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

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Solve: $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$

In the plane find two solutions of the initial-value problem $xu_x + yu_y+ \frac{1}{2}(u^2_x+u^2_y) =u, u(x, 0) =\frac{1}{2}(1−x^2)$. I think we get to use the method of characteristics But I am not ...
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1answer
20 views

Integrals with functions as bounds

How to calculate integral such as $$\int_{g(χ)}^{φ(χ)} f(s) \, ds$$ where $F'(s)=f(s)$ Integrals like this appear often in PDE's .I'd like to know the whole theory i mean if there is a formula how ...
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0answers
4 views

When is a symmetric hyperbolic system a second order scalar equation?

Given a linear symmetric operator of the form $$A^\mu \partial_{\mu}v+Bv=F$$ where $v\in C^{\infty}(\mathbb{R}^m ,\mathbb{R}^n)$ is a solution of the system and $A^\mu, B$ are $n \times n$ matrices ...
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1answer
73 views

Uniform norm $ \|u\|_{C(\overline{U})}$ in PDE

Let $U\subset \Bbb{R}^n\to\Bbb{R}$ be an open set (not necessarily bounded) and $u:U\to\Bbb{R}$ be a bounded continuous function. In Evans's PDE textbook, the author defines a norm $$ ...
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0answers
11 views

PDE, radially symmetric function

I got to show that the minimal surface $$(1): \nabla (\frac{\nabla u(x,y)}{\sqrt{1+|\nabla u(x,y)|^2}})=0$$ for radially symmetric functions can be wrtten as $$(2): \partial ...
1
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1answer
19 views

Estimation of gradients in Poisson's equation

I am trying to show the following result. Let $D\subset\Bbb R^3$ be a bounded open set with smooth boundary. For any $f\in H^{-1}(D)$, let $\phi$ be the unique weak solution to the following ...
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0answers
9 views

Solution to Laplaces

I am trying to show that the following is a solution to Laplaces equation. $v(x) = c_2*Ln(|x|), x \in R^2$ where $c_2$ is an arbitrary constant. And $x \neq 0$ . Im just stuck on how to get started ...
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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 ...
7
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0answers
56 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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1answer
20 views

Finding the general solution of a partial differential equation [on hold]

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 = ...
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0answers
16 views

Variable seperable method [on hold]

Find a solution of the equation $$\frac{∂^2u}{∂x^2}=\frac{∂u}{∂x}+2u$$ in the form $u=f(x)g(y)$. Solve the equation subject to the conditions $u=0$ and $∂u/∂x = 1+ e^{-3y}$ when $x=0$ and for all ...
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0answers
15 views

Question about asympotic behavior of $\frac{1}{s}\int_0^s u(x,t) dt$ .

I am just reading a paper, in the final theorem, the author wants to prove that $u(x,t)$ converges to some $v(x)$ in the $L^2$ norm as $t$ $\to$ $\infty$. But in the proof, he defines a ...
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0answers
32 views

Seeking help with finding the general solution of this differential equation

I am trying to find the general solution of the following equation. $$\int_0^\infty \frac{\partial f(x, t)} {\partial t} \sin(x \xi) \, dx = \xi \int_0^\infty f(x, t) \cos(x \xi) \, dx -\alpha \xi ^2 ...
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0answers
12 views

Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
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1answer
13 views

Solve transport equations by using Laplace transform

I'm trying to solve rather formally one-dimensional transport equation: $$ u_{t}+cu_{x}=0\quad\text{in $(0,\infty)\times(-\infty,\infty)$} $$ with an initial data $u_{0}$, which is bounded and ...
3
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1answer
650 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
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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 ...
3
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1answer
31 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 $$\Omega:=\{x\in\mathbb{R^d}\mid a<\|x\|_2<b\},$$ ...
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1answer
15 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
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0answers
8 views

Diffusion/Heat equation, weak maximum principle

Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$ and its variational formulation $$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall ...
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1answer
33 views

Kato's inequality

Let u be a smooth function defined in a Riemannian manifold $(M,g)$. The well known Kato's inequality states $$|∇|∇u||^2≤|∇^2u|^2$$ where $∇^2$ represents the Hessian operator of $M$. I would like ask ...
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1answer
18 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 ...
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20 views

Solving $4u_{tt}-3u_{xt}-u_{xx}=0$

Solving $\begin{cases} 4u_{tt}-3u_{xt}-u_{xx}=0\tag1\\u(x,0)=x^2\quad\text{and}\quad u_t(x,0)=e^x\end{cases}$ in $\mathbf R\times\mathbf R_{>0}$ First I factorized and get for the first line; ...
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0answers
7 views

unicity solution of partial differential equation

let the following problem: $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\ u(0,t)=u(l,t)=0\\ u(x,0)=f(x)\\ \dfrac{\partial u}{\partial x}(x,0)=g(x) ...
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0answers
15 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
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0answers
11 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in ...
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0answers
14 views

partial differentia equation and Laplace transformate

let the following differential equation: $$ \begin{cases} \dfrac{\partial^2 y}{\partial t^2}= a^2 \dfrac{\partial^2 y}{\partial x^2}, 0<x<l, t>0\\ y(0,t)=f(t)\\ y(x,0)=0\\ \dfrac{\partial ...
0
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1answer
30 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, ...
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0answers
17 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 $V,x(x=0,y)=0,V(x=10,y)=0,V(x,y=1)=0,V(x,y=0)=0$
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1answer
44 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 ...
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0answers
21 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 ...
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1answer
16 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 ...
3
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1answer
19 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?
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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 ...
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0answers
27 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 ...
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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 ...
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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 ...
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0answers
15 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
38 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 ...
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0answers
13 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 ...
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0answers
54 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 ...
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1answer
43 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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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
22 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
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\\ ...
2
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0answers
29 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 ...
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0answers
11 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 ...
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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 ...
1
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0answers
34 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 $\mathbb{R}^n$ ? ...