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

learn more… | top users | synonyms (2)

0
votes
1answer
17 views

Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
0
votes
0answers
7 views

Domain of Dependence Inequality for the Wave Equation?

Let $(x_0,t_0)\in R^{n+1}$ with $t_0>0$, and let $\Omega$ be the conical domain in $R^{n+1}$ bounded by the backward characteristic cone with apex at $(x_0,t_0)$ and by the plane $t=0$. Suppose ...
3
votes
0answers
19 views

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point ...
1
vote
2answers
47 views

Finding a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$

To find a solution for $x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y}=x^2$ knowing that $u(x,y)=1$ if $xy=1.$ I thought it may be useful to do the change $v=\log x, w=\frac{1}{2}\log ...
6
votes
3answers
48 views

Solving $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$

I want to solve the differential equation $$x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y }=1$$ with the initial condition $u(1,y)=y.$ I'm very unfamiliar with possible methods to ...
0
votes
0answers
19 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
0
votes
1answer
15 views

Uniqueness of solution for seperation of variables solvable PDEs

I am taking first course in PDEs and the only way i know of solving PDEs is separation of variables , and all the equations i saw had unique answers due to the ICs and BCs , but not this one : $$ ...
0
votes
0answers
6 views

General solution of a hyperbolic second order pde

How do I find the general equation to $$\frac{\partial^2 u}{\partial x \partial y} = 0$$ using characteristics. I am confused because I thought hyperbolic equations always had 2 distinct roots.
0
votes
0answers
16 views

uniformly convex domain and uniformly convex function

I want to ask a question about uniformly convex domain: Suppose $\Omega$ is unifromly convex, i.e. for each point $x_0\in\partial\Omega$, with regualrity $C^2$ smooth. Uniform convexity of domain ...
0
votes
0answers
16 views

Boundary value problem for general elliptic equations

I have showed that if $U$ is connected then the only smooth solutions of \begin{align} -\Delta u=0 \;\;\;\text{in } U && \frac{\partial u}{\partial \nu}=0 \;\;\; \text{on } \partial U ...
0
votes
1answer
16 views

Green's Function - PDE

I am completely stumped on Green's function regarding PDE's. There are barely any examples in my book on how to apply it. For example, if a question asks, "find the Green's function ..." to some ...
1
vote
0answers
15 views

One simple question about Fourier transformation of system of PDE's

Let's assume set of equations $$ \tag 1 \frac{\partial \mathbf A}{\partial t} = \Delta \mathbf A + a [\nabla \times \mathbf A] - d\mathbf b_{k} (\mathbf b_{k} \cdot \mathbf A), \quad \mathbf A(0) = 0 ...
0
votes
0answers
10 views

using method of lines to solve a pde heat equation [on hold]

the heat equation to be solved is du/dt=du2/dx2 domain is 0 the RHS is dicretised into sets of odes and solved with any stiff ode solver in matlab thanks
1
vote
0answers
15 views

$L^{2}$ convergence and converence of distribution

Suppose that $f_{n}(x)$ are a sequence of $L^{2}$ functions which converge to a function $f(x)$ in the $L^{2}$ sense. Show that it also converges weakly in the sense of distributions, ie for any test ...
0
votes
0answers
6 views

How to solve a factorized Helmholtz equation?

I am reading a paper on optics an in appendix A2 they split the Helmholtz equation into two parts and write down the solution for one of those parts (link). Helmholtz Equation where the Laplacian is ...
0
votes
0answers
23 views

Green function integration

When I'm trying to find the Green Function of Helmholtz equation for a cube $0≤x,y,z≤L$ $$\nabla^2u+k^2u=\delta(\vec{x}-\vec{x}')$$ where u=0 on the surface. I set to find the green function where ...
0
votes
0answers
11 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
0
votes
0answers
11 views

continuous dependency estimate for viscosity solutions

Let $u^i$, $i=1,2$, be viscosity solutions of \begin{align*} u_t^i + H(Du^i,x) & = 0\quad\mathrm{in}\ \mathbb R^n\times (0,\infty)\\ u^i & = g^i\quad\mathrm{on}\ \mathbb R^n\times \{t=0\} ...
0
votes
1answer
12 views

Solvability of eigenvalue problem with Schwartz data

Fix $a\in\mathbb{R}$ and define the operator $T$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ by sending $\phi$ to $\Delta\phi-a^2\phi$. Then $T$ is clearly a bounded operator.Question is ...
0
votes
0answers
8 views

Find a function to satisfy a necessary condition on a system of pdes

Consider the following set of PDE's $\displaystyle \frac{\partial u}{\partial x}(x,y)=f(x,y,u(x,y))$ $\displaystyle \frac{\partial u}{\partial y}(x,y)=1$ $u(x_0,y_0)=u_0$ Show ...
0
votes
0answers
34 views

Inverse non homogeneous heat problem unicity

Consider the inverse problem: Determine the pair $u = u(x, t)$ and $f = f(x)$ satisfying $$u_{t}(t, x) - u_{xx}(t, x) = f(x) \hspace{0.5cm} t > 0, x \in ]0, L[;$$ $$u(t, 0) = 0 \hspace{0.5cm} t ...
0
votes
0answers
9 views

Higher order numerical PDE schemes near boundaries, implementation in MATLAB

Followup to my previous question. The first order scheme proved unstable for my pde: $$f_t + A y f_x - B x f_y =0$$ So I'm looking to implement a higher order scheme (using these tables). I was ...
1
vote
1answer
31 views

Determine adjoint operator

Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$ $D $ is the space of testfunctions I know that ...
1
vote
2answers
39 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
0
votes
0answers
6 views

How to find multipliers in charpit's method [on hold]

After getting to the subsidiary equations in charpit's method i am unable to see further. It will be a great help if any one can explain how to choose the multipliers i mean is there a criteria for ...
2
votes
1answer
15 views

An identity for the integral $ \int_{\partial B(0,1)}u(x_0+aw)u(x_0+cw)$ with a harmonic function $u$

This is Question 2.18 from Gilbarg and Trudinger, chapter 2. We are given that $\Omega$ is open bounded smooth boundary. Now fix $x_0\in \Omega$ and a constant $c>0$ such that ...
3
votes
2answers
32 views

Vibrating string - separation of variables

$u_{tt}=c^2u_{xx}$ where $u(x,0)=x+\sin(x)$, $u_t(x,0)=0$, $u(0,t)=u_x(\pi,t)=0$. Assume a solution $u(x,t)=X(x)T(t)\not\equiv 0$. This yielded $\lambda_n=\frac{1}{2}+2n$. For $X_n(x)$ I have ...
1
vote
0answers
19 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n ...
0
votes
0answers
16 views

Normal derivative property

I have seen in many papers that to obtain some results about PDEs is used the following argument: If $\phi=0$ in $\partial\Omega$ then $\bigtriangledown\phi=\dfrac{\partial\phi}{\partial n}n$, where ...
0
votes
0answers
18 views

Derivative of Smooth convex function composed with sobolev is sobolev?

This is homework so no answers please Here is what I mean specifically The full problem is $f:\mathbb{R}\to \mathbb{R}$ smooth and convex, bounded $u\in H^{1}(U)$ and $v\in H_{0}^{1}(U)$, is it ...
4
votes
1answer
44 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
0
votes
0answers
18 views

Approximating $v \in W^{1,p}_0(\Omega)$

Let $p > 1$. Let $V=W^{1,p}_0(\Omega)$ and $H=L^2(\Omega)$. Suppose $\{\lambda_i\}$ is a basis in $V \cap H$ which is smooth, and orthonormal in $L^2$. Given $v \in V$, is it possible to find a ...
2
votes
0answers
47 views

Stability of nonlinear system of PDE's

Let's assume system $$ \tag 1 \frac{\partial \mu}{\partial t} = \gamma (\mathbf B \cdot \mathbf E), $$ $$ \tag 2 [\nabla \times \mathbf E] = -\frac{\partial \mathbf B}{\partial t}, $$ $$ \tag 3 ...
0
votes
0answers
15 views

What does instability mean and examples, boundary condition

The Upwind-Scheme for the numerical solution of first order PDE's (homogenous case) of the form $u_t + cu_x = 0$ is given by $$ u_j^{n+1} = \left\{ \begin{array}{ll} u_j^n - \frac{c\Delta t}{\Delta ...
1
vote
0answers
25 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
0
votes
1answer
16 views

Simple problem about Laplace Equation in a domain

Suppose that "$u$", is solution of the problem $$\triangle u=0, r<R $$ $$u_{r}(R,\phi)=f(\phi), 0<\phi\ < 2 \pi$$ Show that $$\int_{0}^{2 \pi}{f(\phi)d\phi}=0$$ I know what this question ...
4
votes
0answers
22 views

Unbounded Entropy Solution to Burger's Equation

I need to deduce that $ u(x) = \left\{ \begin{array}{lr} \frac{-2}{3}(t+\sqrt{3x+t^2}) & t^2+4x>0\\ 0 & t^2+4x<0 \end{array} \right. $ is an unbounded ...
0
votes
0answers
16 views

Helmholtz equation in 3D

I understand solving the Helmholtz equation, $\nabla^2u + \lambda^2u = 0$, when $\nabla^2 = \partial_{xx} + \partial_{yy}$. However, I am not sure of the form I should obtain when $\nabla^2 = ...
1
vote
1answer
24 views

Separation of variables in the PDE $u_{tt}=c^2 u_{xx}$.

I'm stuck trying to work with the constants to the solution to the SOV problem Given the following equation: $u_{tt}=c^2 u_{xx}$ and the following conditions: $u(0,t)=0=u(\pi,t)$, $u(x,0)=0$, ...
2
votes
1answer
19 views

How to use D'Alembert formula for Neumann boundary conditions on a finite interval?

I have a PDE to solve that I am not sure how to do. I know how to solve this using D'Alembert's formula for Dirichlet boundary conditions but I do not know how to solve it for Neumann boundary ...
0
votes
0answers
22 views

Traces of $W^{1,\infty}$ functions

Let $\Omega$ be a Lipschitz domain and $p \in (1,\infty)$. It's known that if $u \in W^{1,p}(\Omega)$ then $u_{|\partial \Omega} \in W^{1-\frac{1}{p},p}(\partial \Omega)$. I'm wondering if the ...
0
votes
1answer
31 views

Voltage Distribution Inside a Cylinder [on hold]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
1
vote
1answer
70 views

Solving a PDE equation

Could you please help me to solve this equation: $$\frac{\partial^2}{\partial x^2}E(x,t)-LC_1 \frac{\partial^2}{\partial t^2}E(x,t)+LC_2 \frac{\partial^4}{\partial x^2 \partial t^2}E(x,t)=0 \qquad ...
0
votes
0answers
11 views

Second order PDE with initial and boundary conditions

I'm trying to solve the following PDE: $u=u(x,y)$ $\left\{ \begin{array}{1 1} \partial_x ^2 u-6\cdot\partial_x \partial_yu+9\cdot\partial_x^2u=x^2+y^2\ \\ \ u(0,y)=0 \\ \partial_x u(0,y)=y ...
0
votes
0answers
5 views

Dirichlet problem with Specific Boundary Condition

How to solve the Dirichlet Equation $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ for Laplace Equation for following boundary condition? $$u(x, 0) = x; \,u(x, \pi) = x( \pi-􀀀 ...
1
vote
1answer
34 views

d'Alembert's Solution: Should anything more be added to it?

I have this initial value problem: $$ u_{tt}- c^2u_{xx} = 0; \,0 < x < 1; \,t > 0; $$ $$ u(x, 0) = 0; \,u_t(x, 0) = 1; 0 <= x <= 1; $$ $$ u(0, t) = u(1, t) = 0; t >= 0: $$ ...
2
votes
1answer
33 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
0
votes
1answer
15 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
0
votes
0answers
25 views

How to prove this property?

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 ...
0
votes
0answers
15 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...