \Psi(x,t) = ...
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Helmholtz decomposition of $v\in (L^2(\Omega))^3$

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\partial\Omega$ and outward unit normal $n$. I want to study the characterize whether a vector function defined on ...
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1answer
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Any critical point $u_0\in M$ of $I|_M$ satisfies $I'(u_0)=\mu\gamma'(u_0)$

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}u_r^2-\xi u^2+\ln(1+u^2)\right\}r\,dr,$$ $\xi\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in ...
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Sequence of compactly supported functions approximating $x^2$

I encountered this question as part of a proof I am working on and was wondering whether anybody has an explicit way to construct these sequences: 1.) Is there a sequence of positive compactly ...
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A convergence in sobolev spaces involving time

Suppose $\Omega $ is bounded and $$‎‎u^m‎‎\xrightarrow[]{w^*}‎ u ‎\quad \text{in} \,\, ‎L^{\infty} ‎(‎\mathbb{R}^+,H^2(‎\Omega‎)‎\cap ‎H_0^1(‎\Omega‎))$$ and ‎$$‎‎u^m_t\xrightarrow[]{w^*}‎ u_t ...
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Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
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Does weak convergence of $u_m(t)$ in $L^2(0,T;X)$ imply weak convergence of a subsequence of $u_m(t_0)$ in $X$ for a.e. $t_0$ in $[0,T]$?

In a book I'm reading (Navier Stokes Equations, by Constantin and Foias), the authors construct a sequence $u_m$ of functions in $L^2(0,T;V)$ which converge weakly to $u$ in $L^2(0,T;V)$. They then ...
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Equation about fluxes of the Green's functions of the Laplacian on intersecting domains

I've got a problem with the divergence theorem, hope someone could help me. So this is the set up: Let $\Omega$, $\tilde{\Omega}$ be domains with $0\in \Omega\cap\tilde{\Omega}$ with boundaries ...
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93 views

How to solve this Partial Differential Equation? [10]

$${\partial z \over \partial x}f(x,y,z)+{\partial z \over \partial y}g(x,y,z)=h(x,y,z) $$ $${\partial z \over \partial x}F(x,y,z)+{\partial z \over \partial y}G(x,y,z)=H(x,y,z) $$ $$f(x,y,z)={x-x_1 ...
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Physical meaning of the various types of boundary conditions for a vibrating string

I wonder what is the physical meaning of Dirichlet, Neumann and Robin boundary conditions for a vibrating string? Or link to other applications?
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Partial Differential Equations, Fritz John, Page 18, Problem 6

Consider the partial differential equation: $$\frac{\partial R(u)}{\partial y} + \frac{\partial S(u)}{\partial x}=0$$ where $u$ is a function of $x$ and $y$ and $S'(u)=uR'(u)$. We call a function ...
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Existance and Uniqueness Theorem for PDE's

I am looking for a general uniqueness theorem for linear second order PDE's. I would like to have general conditions for the boundary conditions. For example, a very general formulation might look ...
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1answer
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PDE - $y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}$ - how to derive the general solution

$\mathbf{y^2 \frac{\partial^2 u}{\partial x^2}=x^2 \frac{\partial^2 u}{\partial y^2}}$ is a hyperbolic PDE where $\xi =y^2+x^2$ $\eta =y^2-x^2$ which gives ...
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Pertubation of Euler equation of inviscid flow in cylindrical coordinates

Guys I tried to derive pertubation of Eulers equation of inviscid flow but I suspect that I made an error somewhere as the derivation was long. Can someone help identify the error. The following is ...
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maximum principle in a harmonic and positive function

$\begin{equation}u=\frac{1-(x^2+y^2)}{(1-x^2)+y^2}\end{equation}$ is harmonic and positive in $x^2+y^2<1$ because $u=0$ in $x^2+y^2=1$ except in $(1,0)$. Is the maximum principle valid for $u$?. ...
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Definition of the sup-norm in the Qing/Fanghua

I am currently studying the book "Elliptic Partial Differential Equations" - Qing / Fanghua, COURANT, in chapter 2, section 2.3. A priori estimates , the autors define by $\alpha$ the sup-norm of ...
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Find explicit value of ${{u}_{x}}$ and ${{u}_{y}}$ of Burgers equation

Consider the first order quasi-linear equation with initail condition for a function $u(x,y)$ of two variables $x, y$ : $$\left\{ \begin{align} & {{u}_{y}}+u{{u}_{x}}=0 \\ & u\left( x,0 ...
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How to solve this system of Partial Differential Equations [5]

$${\partial z \over \partial x}f(x,y,z)+{\partial z \over \partial y}g(x,y,z)=h(x,y,z) $$ $${\partial z \over \partial x}F(x,y,z)+{\partial z \over \partial y}G(x,y,z)=H(x,y,z) $$ ...
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1answer
34 views

Proving $\|u\|_{L^\infty(0,T;H)}\leq C$ from a given hint.

My question concerns to the problem 6, chapter 7, of Evans PDE book (2nd edition). In the book a hint is given but I couldn't get a solution from it. On the other hand, I got a solution without ...
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1answer
46 views

Identifying the constant of integration

A short extract from a book of mine states that: If $$\color{red}{A(x,y)\frac{\partial p}{\partial x}+B(x,y)\frac{\partial p}{\partial y}=0\tag{A}}$$ where ...
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35 views

PDE - derivative in the boundary condition

PDE $\mathbf{8u_{xx}-6u_{xy}+u_{yy}+4=0}$ with the canonical form $u_{\xi \eta}=1$ and the general solution $\mathbf{u=\xi\eta+F(\eta)+G(\xi)}$ where $\xi=x+2y$ and $\eta=x+4y$ and BC's $u=cosh ...
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elliptic PDE $\mathbf{u_{xx}+x^2u_{yy}=0}$ - general solution

I got the canonical form of the the equation but I am not sure how the general solution should be derived in this case. Should all components be integrated twice w.r.t $\xi$ ? how about $u_{\eta ...