0
votes
0answers
29 views

Method of Characteristics for a PDE

I'm working through a problem at the moment, and I've got an answer, but it seems far too complicated... I've been asked to use the method of characteristics to solve the following PDE; $$x^2 u_x ...
1
vote
0answers
20 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
0
votes
0answers
40 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
0
votes
2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
0
votes
1answer
24 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
0
votes
0answers
39 views

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: ...
0
votes
0answers
17 views

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 ...
1
vote
1answer
21 views

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 ...
1
vote
0answers
24 views

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. ...
1
vote
1answer
63 views

Solving a master equation with linear coefficients

I have the following PDE: $$ \partial_t P(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t). $$ Mathematica suggests that the solution is $$ ...
0
votes
0answers
13 views

obtain the continuous form of a master equation

I have the following master equation: $$ \partial_tP(x,y,t)=-[x+(1-x-y)]P(x,y,t)\\ +(x+\epsilon)P(x+\epsilon,y-\epsilon,t)\\ +(1-x-y+\epsilon)P(x,y-\epsilon,t) $$ where $\epsilon$ is a small number, ...
2
votes
2answers
57 views

Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...
1
vote
1answer
36 views

Solution to $u_t+\Delta^2u+\Delta u=0$

Suppose there exists a solution to *$u_t+\Delta^2u+\Delta u=0$ of the form $u(x,y,t)=c(t)e^{i\pi(x/4\pi+y/4\pi)}$. I need to find such a function $c(t)$. Plugging $u(x,y,t)$ into *, I got ...
0
votes
2answers
26 views

Finding polynomial satistying potential equation and boundary conditions

Can someone help me with this problem? I know that this polynomial is a solution of Poisson's equation.
1
vote
3answers
47 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
0
votes
0answers
43 views

Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
0
votes
0answers
25 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
2
votes
0answers
97 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
0
votes
1answer
19 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
0
votes
1answer
25 views

Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...
0
votes
0answers
22 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
4
votes
2answers
76 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
1
vote
0answers
46 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
1
vote
1answer
81 views

linearize a nonlinear ode

Could anyone suggest me how to linearize the following system of nonlinear odes (special attention to (2) \begin{align} -cU'&=-U''+UV\tag{1}\\ -cV'&=-k(k+1)V^{k-1}(V')^2+(k+1)V^k ...
1
vote
1answer
116 views

Existence and uniqueness for a system of first-order PDE

Let $y$ be a scalar and ${\bf t}=(t_1,\ldots,t_K)$ and \begin{align*} \frac{\partial y({\bf t})}{\partial t_k} &=f_k(y({\bf t}),{\bf t}) \qquad k=1,\ldots,K\\ y(t_{10},\ldots,t_{K0}) &=y_0 ...
3
votes
2answers
36 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
2
votes
2answers
72 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
1
vote
1answer
27 views

What is the difference between bounded boundary and a bounded domain?

This I found while studying sobolev spaces. In some results they have assumed that $\Omega\subseteq \mathbb{R}^n$ is with bounded boundary. Somebody told me the difference between the set with bounded ...
0
votes
1answer
55 views

Help needed in solving a differential equation

Please help me in solving: $$a^2z+\frac{\partial^2z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=0$$ ($a$ is a constant) I plugged this in Wolfram Alpha and it outputs that this is a second ...
1
vote
0answers
37 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
3
votes
0answers
75 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
2
votes
0answers
69 views

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 ...
1
vote
0answers
21 views

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 $g$ ...
5
votes
0answers
89 views

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 ...
2
votes
3answers
57 views

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 ...
1
vote
0answers
28 views

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 ...
2
votes
3answers
132 views

Computing the spherical mean and showing it satisfies PDE

Compute the spherical mean of the function $h : \mathbb R^3 \to \mathbb R$ with $$ h(x,y,z) = x $$ and show that it satisfies the differential equation $$ u_{rr} + \frac{2}{r} u_r = u_{xx} + u_{yy} ...
1
vote
3answers
68 views

Linear PDE with Variable Coefficients Help

I would like to find the closed-form solution (for $x > 0$) to $$\frac{\partial u}{\partial t} = a \frac{\partial^2 u}{\partial x^2} +b \frac{\partial u}{\partial x} + (cx+d) u,$$ $$u(0,\,x) = ...
1
vote
1answer
43 views

How to solve $e^yu_x-u_y+u=xe^y$?

Solve $e^yu_x-u_y+u=xe^y$ $u_x-\frac{u_y}{e^y}=x-\frac{u}{e^y}$ $\frac{b}{a}=\frac{dy}{dx}=-\frac{1}{e^y}$ $e^y=-x+c$, $\eta=e^y+x,\xi=x$ $e^y=\eta-\xi$ ...
0
votes
1answer
30 views

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 ...
1
vote
0answers
20 views

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 ...
0
votes
1answer
55 views

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 ...
0
votes
0answers
24 views

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 ...
0
votes
0answers
86 views

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 ...
1
vote
1answer
61 views

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 ...
1
vote
1answer
29 views

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 ...
0
votes
1answer
31 views

Easy way to see general solution of PDE expressed in different form?

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 ...
0
votes
2answers
43 views

Questions about the Laplace's equation in polar coordinates

The Laplace's equation in polar coordinates at a cyclic disk: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}, \ \ \ 0 \leq r \leq a, \ \ \ 0 \leq \theta \leq 2 \pi$$ $$u(a,\theta)=h(\theta), \ ...
6
votes
0answers
123 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

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 $V(x)$ ...
1
vote
1answer
28 views

smoothness of harmonic functions confusion in proof

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.