# 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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### 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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### 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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### partial differential equations (generalized functions) [closed]

Find all solutions in the $D'$ equation $x^3y''-2y=x$
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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, ...
$V\frac {\partial u}{\partial x} = \frac {\partial ^2 u}{\partial y^2}$. Where $u$ is a function of $x,y$. I've completely forgotten how to solve PDE's of this type!