Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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An inequality: from the complex to the real case.

Let $p\in(1,2]$ and $q\in[2,\infty)$ be its conjugate exponent, then for $z,w\in\mathbb{C}$ the following inequality holds $$ \Large \left|\frac{z+w}{2}\right|^q+\left|\frac{z-w}{2}\right|^q\leq\left[\...
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What conditions are required to guarantee that my matrix is skew-Hermitian?

Consider the equation $$ \frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2} $$ with $$ \boldsymbol{T}=(T_1,T_2,...T_N)^T $$ Let $$ T_i=\sum_{j=1}^...
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Estimates on Derivates for One Dimensional Heat Equation

$\textbf{The problem}$: Let $[-r_0,r_0]$ be a segment in $\mathbb{R}$, let $T>0$ and $u$ be a smooth function satisfying: \begin{align} \begin{cases} u_t-\Delta u = 0 & \qquad \text{on $[-r_0,...
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1 vote
1 answer
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$u$ be the sol of $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}= f(x)$ if $f\geq 0$ then u is constant and $f=0$

Let $\Omega$ be a bounded domain with smooth boundary . Let $u$ be a solution of the problem $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}=f(x)$ and $\frac{\partial u}{\partial n}=0$ . Assume that $...
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0 answers
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Solving an IVBP for a metal rod dipped from warm to cold water

I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees. I prepare the diffusion ...
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summer reading suggestions

I have just finished 2nd year of my maths degree and I was wondering if anyone had any good recommendations for summer reading, I want to make sure I keep practising so that I don't feel underprepared ...
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1 vote
0 answers
28 views

The method of separation of variables in solving a PDE

Suppose we have a PDE and we can solve it with the separation of variables and let's say the variables are $x,y,z$. Is the factorized function $f(x,y,z)=g(x)h(y)l(z)$ the only possible solution? Are ...
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1 vote
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Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
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Does Homogenity implies first order PDE

I have a pde say $$f(x,y,z, a,b)=0$$ with $x,y$ being independent variable and $z$ being the dependent variable. If i can write $z=f(x,y,a,b)$ and if $f$ is a homogeneous function then can i say that ...
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1 vote
1 answer
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Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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  • 555
2 votes
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Overdetermination of the Maxwell-Equations

I‘ve heard people talk about the overdetermination of the Maxwell equations, which are of course: $$ \nabla \cdot E =\frac{\rho}{\epsilon}\\ $$ $$ \nabla \cdot B=0 \\ $$ $$ \nabla \times E=-\...
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3 votes
0 answers
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Solve the PDE $(xz-y)p+(yz-x)q=xy-z$ using lagrange method

I have the following PDE, $$(xz-y)p+(yz-x)q=xy-z$$where $p=z_x,\quad q=z_y$ Now having a hard time to get two solution from, $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{xy-z}$$ I can't think of any ...
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  • 491
1 vote
1 answer
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Why $w(0,t)=w(L,t)=0\Longrightarrow w_t(0,t)=w_t(L,t)=0$

Let $w$ be a $C^2$ function in two variables, $x$ and $t$. The domain of $x$ is $[0,L]$ whilst the domain of $t$ is $t\geq 0$. Suppose that $w(0,t)=w(L,t)=0$. The apparently $w_t(0,t)=w_t(L,t)=0$. I ...
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  • 1,273
0 votes
1 answer
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Numerical viscosity from the Crank-Nicolson method

Consider the following nonlinear diffusion PDE \begin{equation}\frac{\partial U}{\partial t} = \frac{\partial^2}{\partial x^2}[\Phi(U)], \end{equation} where $\Phi$ is a smooth nonmonotone function of ...
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1 vote
1 answer
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Numerical Differentiation Table

The following data was collected by measuring the distances in kilometres that a moving object travels over time (t) in seconds t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 s 0.0 9.0 20.0 34.0 48.0 64.0 80....
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Regularity for the wave equation on the half line

I would like to know the regularity of the wave equation under simple conditions. Consider the wave equation on the half line: $$ \begin{cases} u_{tt}-\triangle u = 0, \quad (x,t) \in (0, \infty]\...
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  • 199
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1 answer
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does $u(1/2,1/2)>v(1/2,1/2)$

given $u_t=u_{xx}+e^t\sin(x)$ $u(x,0)=x^2$ $u(0,t)=u(1,t)=t^2$ and $v_t=v_{xx}+e^t\sin(x)$ $v(x,0)=x$ $v(0,t)=v(1,t)=t$ Does $u(1/2,1/2)>v(1/2,1/2)$? My attempt: Define $w=u-v$ and the system of ...
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1 vote
0 answers
30 views

How to solve this partial differential equation (heat-diffusion equation)

I'm having trouble in solving a specific partial differential equation. It writes: $$ \dfrac{\partial p}{\partial t} = c \left( \dfrac{\partial^{2} p}{\partial x_{1}^{2}} + \cos^2\left(\theta\...
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0 answers
16 views

Finite Element Method of Diffusion equation

I'm trying to understand what my professor wrote, maybe it's a stupid question.. I lost him in the step from $1D$ diffusion equation: $\frac{d}{dx}(g\frac{du}{dx})=f(x)$ where $f$ is a given forcing ...
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0 answers
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confusion between finite difference methods and finite volume methods for PDEs

I am new to numerical methods for PDEs, but I am seeing some confusing perspectives in two different common textbooks: Langtangen's book on Finite Differences and Leveque's book on Finite Volume ...
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0 votes
1 answer
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stability of a numercial scheme for a hyperbolic system?

This is related to my question here Lax-Wendroff scheme stability analysis for a linear system of conservation laws , I hope it will reach more readers. Consider the numerical scheme given by the ...
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4 votes
1 answer
55 views

how to show an inequality from P.D.E

Let $\Omega$ a regular bounded open subset of $\mathbb{R}^N$. Let $T>0$, $u_0 \in L^2$, $b \in L^{\infty}(]0,T[\, \times\, \Omega)^N$, and $c \in L^{\infty}(]0,T[\, \times\, \Omega)$. We consider ...
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  • 2,850
0 votes
1 answer
25 views

Proper ways/strategies of observing and transforming ODE / PDE

I encountered with a lot of problem about ODE, many of which needs a proper observing or transforming before continuing, but to me, seeking the "key" out is very hard, for an observing ...
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  • 774
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0 answers
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How to find transmitted and reflected waves due to an interface of two media in 2D or 3D?

Consider a time-independent Schrödinger's equation (or an equivalent Helmholtz equation with variable speed coefficient) $$-\Delta\psi(\vec r)+V(\vec r)\psi(\vec r)=E\psi(\vec r),$$ with boundary ...
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  • 6,250
0 votes
0 answers
20 views

Clarification on solution method for system of partial differential equation

I have following function - $$ \max_{x, y} ~ u(x, y)^{3}x + (1-u(x, y))^{3}y$$ FOC: $$u_{x}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +u(x,y)^{3} = 0$$... (1) $$u_{y}(3u(x, y)^{2}x - 3(1-u(x, y))^{2}y) +(1-u(...
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2 votes
1 answer
33 views

solve the pde without any initial condition $xu_x-xyu_y=u$

Solve of the following PDE using method of characteristics: $$(1)\qquad xu_x-xyu_y=u \qquad u(x,x)=x^2 e^x$$ $$(2)\qquad xu_x-xyu_y=u, \forall x,y$$ I assume that $(x,y)$ are functions of a parameter, ...
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  • 491
0 votes
0 answers
19 views

A $C^2$ boundary implies Interior sphere condition

I have problem when I try to read evan’s PDE in chapter 6. That is a $C^2$ boundary implies the interior sphere condition. Could anyone give some details of this proof? I think $C^2$ is used when we ...
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1 vote
1 answer
40 views

Similar to energy estimate in PDE

Let $B_1(0)$ be the unit ball in $\mathbb R^3$ centered at the origin. Assume that the function $v$ is a smooth function defined on $\mathbb R^3$ with $v_r = \frac{x\cdot\nabla v}{|x|}\in L^2(B_1(0))$....
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0 answers
18 views

Liouville's equation $\Delta u=K e^{ u}$ when $K<0$.

I'm quite interested in the elliptic PDE like $\Delta u = K e^{u}$ when $K<0$, when $K>0$, it's very easy talk about the existence of the solution but when $K<0$ it seems that we lose some ...
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0 votes
1 answer
26 views

Symbol and principal symbol of differential operator

The symbol of a differential operator is defined as $L(x,p):=\sum_{|\alpha| \leq m}a_\alpha(x)p^\alpha$ and the principal symbol $L^p(x,p)$ is defined similar but with $|a|=m$. What would be the ...
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0 answers
11 views

Prove that a linear partial differential equation of first order can't be elliptic

A linear PDE of first order has the general form $$a(x,y)u_x+b(x,y)u_y-c(x,y)u-d(x,y)=0$$ A PDE is elliptic if the symmetric matrix of the coefficients of the highest derivatives has a determinant ...
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0 answers
25 views

Inequality $\theta\|{-\Delta u}\|_{L^2(U)}^2 \leq (Lu,-\Delta u),$ for elliptic operator

Let $U$ be the bounded smooth open subset of $\Bbb{R}^n$, with $u \in H^2 \cap H^1_0$. Let $L = \sum_{ij} (a_{ij}(x) u_{x^i})_{x^j} + \sum_k b_k(x) u_{x^k} + c(x) u$ be a general linear differential ...
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3 votes
0 answers
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Compact supports of initial data implies compact supports for all $t$ in semi-linear wave equation $-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$.

Problem: Let $u(t,x,y)$ be a smooth real function defined on $\mathbb R \times \mathbb R^2$ where $t \in\mathbb R$ and $(x,y) \in\mathbb R^2$. We assume that it solves the following semi-linear wave ...
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  • 774
0 votes
1 answer
30 views

Solve partial equation

$x z_x + y z_y =x+y+z$ Solve for z if $z(x,x+1)=2x+1$. I have used characteristic method to get $y=cx$. How to find the second solution to get the general solution?
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  • 93
0 votes
0 answers
10 views

Nonlinear first order PDE

Let $V$ is a function of $x$ and $y$ and $x,y:[0,t]\to\in\mathbb{R}$ are smooth. Now consider the nonlinear PDE$$(x^2-x^2(0))e^{2y}\frac{\partial V}{\partial x}+x(1-e^{2y})\frac{\partial V}{\partial y}...
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  • 61
4 votes
1 answer
39 views

Prove the uniqueness of $u\in H_0^1(\Omega)$ with $\Delta u=\vert u\vert^{q-1}u+f$ in $\Omega$ with $\Omega$ as a bounded domain with smooth boundary.

Problem: Let $\Omega\subset\mathbb R^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f \in L^p(\Omega)$, there exists a unique $u\in H_0^1(\...
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1 vote
0 answers
42 views
+50

Showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space

This question has been asked here showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space However I came across some problem when following ...
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0 votes
1 answer
42 views

Elliptic regularity for Poisson equation

Let $U \subset \Bbb{R}^n$ be an open bounded smooth domain, let $u \in H^1_0(U)$ satisfy the Poisson equation: $$-\Delta u = f \tag{*}$$ with $f \in L^{2}(U)$, by the classical elliptic regularity ...
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0 votes
0 answers
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Classification of differential equations as ODE and PDE

$1. \frac{df}{dx}+\frac{dg}{dy}=f+g$ This isn't ODE since two independent variables are involved. This should not be PDE (I think) as no partial derivatives are involved. So in which category this ...
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11 votes
0 answers
102 views
+200

Are these equations "properly" defined differential equations?

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it Intro Recently, in this answer I figure out that the following autonomous ...
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  • 761
0 votes
1 answer
22 views

Purely-imaginary Harmonic function on hyperbolic space is constant

Suppose I have a function $$ f: \mathbb H^3 \to \mathbb C $$ that is harmonic and whose image can be shown to be purely imaginary. Is it possible to deduce that it is in fact constant?
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  • 777
1 vote
1 answer
25 views

Sobolev inequality for cubes

Consider the fallowing result from Evans, 2010, page 279: Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then ...
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0 votes
0 answers
23 views

Modeling body temperature in a continuous framework

I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be ...
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0 votes
0 answers
25 views

Solving partial differential equation of more than 2 independent variables

If we have a function f(x,y,z) and have this equation $af_{x}+bf_{y}+cf_{z} = 0$ I have 2 questions :How do we transform the variables x,y,z to ξ,n,k to solve the differential equation? I mean $1)\...
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3 votes
1 answer
46 views

Spectrum of the operator on $L^2[0,1]$

Consider the operator T on $L^2[0,1]$, given by $T(f(x)) = \int_{1-x}^1 f(y)dy$. I want to find the spectrum of this operator. I know the only possible candidates are 0 and non-zero Eigen values of T, ...
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  • 399
2 votes
1 answer
34 views

Apply Gronwall's inequality to the following inequality

Assume the norms in the following inequality make sense, $U\subset \Bbb{R}^n$ open and $u: U\times [0,T] \to \Bbb{R}$ , with the relation holds: $$\frac{d}{d t}\left\|{u}\right\|_{L^{2}(U)}^{2}+\theta\...
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-1 votes
0 answers
16 views

Solving non linear partial differential equation

Can we solve non linear partial differential equations algebraically?If yes how?What transformations do we have to make to the original equation?
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1 vote
1 answer
55 views

Solutions of PDE

I am looking for solutions $g \colon \mathbb R^2 \times \mathbb R^2 \to \mathbb C$ of the PDE $$\partial_{x_1} g(x,y) + \partial_{y_1} g(x,y) = a(y_1+x_1+i(x_2-y_2))g(x,y), \\ \partial_{x_2} g(x,y) + \...
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-1 votes
1 answer
18 views

Find $U(x,y)$ given boundary conditions

I have this equation: with a and b not equal to 0. If we solve the partial differential equation: Now I am given the boundary conditions $U(x,0) = x^2$ .How do I continue?
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0 answers
14 views

Does the von Neumann equation hold for continuous functions of the density matrix?

Let $\rho$ be a density operator obeying $$i\hbar \partial_t \rho = [H,\rho]$$ where $H$ is a time-dependent Hamiltonian and $\rho(x,0)=\rho_I(x)$. The evolution is given by $$\rho(t) = U(t) \rho_I U(...
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