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

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1answer
12 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|), ...
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5 views

Dirichlet problem in exterior domain and uniqueness

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 $\lim_{r \rightarrow 0} u(x) =0$. ...
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1answer
74 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...
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1answer
38 views

Which 2D domain with fixed area has the lowest laplacian eigenvalue?

I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?
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1answer
362 views

What is the difference between single and double layer potential

I want to know the difference between single layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
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0answers
117 views

Using the Kuramoto-Sivashinsky operator applied on the Korteweg–de Vries Soliton as a filter for image processing

When the Kuramoto-Sivashinsky operator (Kuramoto-Si) is applied to the Korteweg–de Vries Soliton (Soliton) we obtain a very interesting filter which is able to process an image via convolution. An ...
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1answer
129 views

Solving cauchy hyperbolic second order pde

I'm currently taking a course in partial differential equations. I'm trying to solve the following problem (which is, as far as I can tell, a bit above the level of the course): $$\begin{align} ...
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10 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 ...
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1answer
15 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 ...
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2answers
34 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 ...
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0answers
10 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 ...
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1answer
228 views

Inequality between Neumann and Dirichlet eigenvalues

Let $\Omega$ be a fixed, smooth and bounded domain in $\mathbb{R}^n$. If we denote with $\{\lambda_n\}_{n \ge 1}$ the nondecreasing sequence of eigenvalues of the Dirichlet problem $$\left\{ ...
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0answers
20 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 ...
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1answer
25 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\\ ...
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3answers
34 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 ...
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0answers
9 views

Reduction of order for PDE

I am trying to reduce the order of a PDE, as I was not able to do this in Maple (Maple: Substitution in PDE to reduce order) I am now trying the transformation in Mathematica. For the procedure I ...
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1answer
19 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 ...
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0answers
6 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 } ...
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3answers
536 views
+50

The solution of Cauchy-Riemann equation

Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?
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0answers
31 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 ...
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0answers
24 views

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 ...
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0answers
19 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: ...
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1answer
82 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
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1answer
364 views

composition of convex function with harmonic function.

Consider $u:\Omega \to \mathbb R$ be harmonic and $f$ be convex function. How do i prove that $f\circ u$ is subharmonic? It seems straight forward : $\Delta (f\circ u (x)) \ge f(\Delta u(x))=0 $. Is ...
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0answers
10 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 ...
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1answer
9 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 ...
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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 ...
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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 ...
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1answer
31 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 ...
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16 views

Solving a specific second order ODE

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 $\alpha(t,x) = \frac{4\epsilon(1+\epsilon t)}{(1+\epsilon t)^2 - \epsilon^2 x^2}$. ...
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2answers
46 views

PDE (linear, nonhomogeneous, first order)

Problem: $$\frac{du}{dx} + \frac{du}{dy} + u = e^{x+2y}$$ I have tried many methods and non have given me the correct result. The best lead I had was to change the coordinates and I got: ...
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2answers
20 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 ...
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2answers
19 views

Sketching solutions to IVP

Consider the following initial value problem (IVP): $$u_t + \cos(t)u_x = −u, \, \, \, \, (x,t) ∈\mathbb R×(0,∞)$$ $u(x,0) = u_0(x)$, $x ∈\mathbb R$, where $u_0 : \mathbb R → \mathbb R$ is a prescribed ...
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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 ...
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1answer
30 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 ...
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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 ...
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0answers
79 views

Difference between $C^\infty (U)$ and $C^\infty (\overline U)$

I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use ...
2
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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 ...
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2answers
99 views

What does the notation $C(\bar U)$ mean for $U\subset\Bbb{R}^d$ open?

Let $U$ be an open subset of $\Bbb{R}^d$. In Evans's PDE book, $$ C(U)=\{u: U\to\Bbb{R} \mid u\ \hbox{continuous}\} $$ and $$ C(\bar U)=\{u\in C(U)\mid u\ \hbox{ is uniformly continuous on bounded ...
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1answer
222 views

Extension of partial derivatives and the definition of $C^k(\overline{\Omega})$

The following is an excerpt from Folland's Introduction to Partial Differential Equations: Let the open set $\Omega\subset{\mathbb R}^n$, and $k$ be a positive integer. $C^k(\Omega)$ will denote the ...
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1answer
521 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of a PDE textbook(e.g. Folland's Introduction to Partial Differential ...
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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 ...
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13 views

solve div( (divE) *E )=0 in two-dimension, E is a vector [on hold]

solve div( (div E)*E)=0 in two-dimension, E is a vector, using numerical method
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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
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1answer
60 views

Diffusion equation with advection and decay

I'm trying to solve the following initial value problem. $\begin{cases} u_t + u_x - u_{xx} &= -u, \quad \text{on} \quad \mathbb R \times \mathbb R_+\\ u(x,0) &= \frac{1}{4\pi} ...
2
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1answer
29 views

Poincare's inequality with difference quotient

For the classical Poincare's inequality, if $u \in H^1_0(\Omega)$, then $$\int_\Omega u^2 \,dx \le C \int_\Omega |\nabla u|^2 \,dx.$$ Do we have something similar with the difference quotient? That ...
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36 views

How to solve non-symmetric, generalized saddle-point problem

From a discretization of an incompressible Navier-Stokes equation I get the following matrix $$M = \begin{pmatrix}A & B_1^\intercal \\ B_2 & C\end{pmatrix}$$ Because my density is ...
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0answers
6 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 $x =1, y= 1$.
2
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2answers
78 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 ...
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2answers
48 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} = ...