Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [eulers-method]

Euler's method is a numerical method to solve first-order first-degree differential equations with a given initial value.

Euler's method is a numerical method to solve first-order first-degree differential equations with a given initial value. It is the simplest Runge-Kutte method.

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How to prove that $\frac{1}{1^2}+\frac{1}{2^2}+\dots+\frac{1}{n^2}+\dots=\frac{\pi^2}{6}$ using the spiral right angle triangle method?

I see this formula given below on You tube video of mathologer channel and then I try to find some new method to prove it: $$\sum_{n=1}^\infty \frac1{n^2} = \frac{\pi^2}6$$ I tried to prove it geometrically like this Our attempt: (1) First I tried…
user999691
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Local vs global truncation error

I was reading about local and global truncation error, and, I must be honest, I'm not really getting the idea of the two and what's the difference. Lets focus on the forward Euler method in particular. According to the book I'm reading the global…
user168764
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Is gradient descent nothing other than discretized gradient flow?

I can find definitions of the gradient flow of a scalar field $f$ as $$\frac{d \xi}{dt} = - \nabla_\xi f$$ in here and here. Gradient descent can be used to find a minimum in $f$ and can be written as $$\xi_{i+1} = \xi_{i} - \lambda \nabla_\xi…
yavagi
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Reference for Shooting Method

Consider the following setup. We have a second order boundary value problem: $$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$ A numerical approach is to almost first write as two first order…
JP McCarthy
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Are S.I. Euler and Verlet the same thing?

Verlet as given by Wikipedia: set $\vec x_1=\vec x_0+\vec v_0\,\Delta t+\frac12 A(\vec x_0)\,\Delta t^2$ for ''n=1,2,...'' iterate $\vec x_{n+1}=2 \vec x_n- \vec x_{n-1}+ A(\vec x_n)\,\Delta t^2.$ S.I. Euler as given by Wikipedia: $v_{n+1} = v_n +…
Jorge Rodriguez
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A nice way to do Euler's method on a calculator?

As part of the calculator paper for IB (International Baccalaureate), we may be asked to do Euler's method, for say, 10 iterations. While it is feasible to do with a calculator (slightly easier if you have 2 calculators), it is quite a nuisance, and…
Shuri2060
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Convergence of Euler scheme for ODEs

Consider the initial value problem for the ODE \begin{align} \frac{dy}{dt}&=f(y), \\ y(0)&=y_0, \end{align} where $f$ is a Lipschitz continuous function on $\mathbb{R}.$ Since $f$ is globally Lipschitz, the IVP admits a unique solution globally on…
Veronica
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Euler's method, Multiple choice does not match my answer.

This is the original question. Use Euler's method with h=0.2 to estimate y when x =1 if $y' = (y^2-1) /2 $ and y(0) = 0 A. 7.690 B. 12.730 C. 13.504 D. 90.676 My answer follows. n= 5, h= 0.2 a= x_0= 0, b=1, y0= 0 I'm using the formula $y_{(n+1)} =…
nickalh
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Confusion about the semi-implicit Euler method

On Wikipedia, the page about symplectic integrators talks about how the semi-implicit Euler method is a first-order symplectic integrator. But when I read the page about the semi-implicit Euler method, they do not show the same algorithm. On the…
Bloc97
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Existence of the Lyapunov spectrum for discretized ODEs?

It is a tedious, straight and narrow clarification of concepts, but still helps. When we discretize a continuous dynamical system/ODE ${\bf y}' = {\bf F}(t,{\bf y})$, where ${\bf y}={\bf y}(t)$ is a function of $t$, e.g. using the Euler method,…
Charlie Chang
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Reason for multiplication of function with step size (and subsequent addition) in Euler method

What is the reason behind the multiplication of the function's derivative with the step size (and the subsequent addition) in the numerical Euler method? $$ y_{n+1} = y_n + hf(t_n, y_n) $$ I can't figure out why exactly this would work for…
Chris BT
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Is it possible for me to model mathematically the revolution to sharpen a pencil using complex number?

I was planning to do research on complex number since I find the topic intriguing, How can I conduct a research by modelling mathematically to calculate the required number of rotation for a pencil in a circular motion when sharpening the pencil…
Acetrow
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Taylor polynomials in Horner Form

Professional high school math teacher/ amateur mathematician here looking for feedback on my attempt to to derive the general case of a Maclaurin polynomial using finite difference methods. Here's my overall approach (not a proof). Consider an…
Bob Tivnan
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Is Backward-Euler method considered the same as Runge Kutta $2^{\text{nd}}$ order method?

I have a book that quotes: Euler's method, Modified Euler's method and Runge's method are Runge-Kutta methods of first, second and third order respectively. The fourth-order Runge-Kutta method is method is most commonly used and is often…
paulplusx
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Euler method and bisection method

I'd like to solve the equation $$ \phi''(x) = \lambda \sin (\phi(x)) $$ where $x \in (0,L)$, $\phi'(0) = 0$, $\phi'(L) = 0$. Let $ \psi = \phi'$ and $$ \phi'(x) - \psi(x) = 0$$ $$ \psi'(x) - \lambda \sin (\phi(x)) = 0$$ for $x \in (0,L)$ and…
DMan
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