# 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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### How to solve this 2D Laplace equation $-\Delta u=0$ with $u(x,y)=y^{2}$ on boundary

$$\mbox{I want to solve}\quad\Delta u = 0\quad\mbox{with}\quad u(x,y) = y^{2}$$ on boundary: The region is a $2D$ ball centered at $0$ with radius $1$. I want to write it in polar coordinates but it ...
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### How to prove the maximum principle about the first eigenvalue?

Let $\Omega\subset\mathbb{R}^{n}$ is a bounded domain. Let $u\in C^{2}(\Omega)\cap C(\bar{\Omega})$ satisfy $$-\Delta u+c(x)u\leqslant 0\ \ \text{in}\ \ \Omega.$$ We can assume $c(x)$ is sufficiently ...
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### Solution of linear dynamical systems without Fourier transforming.

I'm trying to understand an exercise for my exam of mathematical methods for physics, in which it is asked to find the general expression of the Green function of the following system without using ...
1 vote
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### Is there always a sequential nature of time-stepping? [closed]

Pretend someone is solving a partial differential equation using something like the finite element method. Lets say they are calculating the propagation of a seismic wave through inhomogeneous terrain ...
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### Could you please recommend some solutions manual about PDE?

the table of contents of the textbook is given in the link, and I am searching for some exercises set for these contents, but I can't find any one that is suitable, could you please recommend some ...
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### Search for a general solution for a hyperbolic equation with a boundary condition on a sphere

I need help solving an interesting problem on partial differential equations. I got stuck with the problem of finding a general solution to the equation $$u_{xy} + u_{xz} + u_{yz} = 0$$ with a ...
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### How to use Fourier method to solve the Poisson equation $-\Delta v=x y$?

Suppose that $B_{1}$ is a ball centered at $0$ with radius $1$, consider the equation $-\Delta v=x y$ and $v=0$ on boundary, I want to use Fourier method to solve it, but the Fourier transform is ...
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### Does $i\partial_t u = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?

Consider the initial-value problems in $d=1$ $$\begin{cases} i\partial_tu = \Delta^2 u \\ u(x,0)=u_0 \end{cases}$$ and $$\begin{cases} i\partial_t u= \Delta u \\ u(x,0)=u_0, \end{cases}$$ Solutions to ...
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### Initial and boundary conditions for parabolic PDEs

Consider a parabolic PDE of the form \begin{align} \phi_t=f\left(\phi,\phi_x,\phi_{xx} \right), \end{align} with $f$ some reasonable function, and if needed for the argument below linear $\phi_{xx}$. ...
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### Solving Function Optimization and Functional Equations

I am trying to solve the following mathematical problem. The goal is to compute the functions $p(b), s(v)$, and the probability density function $h_B(b)$ (or the corresponding cumulative distribution ...
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### Good PDE textbooks that contains the usage of distribution theory to study wave equations

I have learned some basic PDE theories, and now I want to learn further properties of wave equations by using distribution theory, which contains fourier transform, convolution of distribution and so ...
1 vote
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### Fundamental solution of Poisson equation on the torus

The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. this MathSE post): \begin{align} &\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y) \quad \Rightarrow ...
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### Why $u_{xt}$ differs from $u_{tx}$ when transforming $x$ and $t$?

I am trying to solve a linear second order PDE with non-constant coefficients. To this end, I am taking a transformation in the independent variables $T: (x,t)\rightarrow(\xi,\eta)$, which I'll denote ...
1 vote
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### Classification of this nonlinear PDE into elliptic, hyperbolic, etc.

I wanted to know how one would classify a nonlinear PDE into elliptic, hyperbolic or parabolic forms. The particular PDE I would like to know about would be \begin{align} \partial_t u &= D(\...
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Motivation A section on the Wikipedia page (here) of ordinary differential equations states the following. Reduction to a 1st Order System Any differential equation of order $n$ $$F\left(x,y,y',y'',\... 3 votes 0 answers 52 views ### Details in Zuazua's paper "Exponential decay for the semilinear wave equation with locally distributed damping" CPDE 1990 When I read Zuazua's paper "Exponential decay for the semilinear wave equation with locally distributed damping" CPDE 1990, I had a problem with inequality estimation. That is (1.10)\|\psi\... 2 votes 1 answer 44 views ### Density of similiar Sobolev space Consider the space of functions defined as,$$ D = \{f \in L^p(0,\infty): f \in AC_{loc}(0,\infty) \text{ and } xf'(x) \in L^p(0,\infty)\}, $$where AC_{loc}(0,\infty) is the set of locally ... 0 votes 0 answers 26 views ### Central Difference for Fractional Derivative Consider the following approximation: For 0\leq \alpha \leq 1,~k=t_n-t_{n-1},$$ \begin{aligned} \frac{\partial^\alpha V\left(S_l, t_n\right)}{\partial t^\alpha} & =\frac{1}{\Gamma(1-\alpha)} \...
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Given a system of two linear homogeneous first-order hyperbolic PDE's, $n_{\theta} + An_{\xi} = 0,$ I need to show the right eigenvectors of $A$ show in the direction of the characteristic lines. The ...
A 1D Fokker-Planck equation within a constrain region is uniquely characterized by three functions and a boundary conditon: A drifting term $\mu(x,t)$. A diffusion term $\sigma(x,t)$. An initial ...