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

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0
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1answer
17 views

Let $w = \log(u^{2} + v^{2})$ where $u=e^{(x^{2}+y)}$ and $v= e^{(x+y^{2})}$

Then $\frac{\partial w}{\partial x}$ for $(x=0,y=0)$ is ? I got answer as 0 since on partial differentiation I got, ...
2
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0answers
14 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R^d}\mid a<\|x\|_2<b\},$$ ...
-1
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0answers
16 views

Given the following partial differential equation, determine whether the separation constant $\beta = 0$ [on hold]

Determine whether the separation constant $\beta=0$ would lead to a viable solution: $\nabla^2 V=0$ with conditions $V,x(x=0,y)=0,V(x=10,y)=0,V(x,y=1)=0,V(x,y=0)=0$
1
vote
1answer
40 views

Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
0
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0answers
19 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
1
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0answers
12 views

Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
0
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1answer
15 views

Stability (wrt parameters) of elliptic partial differential equation

consider the equation $$\mathcal Lu=f \quad \text{in } \Omega $$ With some appropriate boundary condition, $\Omega$ regoular as you like, $ \mathcal L$ to be defined by $$\mathcal ...
2
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1answer
15 views

where does the pdes mathematocal classification names come from?

PDEs are classified into hyperbolic, parabolic and elliptic. where do these names come from? Do they have anything to do their geometric shapes?
1
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2answers
255 views

finding solution to partial differential equation

what is the best way to solve a partial differential equation: $$ (1-ax)(∂^4 y)/(∂x^4)+2a (∂^3 y)/(∂x^3)=0 $$ like in ordinary differential equations I tried the power series method (I'm not very ...
0
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0answers
13 views

Finding the general solution of a partial differential equation

Find the general solution of the following PDE $$\frac {\partial^2 z} {\partial x^2} - \frac {\partial^2 z} {\partial y^2} + \frac {\partial z} {\partial x} + 3\frac {\partial z} {\partial y} -2 = ...
0
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0answers
25 views

General solution of a 2nd order inhomogeneous PDE [on hold]

Find the general solution(Complementary function + Particular integral) of the following PDE $$\frac {\partial^2 z} {\partial x^2} - \frac {\partial^2 z} {\partial y^2} + \frac {\partial z} {\partial ...
0
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1answer
59 views

HELP to solve - PDE First order

I have this equation $$ u_x+uu_y=0$$ by the book "Handbook of First order Partial Differential Equations - page 290." The general solution is $F(ux-y,u)$, where $F$ is a arbitrary function. I try ...
0
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0answers
13 views

Kato's inequality

Let $u$ be a smooth function defined in a Riemannian manifold $(M,g)$. The well known kato's inequality states $$|\nabla|\nabla u||^2\leq |\nabla^2 u|^2$$ where $\nabla^2$ represents the Hessian ...
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0answers
2 views

How to find spacial periodicity

I am struggling with the following question. Consider $$U(x,t)=2\cos\left(kx-\sqrt{\frac{a}{b}}k^2+(\Delta k)^2)t\right)\cos\left((\Delta k)(x-2\sqrt{\frac{a}{b}}kt\right)$$ and suppose that $\Delta ...
1
vote
1answer
21 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 ...
6
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0answers
48 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
13 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
2
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1answer
36 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
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0answers
12 views

linearized KDV equation

Consider the PDE $$u_t +4u_x +6u_{xxx} =0$$ for the function $u = u(x, t).$ (a) Determine for what wave speeds $c$ there exist bounded travelling wave solutions of $u$, and find the form of the ...
2
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0answers
40 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 ...
1
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1answer
39 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
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0answers
15 views

Question about trace operator

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it ...
3
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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 ...
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0answers
15 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
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1answer
37 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\\ ...
2
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0answers
26 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
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0answers
10 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 ...
0
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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 ...
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0answers
29 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
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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} ...
0
votes
1answer
14 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 ...
2
votes
2answers
39 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\\ ...
2
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0answers
52 views

Relation between a linear second order differential equation and Riccati special differential equation

Consider the following differential equation \begin{equation} \frac{d}{dx}\left[N(x)\frac{dw}{dx}\right]+\sigma^2\rho(x)w=f(x,\sigma),~~ 0<x<l, \end{equation} $0<N\in C^1(0,l)$, $0<\rho\in ...
1
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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 $\lim_{r \rightarrow 0} u(x) =0$. ...
0
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0answers
8 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
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|), ...
2
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1answer
76 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} + ...
1
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1answer
40 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?
0
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1answer
366 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 ...
1
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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 ...
3
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1answer
131 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} ...
0
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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 ...
3
votes
2answers
40 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 ...
0
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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 ...
3
votes
1answer
229 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
22 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 ...
1
vote
3answers
45 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
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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 ...
0
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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 } ...
2
votes
3answers
540 views

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?