# Tagged Questions

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### Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H$ is singular when $x=\alpha$ and/or $x=\beta$, hence my question: ...
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### Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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### Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
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### existence and uniqueness of volterra integral equation of the first kind

$$\int_0^t k(s,t)f(s)ds=g(t)$$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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### Solution to Schrödinger equation $\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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### System of ODEs and DAE system

Let us consider the following system of ODEs: $$y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0$$ and the following one: $$y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0.$$ $f$ and $g$ ...
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### Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$\partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
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### 2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
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### Solving $u_{\xi\eta} = 0$ and differentiability conditions on solutions

After transformation someone often encounters the PDE $$u_{\xi\eta} = 0$$ but I am quite confused about the differentiability conditions of its solution (for example in this post I read different ...
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### How to solve $e^yu_x-u_y+u=xe^y$?

Solve $e^yu_x-u_y+u=xe^y$ $u_x-\frac{u_y}{e^y}=x-\frac{u}{e^y}$ $\frac{b}{a}=\frac{dy}{dx}=-\frac{1}{e^y}$ $e^y=-x+c$, $\eta=e^y+x,\xi=x$ $e^y=\eta-\xi$ ...
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### Duhamels principle

I have a problem with the following exercise (I know how to proof for $u_{tt}$ but for $u_t$ I met a problem, for $u_{tt}$ proof works as we can assume that $v(x,t,t)=0$ and we add it artificially to ...
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### Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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### The analytical solution for advection-diffusion equation with source term.

We have: $$\frac{\partial w}{\partial t} + a(x) \frac{\partial w}{\partial x} - v \frac{\partial^2 w}{\partial x^2} = f(t)$$ within a domain $x \in [0,1]$ Simplest Sample is $a(x) = 1$ (constant) and ...
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### Maximum principle in PDE [duplicate]

I was told Maximum principle is a common method in proving uniqueness of the solution to certain PDE. Could anyone explain 1) How does Maximum principle work in the context of PDE theory? 2) How ...
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### Prove a differential equation:

The partial differential equation $$\frac{d^2u}{dt^2}=c^2\left(\frac{d^2u}{dx^2}+\frac{d^2u}{dy^2}+\frac{d^2u}{dz^2}\right) \tag{*}$$ is the three dimensional wave equation. In the case of ...
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### Partial derivatives-Why does this stand?

In my notes there is the following: $$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix} u_{\xi}=0 \Rightarrow u=g(\eta)\\ u_{\eta}=0 \Rightarrow u=f(\xi) \end{matrix}\right.$$ I haven't understood ...
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### Explaining the one-dimensional continuity equation with respect to density evolution

I've got a rather abstract question So the continuity equation for a one-dimensional continuum is: $$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho v)=0$$ and we can expand ...
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I think this question should be easy and shouldn't require me to solve the entire PDE for a general solution. Basically, how would you see immediately that the general solution of: $$y^2 u_{xx}-2xy ... 2answers 43 views ### Questions about the Laplace's equation in polar coordinates The Laplace's equation in polar coordinates at a cyclic disk:$$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}, \ \ \ 0 \leq r \leq a, \ \ \ 0 \leq \theta \leq 2 \piu(a,\theta)=h(\theta), \ ...
I'm trying to solve a 1D time-dependent Schrodinger equation: $$i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t)$$ where $V(x)$ ...
can you explain that last two steps? how that $$na(n)r^{n-1}$$ disappeared in next integral?I noticed the transformation of variables but still not able to figure out properly.