Skip to main content

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.

Filter by
Sorted by
Tagged with
2 votes
3 answers
63 views

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 ...
0 votes
0 answers
14 views

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 ...
0 votes
1 answer
32 views

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
1 answer
42 views

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 ...
0 votes
0 answers
19 views

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 ...
0 votes
1 answer
30 views

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 ...
0 votes
1 answer
37 views

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 ...
4 votes
0 answers
78 views

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 ...
1 vote
2 answers
47 views

Prove that $ \int_{\Omega} \Vert \nabla u\Vert^2 -6F(u)dx=-\int_{\partial \Omega} \Vert \nabla u\Vert^2 x\cdot \vec{n}d\sigma$

Let $\Omega\in \mathbb{R}^3$ be a compact region, $f \in C(\mathbb{R})$ and $F(r)=\int_{0}^{r}f(t) dt$. If $u\in C^2(\Omega)$ satisfies $\Delta u(x)+f(u(x))=0,x\in \Omega$ and $u(x)=0,x\in \partial \...
0 votes
0 answers
22 views

Growth rate of perturbations in linear stability analysis (expantion in powers of wavevector) [closed]

I have done a linear stability analysis for a system of coupled PDEs. The growth rate of perturbations, $f(\lambda)$, satisfies an equation $f(\lambda)=0$. Now I want to find the leading order terms ...
0 votes
0 answers
55 views

Method of Characteristics for this PDE and what's the problem with its domain?

$\dfrac{\partial u}{\partial x}y -\dfrac{\partial u}{\partial y}x=1$ with $u(x,0)=0$ and $x\geq0$ By using the method of characteristics, I get $u=t$ and $x^2+y^2=x_0^2$. The answer is $u=-\arctan(y/x)...
0 votes
0 answers
42 views
+50

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 $\phi_{xx}$. ...
0 votes
1 answer
78 views

$\frac{\partial \ ^2 u}{\partial x^2}$+$\frac{\partial \ ^2 u}{\partial x \partial y}$+ $\frac{\partial \ ^2 u}{\partial y^2}$=0 [closed]

I am new to PDE and this question can be solved by separation of variables. $\text Attempt$: Assume $u(x,y)= X(x)Y(y) $, then I get $\frac{X''(x)}{X(x)}+\frac{X'(x)Y'(y)}{X(x)Y(y)}+\frac{Y''(y)}{Y(y)}=...
1 vote
3 answers
126 views

Laplace's equation in Polar coordinates on the semi-annular domain

Solve the following boundary value problems for the Laplace equation on the semi- annular domain $D = { 1 < x^2 + y^2 < 2,\; y > 0 }:$ $ u(x, y) = 0, x^2 + y^2 = 1\; \text{or}\; 2, u(x, 0) = ...
0 votes
0 answers
39 views

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 ...
1 vote
0 answers
35 views

My solution to the fourier transform of multi-dimensional wave equation

Our definition of Fourier transform:$\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x) e^{-i x \cdot \xi} d x$ now consider the wave operator $$ \square:=-\partial_t^2+\partial_{x^1}^2+\cdots+\partial_{x^d}^2=-\...
0 votes
0 answers
53 views

Viscosity solution $u$ and the equation of $e^t u$

The function $u_0$ is called the viscosity subsolution of the equation $$\partial_t u+\Delta u=0$$ with $u(x,0)=u^*(x)$ if for any $(x_0,t_0)\in\mathbb{R}^d\times[0,T]$ and any test function $\varphi \...
-1 votes
0 answers
29 views

Fundamental theorem of space curve_Existence Theorem [closed]

I was trying to learn the proof of the Existence Theorem for space curves. Can someone give me a specific example of two different curves and tell me how to prove the existence theorem for them curves....
0 votes
1 answer
31 views

Solving a heat equation, with puzzling passages.

I have the heat equation $$ u_t=\frac{1}{2}u_{xx},\quad x\in \mathbb{R}\\ u(0,x)=x^2 $$ I tried to solve it with these passages: $$u(t,x)=\int_{-\infty}^{+\infty} G(t,x-y) \phi(y) dy$$ $$u(t,x)=\frac{...
0 votes
0 answers
24 views

Evans Chapter 8 variational method [closed]

Consider the functional $$ I[u] = \int_U \left( m(x) + \nabla u \cdot A(x) \nabla u \right)^{\frac{\alpha}{2}} dx $$ where $U$ is an open bounded domain in $\mathbb{R}^n$, $m$ and $A$ smooth on $\...
0 votes
0 answers
16 views

Kernel of a linear differential operator in the space of polynomials

Consider the differential operator F = p * D_x + q * D_y, where p, q are polynomials in Q[x,y], Q - the field of rational numbers, D_x, D_y are partial derivatives on polynomials by x and y ...
2 votes
0 answers
29 views

Using a cutoff to show an interior gradient estimate for a harmonic function

I am trying to follow a calculation shown in Chapter 1.4 of Han and Lin's Elliptic PDE textbook. Let $u$ be a harmonic function on the unit ball $B_1$ in $\mathbb{R}^n$, and let $\eta \in C^\infty_c (...
2 votes
1 answer
39 views

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
1 answer
60 views

Studying stability of pde

I have a problem with studying the stability of this PDE. $$U_t = U_{xx} + f(U).$$ Let $U^{*}(x)$ be a solution for this equation. As conditions we have ${U^{*}}'(x) > 0$ for $x < x_{0}$, ${U^{*}...
2 votes
0 answers
37 views

Solving a Fokker-Planck equation with discontinuous drift coefficients

Recently I'm trying to solve a Fokker-Planck equation corresponding to a piecewise-linear diffusion process with discontinuous coefficients. Namely, I'm dealing with the equation \begin{equation}\...
4 votes
1 answer
1k views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
0 votes
0 answers
13 views

Energy estimate for the harmonic map heat flow

I am working through Struwes paper "On the evolution of harmonic maps in higher dimensions" (https://projecteuclid.org/journals/journal-of-differential-geometry/volume-28/issue-3/On-the-...
0 votes
1 answer
27 views

Elliptic problem of Euler Lagrangian form

Let $f, g \in C_c^\infty(\mathbb{R})$ be given, $c$ a positive constant, and suppose that $w$ solves the following initial value problem for the wave equation on $\mathbb{R}$: \begin{cases} c^{-2} w_{...
2 votes
0 answers
41 views

Asymptotic behavior of the solution of homogeneous heat equation with Neumann boundary condition.

For $$ \begin{cases}u_t=\Delta u & \text { in } \Omega \times(0, \infty), \\ \frac{\partial u}{\partial \nu}=0 & \text { on } \partial \Omega \times(0, \infty), \\ u(x, 0)=\phi(x), & x \in ...
0 votes
0 answers
32 views

How to obtain a PDE from a solution

I am trying to do something similar to this question but for a PDE. Specifically, I have these some equations and also know the answer as it's from a paper but cannot understand it or redo it. ...
0 votes
0 answers
21 views

Numerically solving the Advection-diffusion equation with no-flux boundary condition results in violation of mass conservation

I am trying to solve numerically the advection-diffusion equation of the following form $$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$ ...
4 votes
0 answers
135 views
+200

The uniqueness of the damped Sine-Gordon.

The damped Sine-Gordon equation given : \begin{equation} \partial_{tt} u + \alpha \partial_t u - \Delta u + \beta \sin(u) = 0 \end{equation} for an unknown $u : D \times [0, \infty) \rightarrow \...
0 votes
0 answers
22 views

deriving differential equation from difference of PDE solutions

Consider defining $$f(x,t)=g(x,t)-h(x,t)$$ where the PDEs describing $g(x,t)$ and $h(x,t)$ are $\it{known}$ but the solutions themselves are $\it{unknown}$. It could be impractical to solve the PDEs ...
1 vote
1 answer
962 views

PDE subscript notation

What does this notation mean in a PDE: $(u_xu_t)_{xx}$? For example, I have checked that $(u_xu_t)_x=u_{xx}u_{xt}$ so the product rule does not apply. However, it applies in the former case. What's ...
1 vote
0 answers
23 views

Finite difference scheme for non-standart PDE (bicylindrical coords)

Mathematical formulation of the problem: There is the Aifantis equation for the elasticity gradient $$ \eta =\lambda \left( \operatorname{tr}\ \varepsilon \right)l+2G\varepsilon -c{{\nabla }^{2}}\...
0 votes
0 answers
37 views

Error in the energy norm for inhomogeneous Dirichlet boundary conditions

Short description When conducting FEM analysis with inhomogeneous Dirichlet boundary conditions, I compute the error in the energy norm with an expression that should only work for problems with ...
0 votes
0 answers
38 views

Weak solutions of heat equation in Brezis' book

If every classical/strong solution to a PDE with boundary values condition is a also a weak solution, then why does Brezis bother with proving the existence of a weak solution to the heat eqution ...
0 votes
1 answer
19 views

Solution of the parabolic PDE using Green's function

Green's function for the parabolic PDE is defined as: $$\Delta G(\vec{x},t,\vec{\xi},\theta)=\delta(\vec{x}-\vec{\xi},t-\theta)$$ Where $G$ satifies the homogeneous initial and boundary conditions. ...
1 vote
3 answers
1k views

Proof of Nagumo's Theorem of invariance

I didn't find the proof of the following invariance Theorem due to Nagumo (1942). The statement of the Theorem is as next:` Given a continuous system $x'(t)= p(x(t))$, $t\geq 0$ on $\mathbb{R}^n$ ...
0 votes
0 answers
27 views

Heat Equation : Can a singularity develop away from origin?

Consider a radial, nonlinear, 2D-heat equation \begin{align*} u_t &= u_{rr} + \frac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\ u(0,r) &= f(r) \in C^{\infty}([0,1]), \...
0 votes
1 answer
33 views

Troubles with solving a Laplace equation

I'm struggling in solving an exercise about the Laplace equation over the domain $[\frac{\pi}{2}, \pi] \times [0, \pi]$ with the boundary conditions: $f(\frac{\pi}{2},y)=f(\pi,y)=f(x,\pi)=0$ and $f(x,...
2 votes
1 answer
140 views

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 ...
0 votes
0 answers
41 views

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
1 answer
4k views

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(\...
3 votes
0 answers
274 views
+50

Can Every Higher Order PDE be Written as a System of 1st Order PDEs?

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)} \...
1 vote
0 answers
17 views

Proving the right eigenvectors show in the direction of the characteristic lines

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 ...
1 vote
0 answers
33 views

Existence of Fokker-Planck equation under Cauchy boundary condition.

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 ...

1
2 3 4 5
469