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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Solving $ y' = x+y $ with Euler's method

I was going over Euler's method for solving DE and I had an idea: Could we use it to get an exact solution to a DE by considering an infinitesimal step size? This is the main idea: if the ...
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Help identifying this euler-like method for approximating 2nd order differential equation

Context: I'm implementing an engine that simulates n physics bodies, and am trying to understand and refine my 2-body-problem engine first. If $\textbf{r}$ is my position vector, my ode is $\ddot{\...
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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 ...
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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 ...
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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 ...
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Python Code for Eulero Discretization

Write the Euler discretization of the 1-dimensional stochastic equation $$dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$$ For this part I would say all right because it is a purely ...
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Euler integration solution from system of ODE's - already estimated values

I am currently completing an investigation assignment on modelling the growth of a virus inside of the host. There are 3 ODEs that I am using in the system, all determined by change in t. The ...
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Upper bound on stepsize for guaranteeing convergence of forward euler method applied to nonlinear ODEs?

Consider the autonomous $d$-dimensional ODE given by $$ \dot{x}(t)=f(x(t)), \quad t\in [0,T], \quad x(0) = x_0 \in \mathbb{R}^d, $$ where $f$ is, in general, nonlinear. Consider now a discretized grid ...
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Convergence of Euler's Method To Solve Differential Equations

if we use backward euler's method to solve following differential equation , for which values of h the method is convergent? $y^{\prime}(t)=\lambda y(t)+g(t), \quad y\left(t_{0}\right)=y_{0}, \quad \...
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Will a linearized dynamical system be stabile when using it with Euler's Method?

Assume that we have a dynamical system $\dot x = f(x, u)$ and I want to simulate the system with Euler's Method. $$x = x + af(x, u)$$ Where $a$ is a small number. This can create unstability. What if ...
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Adaptive step size for Euler Method - How to create?

I think Euler's Method is a great way to simulate ODE:s. It's not the most accurate, but it's the fastest and simplest. Euler's Method is usaly used with fixed step size, where ...
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How can I apply the Euler–Maruyama approximation to the following SDE?

I'm trying to apply the Euler–Maruyama discretization to a python code using this Wikipedia page Wikipedia page where it says that the SDE $$dX_t = a(X_t, t) dt + b(X_t, t) dW_t, \quad X_0 = x_0 $$ ...
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Taking derivative of ODE for Euler's Method

I find myself having trouble with an exercise from Gautschi's Numerical Analysis on finding the truncation error of Euler's method. I have been given a system: $$y' = Ay, \quad y \in \mathbb{R}^{d}, A ...
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Problem About the Order of Convergence of Numerical Method

Sometimes when I did convergence test for my numerical solver, I can't get the exact order of convergence even I am sure that my code is correct. I would like to ask if anyone else has experienced ...
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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 ...
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2 votes
2 answers
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Convergence of Euler's Method

I am to show the IVP as defined by $$y' = ay + b, \quad y(0) = y_0$$ converges using Euler's method. To do this, I need to show that $$\lim_{h \to 0} u_n = y_n$$ where $u_n$ is the approximated value ...
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Order of convergence in Euler's method

I am writing a program to find order of convergence of the Euler's method for finding numerical solution of ODE with given initial condition. The formula to estimate the order of convergence is given ...
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Euler's method for system of linear ODE's

I have the following system: $$ \begin{aligned} \dot x &= -y \\ \dot y &= x \end{aligned} $$ Given that $(x_i^n,y_i^n)$ are the points obtained for $i=1,2\dots n^2$ using a time-step $h=1/n$ ...
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Convergence of semi-implicit Euler scheme for SDE

I am confronted with the following problem: For $W$ being a one-dimensional brownian motion and $\alpha\in[0,1]$, what are the conditions for the numerical scheme $X_{n+1}=X_n+(1-\alpha)\mu X_n\Delta ...
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Implicit Euler looks different from Runge-Kutta implicit Euler

In theory Implicit (Backward) Euler should be a Runge-Kutta method with tableu given below, however I find that the standard Backward Euler formula and the Runge-Kutta one differ. $$ {\begin{array}{c|...
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Fundamental limit in the problem of energy of the Euler-Method for harmonic oscilations

I was discussing with one friend and he sent me a strange manipulation. It's a common result that the Euler Method isn't good for Harmonic oscillations because the energy isn't conserved. ...
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Combining Multistep and Truncated Taylor methods

Consider solving an ODE $d_t x = f(x, t)$ numerically. The 1st-order Truncated Taylor method is simply Euler's method: $x_{n+1} = x_n + h_n f(x_n, t_n)$. We can "improve" Euler's method ...
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When does/doesn't the sequence from Euler's method exist?

For a differential equation say $y'=ty^3$ with $y(0)=1$ how would we know if the sequence from Euler's method exists or not? So I know this particular equation has solution: $$y=\frac{1}{\sqrt{1-t^2}}$...
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Computation of global error bound for Euler's formula [closed]

I am trying to calculate the global error bound for Euler's method, but I am having trouble. I am given the formula $|y(t_{i}) - u_{i}| \leq \frac{1}{L}(\frac{hM}{2} + \frac{\delta}{h})(e^{L(t_{i}-a)}-...
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Stability of time step for $\frac{dx}{dt} = \mu + \lambda x$.

I want to know the time-step $\Delta t$ for which the explicit Euler method is stable. $$\frac{dx}{dt} = \mu + \lambda x.$$ I know for $\mu = 0$ that we need $\Delta t \leq \frac{2}{|\lambda|}$ where $...
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Simplifying An Exponential Function Which Includes Complex Numbers [closed]

In there j is the complex number. In signals and systems lecture we generally use 'j' instead of 'i'. In my book solution it's includes one equation like that. But i didn't understand this conversion ...
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Approximating $\pi$ using Euler's Method Approximation with $y' = \frac{4}{1+t^2}$

The solution to the initial value problem $\frac{dy}{dt} = \frac{4}{1+t^2}, y(0) = 0$ has the value $y(1) = \pi$. How small would you have to make $\Delta t$ using euler's method to get two correct ...
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How to determine the sign of a square root in Euler's method?

Considering the equations $$ 1 = \left(\frac{dx}{dt}\right)^2 + x^2, ~~~~~~~~~~~~~~\frac{d}{dt}\frac{dx}{dt}=-x$$ I have $\frac{dx}{dt}$ vs $x$ constrained as a circle of radius 1. I want to determine ...
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The continuum limit of Euler's method

Euler's method is generally applied as a numerical technique. But if you choose not to plug in values along the way, you can end up with a very long algebraic expression for $f(x)$ instead of a ...
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Fokker-Planck Equation of an Euler Scheme

Consider the overdamped Langevin equation $$ dX_t= -\nabla V(X_t) dt + dW_t $$ with associated FPE for its density $\rho_\cdot(\cdot):\mathbb{R}\times \mathbb{R}^d\to \mathbb{R}$ $$ \partial_t \rho_t ...
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4 votes
2 answers
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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= ...
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Numerical integration of differential equation in state-space form

I'm using numerical integration methods like Explicit/Implicit Euler, Runge-Kutta to solve a system of linear ordinary differential equations in state-space representation $\dot{x}=A\,x + B\,u$. I ...
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How to Discretize this SDE found in finance?

Continuous-Time In continuous-time form, the "Heston model" is written as $$ dS_t = \mu S_t \,dt + \sqrt{\nu_t} S_t \, dW_t^S \\ d\nu_t = \kappa (\theta - \nu_t)\, dt + \xi \sqrt{v_t} \, ...
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Fully-implicit scheme for reactive transport equation.

I am reading a paper by Zhang 2007 about reactive chemical transport where a fully implicit method was used in solving the transport equation given by, "At $\left(n+1\right)$th time step, ...
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2 votes
1 answer
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What’s the name of this phenomenon occurring when numerically computing the explicit Euler method for this PDE IVP and why does it occur?

I’ve been asked to calculate numerically the solution of this IVP using Euler’s explicit method on Mathlab: $$u_t-2u_{xx}=0$$ $$u(x,0)=\sin (2\pi x), \quad x\in(0,1)$$ $$u(0,t)=u(1,t)=0, \quad t\in(0,...
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Calculating brownian motion in Eulers method

Say for example we have brownian motion term added to an expression $W(t)$ and our $t=2$, what would $W(2)$ be? I' ve tried searching the internet but I can't seem to find a way that gives a formula ...
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1 answer
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Euler's method produces mirrored solution?

I have the differential equation $$ 0.22y'' + y' + 10000y = 2000\pi \cos (2000 \pi t) $$ from this I get the Euler's method system of equations: $$ t_{n+1} = t_n + 0.001 $$ $$ x_{n+1} = x_n + 0.001\...
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What is the error of $x_i$ at a fixed time $t_i$ when using Euler's method for the IVP $\frac{dx}{dt}=x$ with $x(0)=1$?

So we have Euler's method: $x_{i+1}=x_i+hf(t_i,x_i)$ where $x'=f(t,x)$ and $h$ is a small step size. Applying this method to $x'=x$ gives $x_{i+1}=x_i+hx_i=(1+h)x_i$. Solving this difference equation ...
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0 votes
1 answer
381 views

Uniqueness of solution of differential equation

Consider the differential equation $ y'(x)=f(x,y(x)) $ with initial condition $y(x_0)=y_0$. According to some theorem, if $f$ is continuous, then there exists at least one solution that satisfies the ...
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56 views

System of differential equation, Euler's method

Does anyone know if there is any calculator or how can I input this system of differential equation in wolfram? $$\frac{dy_1}{dt}=0.279\left[-0.125y_1+2y_2-1.25y_C+1.25e\right]$$ $$\frac{dy_2}{dt}=-0....
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2 votes
1 answer
122 views

Hamilton equations-Symplectic Euler method

We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
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3 votes
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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 ...
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0 answers
151 views

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 ...
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85 views

Verify order of convergence of the Euler--Maruyama schemenumerically

I have a question about the order of convergence of the Euler-Maruyama scheme and how one verifes this numerically. I have read that the Euler-Maruyama scheme is mean-squared convergent of order 1/2 ...
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1 vote
1 answer
100 views

Improved Euler's Method 2nd order ODE ,not sure if this is right

SO i have this differential equation and actually i am not sure if i have solved it right with Euler's improved method : $z'' = f_{z} - C_{z}*|z'|*z'$ $ z' = u$ $u' = z'' = f_{z}-C_{z}*|u|*u$ Improved ...
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2 votes
0 answers
146 views

Results better without adaptive $h$ than with adaptive $h$ when using RKF45?

I'm trying to solve a fairly simple ODE, $$-y'=t^{-2} +4(t-6)e^{-2(t-6)^2} ,~~~ y(1)=1~\text{ for }~t\in[0,10].$$ Via the Runge-Kutta-Fehlberg method. I don't understand why, but the solutions i get ...
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0 votes
1 answer
28 views

Are the sequences $\frac{ih}{2}((1-ih)^{k}-(1+ih)^{k})$ and $\frac{1}{(1+h^{2})^{k}}\frac{ih}{2}((1-ih)^{k}-(1+ih)^{k})$ bounded?

I want to understand the explicit and the implicit Euler method better. Assume I have the initial value problem $y''+y=0$, $y(0)=0$, $y'(0)=1$, which is of course solved by $y=\sin(x)$, and I convert ...
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1 vote
1 answer
83 views

Convert ODE to first-order form

I am struggling to understand reducing higher order DE to first-order form. Say I have an equation of motion, $$m \frac{d^2}{dt^2} \left( \begin{array}{c} x \\ y \end{array} \right) = -\frac{GmM}{{(...
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1 answer
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Use Euler's Formula to prove the following equation $ y = Ae^{\sqrt{\lambda}ix} + Be^{-\sqrt{\lambda}ix} $

I'm having a difficult time understanding how one would take $ y = Ae^{\sqrt{\lambda}ix} + Be^{-\sqrt{\lambda}ix} $ and end up with $ y = A \cos{\sqrt{\lambda}x} + B\sin{\sqrt{\lambda}x} $ knowing ...
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