# Questions tagged [eulers-method]

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

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### numerical methods and chaos, eulers method in excel

this is my first post. I am trying to figure out this "exploration" for my high level college math class and I think trying to code in excel is messing me up. "1. Consider the “simple” ...
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### Stability of Forward Euler in nonlinear ODE

I have the following ode: $y^{'}= \frac{k}{\sqrt{y}}$ where k is a positive value. Applying the Forward Euler method gives the following: $v^{n}=\frac{\Delta t \ k}{\sqrt{v^{n-1}}}+v^{n-1}$ I'm ...
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### How to integrate arbitrary discrete-time linear and angular body fixed velocity to world space?

I have body fixed angular velocity values and linear acceleration values streaming in to my application. at some interval $\delta t$. I need to get a world position from these, assuming the start ...
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### Absolute stability and numerical robustness.

I am having problem understanding the definition of absolute stability. One definition of absolute stability I have heard is "A numerical solution $w_n$ to a problem is absolute stable for a ...
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### Why are the costate equations solved backwards in time?

I'm trying to find an optimal control to a simple nonlinear SIR model. I am trying to undersand the Pontryagin minimum principle but I don't understand why the costate equations must be solved ...
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### Euler's method for differential equations (estimation)

Is it true that if a curve is increasing, Euler's method will always underestimate an actual solution? So if a curve is either increasing and concave down, or increasing and concave up, we can simply ...
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### Logistic equation - How to go from continuous form to discrete form?

Logistic equation in continuous form: $\frac{\mathrm{d} y}{\mathrm{d} t} = ry(1 - ay)$ (Autonomous Differential Equations and Population Dynamics, equation 6 in Boyce Diprima's book, eleventh Edition)...
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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 ... 115 views

### Runge-Kutta-Munthe-Kaas integration for SE(3)

I am trying to implement in Python the Runge-Kutta-Munthe-Kaas integration for SE(3) for Euler’s method and RK4 for a simple trajectory with constant speed and angular velocity in the body frame. I ...
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### Help me understand why fixed point iteration works for backwards Euler's method

Euler's method for integration can be written as, $$f(x) = x + g(x)$$ Assuming that $g$ has a Lipschitz constant which is $<1$, it is a contraction mapping and therefore has a fixed point by the ...
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 ...