Hi everyone,
WeBWorK assignment 10 is now available – it is due this Tuesday evening at midnight.
Have a good weekend,
Prof. Reitz
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- Error terms in the Richardson extrapolation.The details of what is happening are not important the question I have. We have a solution $X_t(\omega)$ to a stochastic differential equation and an approximation with step size $N$ denoted $Y_t^N(\omega)$. Now a theorem (talay-tubaro; similar to the Richardson extrapolation) tells me that $\mathbb{E}[f(X_T)]-\mathbb{E}[f(Y_T^N)]=\frac{C_1}{N}+ \mathcal{O}(N^{-2})$, this fact leads to an improved scheme with error: […]
- Asymptotic Stability of OriginI need to show that the origin is globally asymptotically stable for the system $$x' =-x+2y-x^{3}+3x^{2}y-3xy^{2}\\ y' =-x+y-y^{3}$$ I thought that finding a Liapunov function would be easy, but it seems there's a lot of cross-terms. For instance, I've tried $V(x,y) = x^2 + y^2$, but there are many terms that appear that I am […]
- What Exactly Is Symplecticity in the Context of Hamiltonian Systems and Numerical Schemes?I've encountered the term symplecticity in the study of Hamiltonian systems and their numerical discretizations, but I'm struggling to fully grasp its meaning and significance. Initially, I thought symplecticity was just a weaker form of the conservation of the Hamiltonian, but I've since learned that this isn't entirely accurate. From my current understanding, in the […]
- Verifying Lipschitz Condition for a Function Involving Trigonometric and Rational TermsI have this ode: $$v''(r)+\frac{1}{2}(v'(r))^2+P(r)v'(r)=Q(r),$$ where $$P(r)=− \frac{k^3r + 12\pi \rho_0 kr \cos(kr) + 4\pi\rho_0(k^2r^2 − 3)\sin(kr)} {r[k^3r − 8\pi\rho_0[\sin(kr) − kr\cos(kr)]},$$ $$Q(r)= \frac{8\pi\rho_0kr(r^2 + 5) \cos(kr) + 8\pi\rho_0(k^2r^3 + k^2r^2 − r − 5) \sin(kr)} {k^3r^3 − 8\pi\rho_0r^2[\sin(kr) − kr \cos(kr)] }.$$ I wrote the equation as systems of ordinary differential equations: $$\begin{align}v'(r) &= […]
- generator of a semigroup properties $A$ and $A-\alpha I$at the moment I'm understanding a proof of a theorem about generating a C0-semigroup. Here's the theorem and the proof with associated definitions to understand it (see below). My question is what Hille-Yosida has shown that the operator $E_{1}^{-1}\tilde{A}_{1}$ generates a C0-semigroup. I don't understand why the author of the article directly deduces the thesis […]
- axisymmetric ODE $h=\frac{1}{r}\frac{d}{dr}\left(\frac{rh'}{\sqrt{1+(h')^2}}\right).$Consider the following equation where $h$ is a function of $y$: \begin{equation*} h=\frac{d}{dy}\left(\frac{h'}{\sqrt{1+(h')^2}}\right). \end{equation*} Multiplying both sides by $h'\equiv dh/dy$, and integrating the above equation, we get \begin{equation*} \frac{1}{2}h^2=-\frac{1}{\sqrt{1+(h')^2}}+c, \end{equation*} where $c$ is a constant of integration. Now consider the axisymmetric version of the same problem where $h$ is now a function of $r$, and […]
- Why is $X e^{\Lambda t}X^{-1}$ the general solution to system of $n$ linear first order ordinary differential equations?I know that a system of n first order linear ordinary differential equations: $$ \begin{align} \begin{bmatrix} \frac{dy_1}{dt} \\ \vdots \\ \frac{d^ny_n}{dt^n} \end{bmatrix} &= \begin{bmatrix} a _{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn}\end{bmatrix} \begin{bmatrix} y_1 \\ \vdots \\ y_n\end{bmatrix} \\ \mathbf{Y}' &= A \mathbf{Y} \end{align} $$ is […]
- condition to fit the derivative inside an operator $\dfrac{d}{dt} Ex(t) = E\dfrac{d}{dt} x(t)$ (abstract ODE)I have an operator $E: X \longrightarrow Y$ where $X$ and $Y$ are Hilbert spaces and $E$ is a linear and bounded operator, but not necessarily inversible. I would like to know under what conditions can we fit the derivative inside the operator? that is to say when $\dfrac{d}{dt} Ex(t) = E\dfrac{d}{dt} x(t)$ where $x(t)\in […]
- Zero matrix for linearized systemI am given the following system, which I have to linearize such that I can describe the local phase portrait. $x' = x^2, y'=y^2$. The origin is the only equilibrium. If I compute the Jacobian of this system, I get $ \begin{pmatrix} 2x & 0 \\ 0 & 2y \end{pmatrix} $. Evaluated at $(0,0)$, I […]
- Finding the Steady-State Solution of the Burgers' EquationI am trying to determine the steady-state solution of the following Burgers' equation: \begin{equation*} u_t+u u_x=\nu u_{x x}, \quad u(x, 0)=\sin (\pi x) \end{equation*} subject to the boundary conditions $u(0, t)=u(1, t)=0$. Could someone provide guidance on how to find the steady-state solution for this equation? Any tips or references for solving such problems would […]
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