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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: $$ u(t) = \frac{1}{1 + t^2/20}. $$ ...
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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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Newton forward and Forward Euler method

in the article "Numerical studies on 2-dimensional reaction-diffusion equations" (Tang/Qin/Weber 1993?) i found the following: we look for $u(x, y, t)$ satisfying $$ \frac{\partial u}{\...
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Detailed exp does Euler's method fail for second order sinusoidal ODE?

I am trying to explain, for a seminar, why this ODE fails with Euler's method. $$ \partial^2_x y= \sin(x) $$ for boundaries, $L=2\pi$, $y(0)=y(L) = 0,\ y'(0) = y'(L) =-1$, which has exact solution $$ ...
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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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Find approximations for the PVI solution

I'm trying to solve the following problem: Find approximations for the PVI solution on mesh [0,1] with h=0.25: $y'=y-\frac{2x}{y}$ $y(0)=1$ I thought about use Euler's method to do it, $U^{n+1} = U^{...
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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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Backward Euler method with spatial derivative terms

Consider the following differential equation: $$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \tag{1}$$ with $u(x,0)=f(x)$. The solution of $(1)$, using MOC, is $e^...
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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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Trying to solve exercise about Euler method

Help to solve this exercise with Euler Method, my problem here problem is that the solutions I calculated are not close to the real values and they diverge. Thanks! $$y'=e^y $$ with conditions $$ 0\...
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Proving that the semi-implicit Euler method is symplectic

I'm having trouble understanding the following proof that the semi-implicit Euler method $$\begin{cases} p^{k+1} = p^k-\Delta t \frac{\partial H}{\partial q}(p^{k+1},q^k) \\ q^{k+1} = q^k + \Delta t \...
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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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euler method results have a huge error

I have The following IVP: $$y'=-20y+20 \cos ⁡t-\sin ⁡t ,~~~ 0≤t≤2,~~~ y(0)=0;$$ With $N = 5$, $h = 0.4$, and $t_i=0.4i$, for $i = 0,1,2,3,4,5$. I am performing the steps as following: 1st: $y1$=$y0+...
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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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Understanding the stability of eulers method for a nonlinear system of differential equations and the relationship with the Jacobian.

So suppose I have the system of nonlinear differential equations: $y'(t)=F(y(t))$ It looks like one equates the drive function to a local taylor approximation about $y(t_n)=y_n$ e.g $F(y(t)) \approx F(...
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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{...
Statistics 's user avatar
1 vote
3 answers
318 views

Problem with numerical solving 2nd order nonlinear differential equation

I have this 2nd order nonlinear ODE: $$ \frac{I_0}{C} \left[ e^{\frac{1}{2nU_T} \left(U_{in}(t) - U_{out}(t) - RCU'_{out}(t) - \frac{R}{R_z} U_{out}(t) -LCU''_{out}(t) - \frac{L}{R_z} U'_{out}(t) \...
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Explanation to Euler's Method

I am trying to figure out Euler's method. The initial value problem is : $$P'(t)=0.7P(t)(1-\frac{P(t)}{750})-20, P(0)=30$$ The time step is set to $Δt=7$ days For the algorithm we have: $f(t,P)=0.7P(...
Athanasios Paraskevopoulos's user avatar
2 votes
1 answer
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Doubts on the solution of a differential equation

I've encountered myself with a differential equation which I'm not sure that I've solved correctly. The situation is the following: I have a set of parametric equations, which I'll call $\vec{r}$ \...
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Explicit/Implicit Euler method. Show that $ \lim_{k \rightarrow \infty} ||y_k||_2^2 = \infty $ and $ \lim_{k \rightarrow \infty} ||y_k||_2^2 = 0 $

We consider the following initial value problem: $ y'(t) = \begin{pmatrix} -y_{2}(t) \\ y_1(t)\\ \end{pmatrix} $ with $ y(0) = \begin{pmatrix} a \\ 0\\ \end{pmatrix} ...
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Solve SIR model using Euler's method in C#

The SIR model is: $$ \begin{aligned} \dot S &= −\beta I S\\ \dot I &= \beta I S − \gamma I \\ \dot R &= \gamma I \end{aligned} $$ I know how to solve SIR with Euler's method in C#. I don't ...
user1084631's user avatar
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Euler method for ODE with noisy derivative

I am interested in numerically integrating a noisy differential equation: $\frac{dx}{dt} = f(x,t) + \epsilon(t)$ where $\epsilon(t) \sim \mathcal{N}(\mu, \sigma^2)$. Is this a RODE or SODE? How does ...
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How does Euler's method work when step size and wanted values are different?

I have a question that goes as such: Use Euler’s method with $dt= 0.1$, to estimate $p$ for these values of $t$, using initial condition $(0,5)$. $t= 2,4,6,8,10,12,14,16$ $\frac{Dp}{dt}= 0.05(p-4)(...
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Local truncation error vs Global truncation error

I know that global truncation error is proportional to $h^p$ while local truncation error is proportional to $h^{(p+1)}$, where $h$ is the step size. But where does this relationship come from and how ...
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Approximating second order differential equation with Euler's method

My equation is $x''=\frac{kx}{m}$, to apply Euler's method it is seperated into a system of equation consists of two first order ODE: $\frac{dx}{dt}=v$ $\frac{dv}{dt}=-\frac{kx}{m}$ Lets say that the ...
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How to find local truncation error in Forward Euler Method without knowing the actual value

In my high school project, I am solving differential equations using the forward Euler numerical method and this is because the equations were too hard to solve analytically. To find the local error ...
Gaussian 123's user avatar
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Forward Euler and 1st-order linear ODE with exponential solution

Let us consider the simple ODE $y'=y$ with $y(0)=1$ on the interval $[0,1]$. The solution is obviously $y(t)=\mathrm{e}^t$. Now, consider a Forward Euler approximation of the solution to the ODE. The ...
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Estimating pi with Euler's method and Runge-Kutta 4

I'm trying to estimate pi by solving the IVP $y'' + y = 0$ where $y(0) = 1, y'(0) = 0$ numerically by defining $\frac{\pi}{2}$ as the first value on t such that $y(t) = 0$ I'm trying to solve this ...
Minea's user avatar
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1 answer
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Where can I find "detailed" error analysis of modified Euler's methods?

I'm studying the local truncation error of each Heun's, Midpoint, and Ralston's methods. For Heun's method, I found a material in here. However, I don't get how the following is derived. $$ f(t+h, y(t+...
Minsik Seo's user avatar
1 vote
2 answers
139 views

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 ...
Are Berntsen's user avatar
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How is the global truncation error and stability criterion of the forward Euler method consistent with each other?

The forward Euler method can be used to solve ODE's with some initial value by using the update rule $$y_{k+1} = y_k + hf(t_k, y_k),$$ where $0\leq k \leq n-1$ and $h$ is the step size $\dfrac{t_n-t_0}...
McGonald's user avatar
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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)...
Daniel's user avatar
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1 answer
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Backwards Euler Method's error converges to step size (Python)

I am trying to programmatically solve the ODE, $$\displaystyle Y'( t) =-Y( t) +\frac{1}{1+t^{2}} +\tan^{-1}( t)$$ with initial condition $Y(0)=0$. I know the analytical solution is $\displaystyle Y( t)...
ibrahim's user avatar
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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, ...
s_o_c_account's user avatar
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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-\...
Elly's user avatar
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how does this example of runge kutta translate to calculating forces

I saw this example of runge kutta online, I use the unity game engine, and am working on making softbody physics (squishy things like a sponge). It was working, until you run it for like a minute or ...
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