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

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What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
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1answer
31 views

Analytical solution to a second order PDE

Hey all here is my equation in a 2D system. $$\nabla^2u(x,y) = -\sin(\pi x)\sin(\pi y)$$ I haven't done anything like this in a while so could use a bit of guidance, how do I go about solving this ...
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1answer
67 views

One dimensional heat equation with multiple eigenfunctions

To solve the heat equation $u_{t}=u_{xx} \,\, (!)$ which is defined for $x\in[0,1]$ and $t>0$. I want to find the solution which satisfies the boundary conditions $u_{x}(0, t)=u_{x}(1,t)=0$ and ...
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1answer
85 views

Will this well enough to serve as a prerequisite to oksendal's book?

Will this well enough to serve as a prerequisite to oksendal's stochastic differential equations: an introduction with applications book? I refer to shiryeav's probability, but i guess it still miss ...
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1answer
18 views

Does anyone know how to solve this differential equation?

here is the equation: $\frac{\partial\alpha(r,\phi)}{\partial r}=\beta\sin\alpha(r,\phi)\cos\alpha(r,\phi)$ $r$, and $\phi$ are cylindrical coordinates. $\phi$ is the angle off the x-axis. So it ...
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1answer
304 views

MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions

According to wikipedia, the equation $$\psi_t + \mathbf{u} \cdot \nabla \psi =0$$ is hyperbolic. However, when I want to solve it in MATLAB using the pde toolbox (link), the general formula for a ...
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1answer
53 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and ...
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88 views

Tricomi equation canonical form and solution.

Consider the Tricomi equation $$u_{xx}+xu_{yy}=0$$ à find thé canonical form but i did not solve it $$\left(v_{qq}+v_{rr}+\dfrac1{3r}v_r\right)=0.$$
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1answer
34 views

Calculus Problem___Prove $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ for any $t>0$

Given $g \rightarrow R$ continuous and bounded, let $$u(x,t)=\frac{x}{\sqrt{4 \pi}}\int_{0}^t \frac{1}{(t-s)^{3/2}}e^{\frac{-x^2}{4(t-s)}}g(s)ds$$. Prove that $\lim_{x \rightarrow 0^+} u(x,t)=g(t)$ ...
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0answers
38 views

How to obtain a general solution of a linear homogeneous first-order PDE?

Writing the equation as $$u_x+a(x,y)u_y+u(x,y)=0$$ I am wondering how to solve this generally. Is there any general method or solution exist?
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2answers
125 views

what is scattering theory?

I often read the the words "scattering theory", "scattering data", "scattering matrix", scattering XXX ... in my math lecture, but I realised that I am not able to define it correctly. A short search ...
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2answers
53 views

Intuition behind the definition of the d'Alembert operator

The d'Alembert operator is defined as $$\square^{2}=\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}$$ My question is what is the intuition behind this definition? My intuition tells me that we should ...
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0answers
65 views

Simple partial differential equation

I know this is a shame but I can't solve this simple partial differential equation. Can someone help me? $$\frac{\partial ^2F(x,y)}{\partial x \partial y}=h(x,y) F(x,y)$$
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1answer
41 views

Vibration on a rectangular Plate

I am trying to solve a problem that has been set for me. I haven't come across a problem like this like, so i need some help getting through it. It is used to model the vibrations of a rectangular ...
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0answers
79 views

Neutron density PDE

On Mathews and Walker's book exercise (8-2) We are given that the neutron density n inside $U_{235}$ obeys the differential equation $$\nabla ^2u+\lambda u=\frac{1}{k}\frac{\partial{n}}{\partial{t}} ...
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107 views

d'Alembert Solution of fixed end

in d'Alembert Solution for fixed end semi infinite string problem with wave equation $u_{tt} = c^2u_{xx}$,we get $0= \frac{f(ct)+f(-ct)}{2} + \frac{\int_{-ct}^{ct}g(s)ds}{2c}$ where $f$ and $g$ are ...
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1answer
77 views

D’Alembert Formula - PDE

Find $u(2,1)$ and $u(3.5,0.5)$ if $u$ solves $$u_{tt}=u_{xx}, \ 0<x<2, t>0$$ $$u(0,x)=x^2(2-x)^2, \ u_t(0,x)=x(2-x), \ 0\le x\le 2$$ $$u(t,0)=u(t,2)=0, \ t\ge0.$$ I can use D’Alembert ...
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41 views

$2\operatorname{D}$ heat equation with non-zero initial condition

I cant find anything about $2\operatorname{D}$ heat equation with non-zero initial condition, need to find the temperature of the rectangle.
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1answer
63 views

Does $a\ln(x^2 +y^ 2 )+b$ satisfy Laplace’s equation?

I can't verify that $F(x,y) = a\ln(x^2 +y^ 2 )+b$ satisfies Laplace’s equation ($F_{xx}+F_{yy}=0$). Here is what I did: \begin{align*} F_x &= \frac{2ax}{x^2 + y^2} &F_y &= ...
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1answer
65 views

Interesting counterexample of Strong Maximum Principle

Suppose $U=\Omega \cup (0,T)$ where $\Omega$ is a bounded domain. Let $u \in C_1^2 (U) \cap C(\overline U)$ satisfy $$u_t \le \Delta u+cu$$ in $U$ where $c \le 0$ is a constant. If $u \ge 0$, then ...
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140 views

Order of Integrating a partial derivative

I have some questions about the process involved when integrating higher order partial derivatives. I was going through a textbook on engineering mathematics on PDEs. If $\frac{\partial^2 ...
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1answer
145 views

What do the operator semi-groups have to do with PDE's?

Can anybody please help me to understand what does the semi-group do with partial differential equations? We started this subject very recently and we are now in the proof of Hille-Yosida Theorem, ...
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1answer
80 views

How to transform parabolic equation into heat equation?

Consider the parabolic equation: $$u_t-k(\Delta u+\sum\limits_{i=1}^n a_i\frac{\partial u}{\partial x_i}+bu)=0$$ where $a_i,b,k$ are constants and $k>0$. How this equation can be transformed to the ...
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1answer
124 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
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27 views

Solution of an IBVP (nonlinear McKendrick PDE)

Consider the following initial boundary value problem: $$u_t(x,t) + [g(x,P(t))u(x,t)]_x = -\mu(x,P(t))u(x,t)~~(1.1)$$ $$B(t)=g(0,P(t))u(0,t) = \int_0^{x_\infty} \beta(x,P(t)) u(x,t)\, dx~~(1.2)$$ ...
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1answer
45 views

Weak solution in Hilbert Space

How would you show that $u(x)=log|x|$ is a weak solution of $-\Delta u+cu=0$ for some $c(x)\in L_{weak}^{3/2}(B)$ and u is not bounded? I did take the derivative of u(x) and then its second ...
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2answers
78 views

$\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\text{ implies } \frac{\partial^2 u}{\partial z \partial y}=0$

I have the following question: Show that if $z=x+ct$ and $y=x-ct$ then: $$\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\text{ implies } \frac{\partial^2 u}{\partial z ...
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179 views

Explicit traveling wave solution for the diffusion equation

Find explicit formulas for $v$ and $\sigma$ so that $u(x,t)=v(x-\sigma t)$ is a traveling wave solution of the nonlinear diffusion equation $$u_t-u_{xx}=f(u)$$ where $$f(z)=-2z^3+3z^2-z$$ and ...
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1answer
141 views

Green's function of Dirichlet problem

Let $G(y,x)$ be the Green's function for the Dirichlet problem for Laplace equation on domain $\Omega$ with smooth boundary. Show that $$K(y,x):=\frac{\partial}{\partial \textbf{n}_y}G(y,x)\geq 0, ...
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0answers
32 views

How to solve an inhomogenous PDE using Fourier Transform

$u_{tt}=u_{xx}+(8-64x^2)e^{-4x^2}$ $u(x,0)=e^{-4x^2},u_t(t,0)=0$ $0<t<\infty,-\infty<x<\infty$ By Fourier Transform ...
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1answer
151 views

Inviscid Burger's Equation

I have a question about the following burger's equation. $u_t + (\frac12u^2)_x = 0 $ with $u(x,0) = sin(x)$ on $[0,2\pi]$ and periodic boundary conditions. When I studied this equation numerically, ...
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1answer
41 views

Differential equation (inhomogeneous )

I have been trying to solve this equation for a while. Is there anyone who can help me to solve this ? Any comment appreciated. $$\frac1r \frac{\partial}{\partial r}\left(r\frac{\partial E}{\partial ...
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1answer
73 views

Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
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1answer
57 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
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2answers
44 views

Fourier Series Transformation

I have a question regarding the following: Compute the Fourier transform of $f(x)=xe^{-2x^2}$, $x\in\mathbb{R}$. The Fourier Transform of $f(x)$ is given by ...
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0answers
42 views

Duhamel's formulation of a pde

Let's consider the following initial value problem $$u_t=Lu+F,\,\,\,\,u(0)=u_0$$ with $L$ a spatial operator. Which are the minimal assumptions on $F$,$u_0$,$u$ to have the equivalence of the problem ...
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1answer
69 views

How to prove such a relationship?

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
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1answer
20 views

About the smoothness of the boundary of support set

If $\varphi \in C_0^2 (\Omega)$, where $\Omega \in \mathbf{R}^n$. Can we infer the boundary of supp $\varphi$ is $C^2$? How to prove?
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1answer
23 views

Dealing with characteristics of First-Order Wave Equation with initial condition of $w(0,t)$

Solve $$({\partial w}/{\partial t})+4{\partial w}/{\partial x}=0$$ with $w(0,t)=\sin(3t)$. I know how to do this with $t=0$ and just $x$ but how do you approach this? I know that $x=4t+x_0$ but ...
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1answer
35 views

Solution of a pde in $H^{-1}$

What does it mean that an equation is verified in $H^{-1}(\Omega)$? For istance what does it mean that the following equation $$iu_t+\Delta u+g(u)=0$$ in $H^{-1}(\Omega)$ for all $t$?
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1answer
110 views

Numerical integration of a 2D advection equation with spacially varying coefficients.

I am interested in a simple algorithm for solving this PDE for $f(x,y,t)$: $$f_t + A y f_x - B x f_y =0$$ With $A,B>0$. The initial condition is some arbitrary smooth function (probably gonna ...
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0answers
66 views

Banach spaces involving time

Let's suppoe $u\in L^2(0,T;H_0^1(\Omega))$ with $u'\in L^2(0,T;H^{-1}(\Omega))$. We know that $$u\in C([0,T];L^2(\Omega))$$. In this result can the set $\Omega$ be the whole $\mathbb{R}^n$ or we need ...
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1answer
79 views

Dealing with Partial Differential Equations and Burger's equation

The problem is: consider Burger's equation, $$u_t +uu_x = 0 $$ $$ u(x,0) = f(x) $$ Where $$f(x) = \begin{cases} 1 - |x-2| &\mbox{if}\,\, 1\leq x \leq3, \\ 0 ...
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1answer
120 views

Discrete Sobolev Poincare inequality proof in Evans book

In page 275 of Evans book, the Poincare's inequality has been proven via contradiction. I was wondering how can one extend this prove to prove Sobolev-Poincare inequality: $||u-u_\Omega||_{L^{p*}}\le ...
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1answer
192 views

Direction of unit vector that maximize directional derivative

Firstly, I am aware that there are quite a few question regarding with "maximizing direction derivative" already being asked. But after scanning through, I am still not able to figure out my question ...
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1answer
150 views

Help Solving coupled linear PDEs by Separation of Variables

I would like to solve the following coupled system of linear PDEs by separation of variables, where a and b are constants: ${\partial{u}\over\partial{t}} = {b-a \over a+b}u + (b+a)^2v + ...
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1answer
32 views

Help following a proof reducing a PDE to coupled ODEs

I'm having trouble following the following proof. Given $$\rho_a(a,t)+\rho_t(a,t)+\mu(P(t))\rho(a,t)=0~~~~(1.1)$$ $$B(t)=\rho(0,t)=\int_0^\infty \beta(a,P(t))\rho(a,t)da~~~(1.2)$$ ...
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1answer
457 views

You can't solve Laplace's equation with boundary conditions on isolated points. But why?

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of "holes" $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I'm pretty sure that in more than one dimension, ...
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1answer
73 views

Prove that a function belongs to Sobolev space

Denote $U=\{x\in \mathbb{R}^2 | |x_1|<1, |x_2|<1\}$ Define: $ u(x) = \begin{cases} 1-x_1 &\mbox{if } x_1>0, |x_2|<x_1 \\ 1+x_1& \mbox{if } x_1<0, |x_2|<-x_1 \\ 1-x_2& ...
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1answer
155 views

Determining when these two waves separate

There's probably something really obvious I should be getting, but I haven't yet developed the intuition for working with the wave equation. Suppose we're given the wave equation $u_{tt} = c^{2} ...