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

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4
votes
2answers
107 views

heat equation with convection and forcing function

How can I solve this: $u_t - x u_x -x^2 u_{xx} = \ln{x}$ $u(x,0) = \sin ((\pi/2) \ln{x})$ $u(1,t) = 0 \quad u_x(e,t)=0$ What I have so far: Since we have homogenous BC consider no forcing term to ...
0
votes
1answer
47 views

heat equation in three dimensions with non homogeneous bc

I'd like to solve the heat equation for a cylinder in 3D, in cylindrical coordinates with no azimuthal dependence. The equation is homogeneous but the bc at the cylinder wall has an arbitrary ...
1
vote
1answer
24 views

Quasilinear equation

I am not quite sure how to deal with discrete IVP Find self-similar solution \begin{equation} u_t=u u_x\qquad -\infty <x <\infty,\ t>0 \end{equation} satisfying initial conditions ...
3
votes
0answers
197 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
0
votes
1answer
44 views

A Simple function identity

Show the following identity, $$ \int_{\mathbb R^d}|\nabla(\frac{f}{G})|^2\,G\,dx=\int_{\mathbb R^d} |\nabla f|^2\,G^{-1} \,dx \,-\,\int_{\mathbb R^d} f^2(\Delta \Phi) \, G^{-1} dx $$ Here, ...
2
votes
2answers
74 views

Product of functions in $H^1(B)$ where $B \subset \mathbb{R}^2$

I'm rather new to Sobolev spaces and finding myself rather deficient of intuition. So when given a problem like the below where I need to "prove or disprove", I'm finding myself stuck. Suppose $B$ is ...
2
votes
0answers
54 views

Infinitesimal Generator of A One Parameter Group

This is a small problem which drives me crazy. Let $\varphi(x,y,t)=(F_1(x,y,t),F_2(x,y,t))$ be a one paramter transformation group on $\mathbb{R}^2$. Let ...
1
vote
2answers
29 views

Method of characteristics- $u=$ constant or $u = f(y)$?

Say $u(x,y)$ is a function of $x$ and $y$ and suppose we have the following pde - $u_x - u_y = 0$ This equation has the following characteristics - $\frac{dx}{1} = \frac{dy}{-1}$ $\frac{du}{dx} = ...
2
votes
0answers
77 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
3
votes
1answer
48 views

Find all Harmonic functions st $|u(x)|\le C|x|^m$

Find all Harmonic functions $u:\Bbb{R}^n \to \Bbb{R}$ st $|u(x)|\le C|x|^m$, for all $|x|\ge 1$ where $C$ is constant and $m\in (0,2)$. I tried to use the same argument as used in the proof of ...
0
votes
1answer
167 views

General Solution of transformed canonical form. 2nd oder PDE

Okay so Ive transformed the PDE. Thats fine. My prof has not given enough examples and I cant find anything on the net for the method he wants me to use to solve the thing..... I dont know how to ...
0
votes
0answers
82 views

How to check the barrier function is superharmonic?

Suppose $n\geq 3$ and $\Omega$ is a bounded domain. In the Perron's method to solve the PDE \begin{equation} -\Delta u = 0 \text{ in } \Omega \quad \text{and } u = g \text{ on }\partial\Omega, ...
1
vote
0answers
118 views

numerical analysis of partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
1
vote
1answer
44 views

Using dummy variable to derive Nth partial Fourier sum

I am working on a problem$^{(1)}$ as follow: Using the standard formulas for the Fourier coefficients, show $$F_N(x) = \frac1{2\pi} \int_{-\pi}^{\pi} \left( 1+2 \sum_{n=1}^{N} ...
3
votes
0answers
111 views

Small question about condition in the Bouchala paper's 2005

I have this condition from this paper: http://ejde.math.txstate.edu/Volumes/2005/08/bouchala.pdf (Strong resonance problems for the one-dimensional $p$-Laplacian. Bouchala, Jiri. Electronic Journal ...
1
vote
1answer
64 views

The wave equation with forcing function

What would the solution to this problem be? $\frac{1}{x} u _t - (x u_x )_x = \frac{1}{x} \ln{x} \quad 1<x<e \quad t>0 $ $u(x,0) = \sin{(\frac{\pi}{2}\ln{x})} \quad (1<x<e) $ ...
0
votes
1answer
51 views

Sturm-Liouville problem eigenvalues

Solve the S-L $-(xX')' = \frac{\lambda}{ x} X \quad , 1<x<e$ $X'(1)-X(1) = 0 \quad , X'(e) + 10 X(e) = 0$ What are the three least eigenvalues $\lambda_1 , \lambda_2 , \lambda_3$ ? How can I ...
2
votes
1answer
1k views

Solve Laplace's equation inside a semi-infinite strip

Solve $$\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0$$ $$0 < x < \infty, \ \ 0 < y < H$$ subject to these boundary conditions: $${\frac {\partial ...
2
votes
2answers
126 views

Proof of an inequality involving gradient function

I'm reading ahead in my course, and I've encountered the following problem; Let $\Omega \subset \mathbb{R}^n$ be a bounded domain such that the divergence theorem holds. Assume that $u \in ...
2
votes
1answer
213 views

Homogeneous wave equation on half line with nonhomogeneous boundary condition.

I am looking to solve the following PDE: $$\begin{align*}&u_{tt} - c^2 u_{xx} = 0, &0 < x < \infty, t>0\\&u(x,0)=0,\quad u_t (x,0)=0 &0 < x < ...
0
votes
1answer
29 views

Where does this result come from?

I'm sorry about the non-specific title, I wasn't sure where this question would fit in... I'm reading through a few notes for my PDE course, and I'm struggling to see where the following result comes ...
0
votes
1answer
28 views

Bound on $H^1_0$ norm of test function

I'm reading through a book of Courant Lecture notes on elliptic PDEs, and I'm uncertain how an inequality is derived. The function $u\in H^1(B_1)$ satisfies \begin{align} \displaystyle\int ...
1
vote
2answers
32 views

Solve $\frac{\partial^2z}{\partial x \partial y}= x^2y$

Question: Find the particular solution of the following PDE using separation: $$\frac{\partial^2z}{\partial x \partial y}= x^2y$$ such that \begin{align} z(x,0)&= x^2 \\ z(1,y) &= ...
0
votes
1answer
30 views

Solving a partial differential equation

What are the steps to solve $$\frac{\partial ^{2}f}{\partial x\partial y}=0$$ Is this just $$\int fdx(\int fdy) = F(x) (F(y)$$
3
votes
1answer
101 views

$H^1$ convergence of eigenfunctions of Schrödinger operators

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
1
vote
3answers
42 views

Find the general solution of the equation $u_t(x,t)+t^{\frac{1}{3}}u_x(x,t)=u(x,t)$.

Find the general solution of the equation $u_t(x,t)+t^{\frac{1}{3}}u_x(x,t)=u(x,t)$. I'm having trouble starting this one. Any help would be greatly appreciated.
1
vote
1answer
73 views

Solve the damped quasilinear wave equation $u_t(x,t)+u(x,t)u_x(x,t)+u(x,t)=0$ with $u(x,0)=f(x)$.

Solve the damped quasilinear wave equation $u_t(x,t)+u(x,t)u_x(x,t)+u(x,t)=0$ with $u(x,0)=f(x)$. Determine if the solution breaks when $f$ satisfies the condition $f^\prime(x)>-1$ for all ...
0
votes
1answer
131 views

Find the solution of the initial value problem and determine the breaking time. Find all shock wave solutions.

Find the solution of the initial value problem $u_t(x,t)+u(x,t)u_x(x,t)=0$ with $u(x,0)=\begin{cases} 2 &\text{if } x<0\\ 1 &\text{if } 0\leq x<1\\ 0 &\text{if } x\geq1\end{cases}$. ...
0
votes
0answers
39 views

Convergence in Bochner space (don't follow an argument, can you explain it to me please)

Define $V:=W^{\beta, 2} \subset H:=L^2$ which is compact and dense. It follows that $L^2(0,T;V) \subset L^2(0,T;H)$. Let $w^\epsilon$ be a sequence which is uniformly bounded in $L^2(0,T;V)$ and in ...
2
votes
0answers
27 views

Partial differential equation problem1 [duplicate]

I'm supposed to solve $u_{xx}-3u_{xt}-4u_{tt}=0$ with initial conditions $u(x,0)=x^2$ and $u_t(x,0)=e^x$. So I factored the problem into $(u_x-4u_t)(u_x + u_t)$ and set each equal to 0 and found the ...
0
votes
1answer
26 views

Can we have $\|\nabla u \|_{L^1(B(0,1))}\leq \|\nabla u-\tilde{c} \|_{L^1(B(0,1))}$ for radially symmetric function?

Suppose $B(0,1)$ is the unit ball in $\mathbb R^2$ and $u\in C^\infty(\overline{B(0,1)})$. Suppose $u$ is radially symmetric, i.e. $u(x)=u(Rx)$ for any $R\in SO(2)$. My question is, do we have ...
3
votes
1answer
59 views

Show that the distribution is of the form $C \delta + f$

I'm trying to solve this problem: Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution $(\phi u )* (\psi u)$ is of the form $C \delta + f$ for some constant C ...
1
vote
0answers
49 views

Strichartz estimates for wave equations

Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
0
votes
1answer
37 views

Interchanging limits in $\lim\limits_{n \to \infty}\lim\limits_{j \to \infty}\int_0^T \langle u_n', w_j \rangle $ (weak time derivative)

Let $V$ be a Hilbert space which is separable. Let $u_n \in L^2(0,T;V)$ with $u_n(t,x) = \sum_{i=1}^n u_{in}(t)w_i(x)$ where $u_{in}$ are absolutely continuous on $(0,T)$ and $w_i$ are a smooth basis ...
2
votes
2answers
56 views

Standard partial differential equation?

Is there any standard equation which looks like this one, $$ u_{t}(x,t) = \alpha \, u_{xx}(x,t) - \beta(t) \, u(x,t) + S(x,t), $$ where $\beta(t)$ is nonlinear in time, and $\alpha$ is a constant? ...
3
votes
1answer
42 views

Show that $2\nabla \sqrt f\,+\,x \sqrt f=0$ (a.e.). $\implies$ $\sqrt f\in \mathcal C_0$. (Derivatives are in weak sense)

Show that $2\nabla \sqrt f\,+\,x \sqrt f=0$ (a.e.). $\implies$ $\sqrt f\in \mathcal C_0$. (Derivatives are in weak sense) Given that $f\in L^1(\mathbb R^d),f\geq 0,\int_{\mathbb R^d}f=1, ...
0
votes
0answers
42 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
0
votes
0answers
45 views

Rearranging initial data of the heat equation.What result do we obtain for the solution?

Suppose we have the heat equation on the whole space or on some open and bounded set U with Dirichlet boundary conditions and some initial data $f$ for $t=0$.If we consider a rearrangement of the ...
1
vote
0answers
68 views

Boundary Condition defined on a Parallel Vector

I am given the following equation $au_x+bu_y = 1$ for $(x,y) \in \mathbb{R}^2$ and $a,b \in \mathbb{R}$. In addition, $u(s \vec{r}) = \sin(s)$ where $s\in \mathbb{R}$. I am trying to find for which ...
1
vote
0answers
116 views

Association of PDE's with Integral Equations?

We know the following associations : Volterra Integral Equations $\leftrightarrow$ Initial Value Problems Fredholm Integral Equations $\leftrightarrow$ Boundary Value Problems My questions are : ...
1
vote
1answer
159 views

Weak formulation of a system of biharmonic pdes

Consider the system of pdes for functions $u,v$ $$\begin{cases} a_0\Delta ^2u+a_1u_{xx}v_{xx}+a_2u_{xy}v_{xy}+a_3u_{yy}v_{yy} = f\\ b_0\Delta ^2v+b_1u_{xx}v_{xx}+b_2u_{xy}v_{xy}+b_3u_{yy}v_{yy} = g ...
7
votes
1answer
107 views

$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}$

Having difficulty in solving the following partial differential equation: $$\cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z+\frac{1}{z}.$$ Will it be easier if we ...
2
votes
0answers
55 views

Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves ...
0
votes
1answer
39 views

Question on Method of Characteristics & Characteristic Curves

Consider the PDE $$ \begin{align} u_t + (vu)_x &= 0,\phantom{u_0(x)} \quad x \in \mathbb{R},\, t>0 \\ u(x,0)&=u_0(x),\phantom{0} \quad x \in \mathbb{R} \end{align} $$ Let $s \to z(s,x,t)$ ...
1
vote
0answers
48 views

Solution of a partial integro-differential equation

I want to find a solution for the following equation $$ \partial_y u(x,y) + \partial_x u(x,y) + \int_{0}^{x} r(x-x') u(x',y) dx' $$ $$ u(x,0)=0 \quad u(0,y)=\delta(y) $$ in $(x,y) \in ...
2
votes
0answers
80 views

Limit under the integral sign and partition of unity

Let $U \subset \mathbb{R}^N$ be a bounded open set and let $\{ U_j \}_{j=1}^\infty$ be an open covering of $U$ such that $U = \bigcup\limits_{j=1}^\infty U_j$. Suppose $\{ \psi_j \}_{j=1}^\infty$ is ...
0
votes
0answers
56 views

Solving PDE with change of variables

I'm having a bit of trouble with solving a PDE using change of variables. I have the PDE $$ u_t - ku_{xx} + au_x = 0, \quad x \in \mathbb{R},\, 0<t<T $$ And if I let $v(x,t) = u(x+at,t)$, by ...
2
votes
0answers
193 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
2
votes
0answers
46 views

Help with this eingenvalues problem

I'm trying to find the Laplacian eigenvalues for the Dirichlet problem in a right isosceles triangle $T\subset\mathbb R^2$ where its smaller side has length $c$. I saw in an article that the ...
1
vote
2answers
50 views

Have I obtained the proper solution to this PDE?

I'm a little stuck on this. Consider $ u_t -(1+t^2)u_x = \phi(x,t) \quad u(x,0)=u_0(x)$ Via the method of characteristics, the total derivative of $u(x,t)$ is $$\frac{du}{dt} = \dfrac{\partial ...