# 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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### Weird error behavior when implementing Euler's method for solving ODEs in Python

I am solving this particular ODE: $$\begin{cases} u'(t)=-2\,tu^2/20, \quad t \in [0,\sqrt{20}]\\ u(0) = 1 \end{cases}$$ which the analytical solution is given: $$u(t) = \frac{1}{1 + t^2/20}.$$ ...
1 vote
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### Transformation of a complex function into a real function via Euler's formula?

I am new as a member but I'll try my best to make it as understandable and easy to follow as possible. I recently had to find a general solution to this homogeneous differential equation: \begin{...
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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 ...
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### Fixed-point iteration and backwards Euler's method

I have just started with numeric methods for differential equations and have come across this exercise: We have $f$ which is Lipschitz-continuous in the second argument with constant $L$ and we are ...
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### Euler Backward method order of accuracy

I solved a question with Euler's backward method with 4 different step lengths. Then I calculated the error and drew a graph using loglog. Now the task is to find the order of accuracy using the graph....
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### Simulation of a pendulum without using the angle with the Euler-Cromer method.

I am trying to create a computer simulation of a simple pendulum with a mass $m$ hanging from an inextensible and negligibly massless string, but without using the angle, only using the positions of ...
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### Signal Processing, Data Compression, and Complex Numbers

A detached exercise in my course asks us to be able to simplify the following summation, where j is an imaginary number. The current material involves lossless and lossy compression, however I am ...
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### An error bound for Euler's method with $Y'(x) = f(x,Y(x))$ when $\frac{\partial f}{\partial y} \leq 0$

My question comes from section 6.2 of An Introduction to Numerical Analysis (2nd ed) by Kendall Atkinson. Specifically, there is a detail in the proof of the corollary to Theorem 6.3 that I do not ...
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### Euler scheme integral of drift inequality

Let \begin{aligned} t_{k}=\dfrac{kT}{n} ,\ \ k\in \left[ 0,n\right]\\ \sigma(x)=\sigma\ constant \ and \ \sigma>0\\ b\ and \ b'\ lipschitz \ continuous \\ for \ s \ in \ [t_k,t_{k+1})\ ,\underline{...
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### Boundary Value Problems Solving with Central Difference

I have differential equation $u''(x)+9u(x) = \cos(2x), \; x \in [0, π/2]$ the boundary conditions are $u(0)=1$, $u(\pi / 2) = -1$, with $h = 0.2$ Now I replaced $u''(x)$ with centered difference ...
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### Global truncation error of backward Euler method

It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like $$\frac{\exp(LT) -1}{L} \frac{Mh}{2},$$ with $L$...
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### Euler method coupled stochastic differential equations with time-derivative

Given a complex variable $\alpha(t)$ following this stochastic differential equation $\dot{\alpha}(t)=-\kappa\alpha+\sqrt{\kappa n_{\rm th}} \eta(t)$, where $\eta(t)$ describes a Winer process, we ...
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### Time Discretisation of a System of PDEs

Suppose we have an arbitrary system of PDEs $$\partial_t u - D_1 \Delta u + a(v) u = f(t)$$ $$\partial_t v - D_2 \Delta v + b(u) v = g(t)$$ We want to discretise the system in time. We use the ...
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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 ...
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### Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method

Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method, knowing that $y(0)=1$, $t=[0,1]$, $h=0.25$ I know the local truncation error (LTE) introduced by the Euler method is given ...
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### The proof for the Euler decomposition of the VaR

For a portfolio-wide profit/loss variable $X= \sum_{i=1}^{n}w_iX_i$ the value-at-risk of $X$ at confidence level $\alpha$ (usually close to 1) is defined as the $\alpha$-quantile of $-X$: \begin{...
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### Euler method without defined function

Before I begin, I must say that reading and quickly understanding the definitions of mathematical theorems is not my strongest suit. We had a test where one of the subjects was Eulers methods. The ...
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### Using Euler's integration to solve a problem that is not written as a derivative

I am brand new to solving differential equations. I am taking a modeling course that is asking me to use Euler's method to solve for the following: $y = \displaystyle \int_{0}^{2} (x^3) dx$ However, ...
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### Implicit Euler to solve this $y(x_1)$

$$y_{n+1} = y_n +hf(h,y_{n+1})$$ How do I use implicit Euler to solve this $y(x_1)$? $$y' = \frac{-x}{y^2}$$ $$y(0) = 1$$ $$h=0.1$$ $$x_1=0.1$$ I have got this far: y_1 = y_0 + hf(h,y_1) = y_0-\...
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