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

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3
votes
1answer
17 views

Poisson Equation: 'Dirichlet' type problem on all of $\mathbb{R}^N$

I've seen a great deal about solving the Dirichlet problem for Poisson's equation \begin{equation} \Delta u = f \quad \mbox{in} \quad \Omega \subset \mathbb{R}^N\\ u = 0 \quad \mbox{on} \quad ...
1
vote
0answers
24 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
1
vote
1answer
38 views

calculate solution of heat equation with method of separation of variables

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ ...
0
votes
0answers
12 views

Solve Initial value poblem.

Utt=4uxx -infinite 0 U (x,0)=0 Ut (x,0)={1,-1 0, any where else , and sketch at t = 1 and t=4 That what I tried u= 1/4 integration g (z) dz and ...
1
vote
1answer
30 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best NUMERICAL method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...
2
votes
1answer
39 views

Solution to second order PDE $u_{xy}-xu_x+u=0$

Given second order partial differential equation $u_{xy}-xu_x+u=0$, where $u=u(x,y)$ find the general solution. I tried to use $u(x,y)=v(ξ(x,y),η(x,y))$ substitution to get ...
1
vote
0answers
36 views

Sobolev spaces on Riemannian manifold

I know one can define the Sobolev spaces on a Riemannian manifold as completions, but is there an equivalent definition that uses weak derivatives, like in the case of open sets in $\mathbb{R}^n$ ? ...
0
votes
0answers
20 views

show that a function is $L^1$-contraction [on hold]

Let $C_c(\mathbb{R})$ denotes the set of all continuous functions $u_0$ with compact support defined on $\mathbb{R}$ such that the initial value problem \ \begin{equation*} \frac{\partial u}{\partial ...
0
votes
0answers
8 views

uniqueness of the classical solution of first order hyperbolic PDE

Consider the initial value problem : \ \begin{equation*} \frac{\partial u}{\partial t}+a(u)\frac{\partial u}{\partial x}=0, \hspace{.2cm} x\in\mathbb{R}, 0<t\leq T, \\ u(x,0)=u_0(x), \hspace{.5cm} ...
2
votes
0answers
31 views

Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
0
votes
1answer
18 views

Hyperbolic energy estimate in Evans PDE book

Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how ...
0
votes
0answers
9 views

Extremum of Laplacian of a function

Let f(x,y,z) be any arbitrary continuous function. Let's denote Laplacian of f by $\nabla^2 f$. 1) How do we denote the extremum of $\nabla^2 f$ mathematically ? 2) how do we solve for such ...
1
vote
1answer
19 views

Uniqueness for Dirichlet problem in exterior domain

I have the following problem: $\Delta u =0$ in $\Omega_e = \mathbb{R}^3 - \overline{\Omega}$, and with condiction $u=0$ on $\partial \Omega$ and $u=o(1)$, that is $\lim_{r \rightarrow 0} u(x) =0$. ...
1
vote
1answer
45 views

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert ...
0
votes
1answer
19 views

Is it possible that a PDE solved by two different analytical methods with same Initial and boundary values give different results?

I have developed two models of same scenario. Both models involve a PDE which is solved with same Initial and Boundary conditions. In one model it is solved with Laplace transform and in other with ...
0
votes
0answers
15 views

Find homogneous solution to pde

I need to find the homogenous solution and fixed points to this pde. I have no idea how to proceed. $$ \partial_t u = r u - (\partial_{xx}u +1)^2u - u^3 $$ It is second order non linear parital ...
2
votes
2answers
40 views

heat equation-uniqueness of solution

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ ...
0
votes
0answers
7 views

Parabolic equation with discontinuous boundary condition

Consider a parabolic initial-boundary value problem in $\Omega\times (0,T]$ $$\frac{\partial u(x,t)}{\partial t}=\mathcal{L}u(x,t),$$ with $$u(x,0)=0, x\in \bar{\Omega} \text{ and } ...
1
vote
0answers
23 views

solve the pde z=pq?

i tried tried it using charpit method $$f=z-pq$$ $$\frac{\partial f}{\partial x}=0,\frac{\partial f}{\partial y}=0,\frac{\partial f}{\partial p}=-q,\frac{\partial f}{\partial q}=-p,\frac{\partial ...
0
votes
1answer
21 views

two dimensional heat equation

Please I really need some help for this exercise, I can't solve it for any ways... I need to prove the maximum principle for the two dimensional heat equation with zero boundary data. Really I need ...
1
vote
1answer
16 views

Computing the Fourier Transform of the square pulse

The function in question is $f(x) = H(a - |x|)$, where the Fourier transform is given by $F(k) = (\frac{2}{k}) \sin(ak)$. Initial attempt: $F(k) = ( H(a - |x|), e^{-ikx} )$ = $- (\delta(a - |x|), ...
8
votes
0answers
68 views

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
3
votes
2answers
41 views

How to solve this second order linear pde?

I have the following pde for $f(t,x,y)$: $a x^2 f_{xx} + bxf_x + f_t - bxy + c = 0$ subject to $f(T,x,y)=0$ for all positive $x,y$, where $a,b$ and $c$ are constants. The equation seems to be ...
2
votes
0answers
55 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
0
votes
0answers
15 views

Regularity of $u$ and $v$ satisfying $u_t - \Delta u = \Delta v - v_t$

I have that there are two functions $u$ and $v$ in $H^1(0,T;L^2)\cap L^2(0,T;H^2)$ satisfying weakly the equation on a bounded domain $\Omega$ $$u_t - \Delta u = \Delta v - v_t$$$ given some ...
1
vote
1answer
10 views

partial differential equation, classical formulation

I got to find a classical equation for those 2 equations: $$\int_{-\frac{1}{2}\pi}^{\frac{1}{2}\pi}(cos(x)u\prime(x) \phi\prime(x) -f(x)\phi(x))dx=0$$ with $$\phi \in ...
0
votes
0answers
11 views

2D weak formulation, stiffness matrix and load vector of Laplace equation

I have following equation: $$ \begin{cases} - \Delta u(x,y) &= 1, & \forall(x,y) \in \Omega = (0;1)^2 \\ u(x,y) &= 0, & \forall (x,y) \in \partial \Omega \end{cases} $$ The problem ...
0
votes
1answer
41 views

Applications of the wave equation

I've recently started to take interest in PDEs and how to solve them, and I'm wondering a bit about real life applications of the wave equation. So far I haven't found anything about practical ...
-2
votes
0answers
35 views

Solving a specific second order ODE [on hold]

I need some one can help me to solve the following equation : $$z_{tt}-z_{xx}-2z_t = \alpha(t,x)(z_x-z_{tx})$$ where $\alpha(t,x) = \frac{4\epsilon x(1+\epsilon t)}{(1+\epsilon t)^2 - \epsilon^2 ...
2
votes
3answers
53 views

partial differential equation-exercice

let in $\mathbb{R}^2$ the equation $$ \dfrac{\partial^2 u(x,t)}{\partial t^2} - \dfrac{\partial^2 u(x,t)}{\partial x^2} = 0 $$ We put: $ \begin{cases} x=\xi + \eta\\ t=\xi- \eta \end{cases} $ and ...
0
votes
2answers
21 views

Laplace transform of the square wave to solve PDE

Solve $$y'' + 3y' +2y = r(t)$$ given $y(0)=0$ and $y'(0) = 0$ where $r(t)$ is the square wave, $$r(t) = u(t-1) - u(t-2)$$ I'm just going to type out the answer as I read it and tell you which ...
0
votes
0answers
16 views

How to solve the following partial differntial equation using fourier transform?

How to solve this equation? $$2\iota n_0k_0 \frac {\partial E_x}{\partial y}=\frac {\partial^2 E_x}{\partial x^2} + \frac{\partial^2 E_x}{\partial z^2} $$ where, $E_x(x,y,z)$, $n_0$ and $k_0$ are ...
0
votes
0answers
20 views

Solving a first order linear PDE on an interval, with initial and boundary conditions

I'm trying to use the characteristic curve to solve the following IBVP: $$u_t+2u_x=0.5,\;0<x<1,\,t\geq0$$ with IBC: $$u(0,x)=1,\,0<x<1;\;u(t,0)=\phi(t),t\geq0$$, where $\phi(t)$ is: ...
0
votes
1answer
16 views

Examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$

The following is the trace theorem in Partial Differential Equations by Evans: Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then ...
2
votes
1answer
21 views

Power series solution to a PDE?

I have the following partial differential equation: $u_t = u_xu_y$ I know that the solution can be formed via power series. I want to find a solution of degree $2$ that satisfies an initial ...
0
votes
0answers
6 views

weakly non linear analysis of Swift-Hohenberg equation:

Do you have any text book or reference that I can look into for the weakly non linear analysis for solving following swift-hohenberg equation? Also if you could help me solve it, that would be great ...
0
votes
1answer
11 views

Diffusion model - sign of boundary condition

I'm trying to compute the concentration of some pollutant in the rectangular pool. The pool is isolated from two sides (hatching in the picture), on the third side there is some cleaner which ...
1
vote
0answers
7 views

Finding the characteristic curves of the given PDE which is passing through the given point.

How to find the number of characteristic curves of the PDE $(x^2+2y)u_{xx}+(y^3-y+u)u_{yy}+x^2(y-1)u_{xy}+3u_{x}+u=0,$ passing through the point $x =1, y= 1$.
4
votes
1answer
33 views

Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
0
votes
2answers
50 views

solve First order partial differential equation

we need to solve this equation: $$(x^2 + y^2)\frac{\partial z}{\partial x} + 2xy\frac{\partial z}{\partial y} = (x + y)^3 z$$ the general solution to this equation is : $$\frac{dx}{x^2 + y^2} = ...
0
votes
1answer
30 views

Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if ...
1
vote
0answers
22 views

Solving a PDE using method of characteristics

I am trying to solve the following PDE $$u_t+y u_x =-(y+\mathbb{H}(y_x))$$ where $$\mathbb{H}(g)=P.V. \int_{-\infty}^{\infty} \frac{g(x')}{x-x'} dx'$$ is the Hilbert Transform, P.V. means ...
-1
votes
0answers
19 views

integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
1
vote
0answers
12 views

How to build a matrix in MATLAB with the next characteristics?

Let $\lambda_1=\frac{k D_u}{2h^2}$ a constant value. How to generate a matrix in MATLAB with the next entries: $A= \begin{pmatrix} 1+\lambda_1 & -\lambda_1 & 0 & 0 & \cdots & 0 ...
0
votes
0answers
16 views

Studying for Prelims…need help with Cauchy-Kovalevsky Theorem?

I have the following PDE that I wish to solve using the Cauchy-Kovalevsky theorem as a way of studying for my prelims in the future. The original problem for confirmation can be found: ...
1
vote
0answers
14 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
1
vote
1answer
21 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
2
votes
2answers
79 views

$f '' - (f ')^2 + f=0$; what is known about solutions?

I'm curious about solutions to the equation $$f''-(f')^2+f=0$$ on the whole real line, as well as solutions which are periodic. Any info about the obvious multivariable generalization would interest ...
0
votes
0answers
11 views

Non-symmetric case, can we get the minimization problem in Lax Milgram therom?

In the Dietrich braess's book 'Finite Elements'. The Lax-Milgram thm is stated under the condition that $a$ is symmetric(There are many book which also state property of symmetric when getting a ...
1
vote
0answers
18 views

Aproximation in $W^{1,p}(U)$ with U disconnected.

Consider $U=(-1,0)\cup(0,1)$. Define $$v(x)=\left\{\begin{array}{rc} 0,&\mbox{se}\quad -1<x<0,\\1, &\mbox{se}\quad 0<x<1. \end{array}\right. $$ Clearly $v\in W^{1,p}(U)$ for each ...