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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19 views

Multivariable Calculus, Laplacian problem

$u$ is a function of $x,y,z,$ and $t,$ but the Laplacian Equation does't involve $\frac{\partial^2 u}{\partial t^2}.$ Why is that? One explanation I have seen before is that the gradient is ...
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4 views

Numerical solution to the kinetic Fokker-Planck equation

I want to find a numerical solution for the Kinetic Fokker-Planck equation which reads $f_t + vf_x = f_{vv}$ where $x \in [0,1], v\in \mathbb{R}^n$ with the specular boundary condition. I'm kind of ...
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General solution of a wave-like PDE

Suppose we have the following PDE: $$u(x, t) = F(4 x - t) + G(2 x - t),$$ where $F$ and $G$ are arbitrary twice differentiable functions. Let $x \in (- \infty, + \infty)$ and $t > 0$ with ICs $$...
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18 views

A trace embedding for fractional Sobolev space

I was reading a paper. There I found for a Lipschitz domain $\Omega \subset \mathbb{R}^n, n\geq 2$, the following holds: there is a continuous trace operator from $H^\beta (\Omega)$ to $L^2 (\partial \...
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15 views

Eigenvalue problem on the real line

The following is a problem in a text (in Portuguese) on Critical Point Theory that I am reading: Find the eigenvalues and eigenfunctions of the problem $$ (P) \quad \begin{cases} - y'' = \lambda ...
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Definition of PDE

Definition of equation in partial derivatives (PDE): If the equation contains partial derivatives of one or more dependent variables, then the equation is called the partial derivative equation (PDE). ...
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Does the extension of an element of $W^k$ by $0$ still lie in $W^k$?

Let $U\subset\mathbb{R}^n$ be an open set and $Z$ a closed subset of $U$ . We denote by $W^{k}(U)$ the Sobolev space of functions whose derivatives (in the sense of distribution theory) up to order $k$...
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Non-linear differential equations and theory of stability

Determine the stability properties of the following solutions: a. \begin{equation} \left\{\begin{matrix} \dot{x} = -y(x^{2} + y^{2})^{-1/2} \\\ \dot{y} = x(x^{2} + y^{2})^{-1/2} \end{...
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PDE-Poisson Equation $u_{xx}+u_{yy}=-1$??

I want to solve the partial differential equation $$u_{xx}+u_{yy}=-1$$ in the region $0<x<1, y>0$ subject to the boundary conditions $u(0,y)=0, u(1,y)=1$ and $u(x,0)=0$.
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How to determine the Green's function for wave equation

$$\begin{align} u_{tt}&=c^2u_{xx}+ Q(x,t), \quad x>0 \\ \\ u(x,0)&=f(x)\\ u_t(x,0)&=g(x)\\ u(0,t) &= h(t) \end{align}$$ Question: How should you determine the Green's function?
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Weak form of Fokker-Planck PDE

I'm trying to derive the weak form for the following Fokker-Planck equation to use in a finite element package. I have $$\frac{\partial P}{\partial L}(L,\eta) = \frac{1}{\theta}\bigg(2\eta\frac{\...
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Finite Difference: How to handle a square wave boundary condition for the diffusion equation?

A semi-infinite rod (L ~ 10 m) with one end at $u(0,t) = f(t)$ and other at $u(L,t) = 0$ has an initial condition given by $u(x,0) = f(t_0)e^{-0.8x} $. $f(t)$ is a square wave with period $2\pi$ and ...
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Interpretation of a Parabolic Partial Differential Equation

Let $\Omega\subset \mathbb{R}^{d}$ ($d\geq 1$) be a bounded domain with a smooth boundary $\partial\Omega$. Let $S, I$ be dependent variables and $x, t$ their independent variables. Additionally, $q:\...
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How do I solve the distributional equation T.x =1?

I am having a bit of trouble solving distributional equations. An example that I am currently working on is to show that the distributional equation $T.x = 1$ has a solution if and only if $T=p.v.(\...
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Is my interpretation of this proof of a maximum principle for the discrete heat equation correct?

I am looking for help on this proof of a maximum principle for the discrete heat equation. The following is from Introduction to Partial Differential Equations (Tveito, Winther). Consider the Heat ...
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how to solve the differential equation $y''-3(t-1)y'+2y=24t-30$? [closed]

center t = 1 solve by power series $y''-3(t-1)y'+2y=24t-30$
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22 views

Show that the full Fourier series of F converge pointwise to F.

Let f be a piecewise continuous function on $[-\pi,\pi]$,with $\int_{- \pi}^{\pi} f(x) dx=0$. Define $F(x)=\int_{-\pi}^{x}f(y)dy$. Show that the full Fourier series of F converge pointwise to F. ...
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26 views

Find a function satisfy certain condition

Find function $f:\mathbb{R}\to\mathbb{R}$ such that $f(2)=2$ and $$\sum_{i=1}^nc_i\frac{\partial(x_1^2+\dots+x_n^2)^{\frac{f(y)}{2}}}{\partial x_i}=\frac{\partial(x_1^2+\dots+x^2_n)^{\frac{f(y)}{2}}}{...
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Solve a differential equation with Chebyshev polynomials

We have a linear operator $L$ which assigns a function $f \in C_{0}^{2} \cap L_{w}^{2}[-1,1]$ a function on $L_{w}^{2}[-1,1]$ space by the following relation $$Lf(x) = (1-x^2) f''(x) - x f'(x)$$ ...
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Partial differential equations, separation of variables [closed]

The vibrations $u(x, t)$ in an organ pipe of length L satisfy the wave equation $∂²u/∂t² = c² (∂²u/∂x²)$ where $c$ is the speed of sound, and are subject to the following conditions: (i) the end of ...
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1answer
17 views

Deriving (using Fourier transform) the Poisson kernel for solving the Dirichlet problem on unit balls

Let's first consider the following Dirichlet problem on the upper half-space $\mathbb H^n=\{(x_1,\ldots, x_n)\in \mathbb R^n:x_n>0\}$. $$ \Delta u =0, u|_{x_n=0}=g(x). $$ Performing Fourier ...
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MCQ on Cauchy problem

$y u_x-xu_y=0,u=g $ on $ \Omega $ has a unique solution in neighborhood of $\Omega$ for every differentiable function g: $\Omega \rightarrow R$ if 1.$\Omega =\{(x,0):x>0\}$ 2.$\Omega =\{(x,y):x^...
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1answer
27 views

Deriving a heat equation solution in a form of power series

I have read in a book Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations that it is possible to formally derive a solution to heat equation in a form of power series....
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Partial Differential Equation, finding integral surfaces.

Find the general solution of the differential equation $$x(z + 2a)p + (xz + 2yz + 2ay)q = z(z + a)$$ where, $p = \frac{\partial z}{\partial x}$, $q = \frac{\partial z}{\partial y}$ Find also the ...
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49 views

PDE organ pipe question

I kind of got stuck in the end. By separation of variables and tons of substituting value I ended up with $$u(x,t) =\sum_{n=1}^\infty D_n\sin(-vt/D_n)\sin(-vx/D_n)$$ where $-v/D_n = c(n-\frac12)\pi/L$....
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What is a duality argument?

I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these ...
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52 views

Solving the time-independent Shrödinger equation

The time-independent Shrödinger equation is $$\dfrac{-\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi.$$ I am then told that the general solution for a uniform potential can be written as the sum of ...
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When is the region between two Lipschitz graphs a Lipschitz domain?

For $f,g : \mathbb{R}^d \to \mathbb{R}$ Lipschitz with $L_1$ and $L_2$ norms, let $\Omega= \{ (a,b) \in \mathbb{R^{n+1}} : f(a) \leq b \leq g(a) \}$. I was wondering under which (hopefully mild) ...
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Implementing Neumann conditions in PDE discretized in space w/ RK4 to solve resulting ODE system?

I'm numerically solving a 1D diffusion equation in $\sigma(x,t)$ with a nonlinear diffusion coefficient $C(\sigma)$, with Neumann conditions at the boundaries and a zero initial condition. In ...
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7 views

Uniform convexity and embedding result in the Heisenberg group

Can somebody please help me with compact embedding result and uniform convexity property of the Sobolev space $W^{1,p}(\Omega)$ where $\Omega$ is a bounded smooth domain in the Heisenberg group? ...
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25 views

Systems of first order PDE [duplicate]

Given \begin{aligned} \frac{\partial u}{\partial x_1} = F_1(x_1,x_2), \\ \frac{\partial u}{\partial x_2} = F_2(x_1,x_2), \end{aligned} Please, how can I go about solving this PDE. I tried separating ...
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Plot $u(x,t)$ for a string of length $10$

Consider a taut string of length $10$ with wave speed $c = 1$ (in suitable units), with a fixed end at $x = −5$ and a free end at $x = 5$. The deflection of the string is denoted $u(t, x)$ for $−5 ...
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1answer
58 views

Method of characteristics for quasilinear PDE $u_x+u_y=2\sqrt{u}$

I'm having trouble with solving the quasilinear PDE $$\begin{cases} u_x+u_y&=2\sqrt{u}, \\ u(x,x)&=g(x) \end{cases}$$ via method of characteristics as in this paper. My attempt: First I ...
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29 views

Is it possible to get an accurate solution for this PDE?

I have a complicated multi-dimensional Fokker-Planck PDE of the form $$\frac{\partial P}{\partial L}(L,g,h,p) = \frac{p^2}{2}\frac{\partial^2 P}{\partial g^2}(L,g,h,p) + pg\frac{\partial^2 P}{\...
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Reduce the given differential equation into its canonical form

Reduce the equation $x^2r – 2xys + y^2t – xp + 3yq = \frac{8y}{x}$ to canonical form.
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Prove that the function $u= \|x\|{_{2}^{2-n}} $ has $\Delta u = 0 $?

I have been trying to solve this for hours but without success. I tried to write down the definitions which means that our function can also be written as $$u= \frac{\sum_{j=1}^{n} |x_{j}|^{2}}{\sqrt{...
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19 views

Consistency and Monotonicity of the flux [closed]

Consider a conservation law with a strictly convex flux $f$ and the following corresponding numerical fluxes: (i) the Rusanov flux, (ii) the Godunov flux. I need to show that both the numerical ...
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1answer
39 views

Deriving the time-dependent solution of the Schrödinger equation

I have the Schrödinger equation: $$\dfrac{-\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \dfrac{\partial{\Psi}}{\partial{t}},$$ where $m$ is the particle's mass, $V$ is the potential energy operator,...
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1answer
28 views

Solving PDE $u_{t} = (1-s)\mu u_{s} + (s-1)\alpha u$ with method of characteristics.

I am trying to solve the following PDE using the method of characteristics, $$u_{t} = (1-s)\mu u_{s} + (s-1)\alpha u,$$ $$u(s,0) = s^i$$ We can reduce the PDE to the following set of ODEs along the ...
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24 views

Euler-Lagrange equations in 2D with line-wise constraint

I need to tackle a variational optimization problem in two variables with an integral-constraint with respect to one variable and a point-wise constraint with respect to the other variable. One may ...
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11 views

Proof of uniqueness of solution of the Poisson's equation for given boundary conditions

I would like to show that the Poisson's equation, i.e., $\nabla^2 \Phi = \rho$, has a unique solution for given boundary conditions, namely, Dirichlet and Neumann boundary conditions. To this end, ...
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18 views

Evaluate PDE for function involving u(x,y) [closed]

$V\frac {\partial u}{\partial x} = \frac {\partial ^2 u}{\partial y^2}$. Where $u$ is a function of $x,y$. I've completely forgotten how to solve PDE's of this type!
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Where to find reference for the energy method $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm}$

Why do I need to multiply by the function w in the energy method to guaranty at most one solution? This is the example $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm} u=0 \hspace{0....
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25 views

What is the exact definition of $W^{3/2 , 2}$?

I am self-studying some PDE, and I met the following sentence. "the space $W^{3/2 , 2} (U)$ is the interpolation space $[W^{1,2} , W^{2,2}]_{1/2}$..." I am familiar with the definition of (...
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15 views

sign of time partial derivative at a global max

I've encountered this concept in my PDE class but I believe my issue relates to some fundamental misunderstanding of multivariable calculus that I probably should have understood years ago. Suppose I ...
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11 views

How can I tell if a simualtion of a spherical wave equation implements damping?

In this link you can see the following simulation of a spherical wave: The MATLAB code is provided here. ...
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26 views

Solutions for $\frac{\partial^2 u}{\partial t^2}=-\frac{1}{\hbar^2}\left(\frac{\hbar^2}{2m}\Delta(\cdot)-V(\vec{x})\cdot\right)^2u$ [closed]

If $A(\vec{x},t)+iB(\vec{x},t)$ is a solution for $$\frac{\partial^2 u}{\partial t^2}=-\frac{1}{\hbar^2}\left(\frac{\hbar^2}{2m}\Delta(\cdot)-V(\vec{x})\cdot\right)^2u$$ Is corret that $A(\vec{x},t)$...
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1answer
29 views

Why is the non-linear wave equation $u_{tt} = \operatorname{div}(a(Du))$ quasi-linear?

I came across the PDE \begin{align*} u_{tt} - \operatorname{div}(a(Du)) = 0 \end{align*} where $a:\mathbb{R}^n \rightarrow \mathbb{R}^n$, $Du$ is the gradient of the unknown $u$ and $\operatorname{...

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