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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Stability: Backwards vs (normal) Euler method

Why is backwards Euler method more stable than the (normal) Euler method? The only thing that changes in the creation of each method is that the approximation of the derivative is taken "from behind" (...
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Euler method to geometric brownian motion stochastic diferential equation

I am doing Mark Joshi's Concepts and Practice of Mathematical Finance, project $1$ Vanilla options in a Black-Scholes world. At the final part of the project, we have the following: One further ...
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Determining the maximum stepsize for using the forward euler method

I have been given this question: Assume $t \in (3/12,8/12)$ and $W_0 = 1.8$ What is an upper bound for the time step in this scenario? Give an exact value. I haven't been given a differential ...
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27 views

What are reasons to choose between explicit midpoint method and improved Euler method?

What are the reason's to choose between the explicit midpoint method and the improved Euler method in solving an ordinary differential equation numerically? Explicit midpoint method: $y_{i+1} = y_i + ...
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Absolute stability of Forward Euler for non linear ODE

I have an apparently simple problem and probably I'm missing something about the theory. I need to compute the absolute stability interval for Forward Euler method applied to: $$ y' = 10 \sqrt y $$ ...
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Approximate solution for Higher order differential equation

I am trying to find the approximate solution of a third-order time-discrete differential equation by forward and backward Euler method. I would be very thankful if anybody here could share some ...
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25 views

Euler's Method stability for logistic equation

I have the ODE: $\frac{dP}{dt} = kP(1-\frac{P}{L})$ where $k$ and $L$ are constants. I need to find the stability range for step size values for which Euler's method is stable for the above ODE. I am ...
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Explicit and Implicit Euler-Method for $\dot{y}(t) = - \lambda y(t), y(t_0) =y_0$ and $\dot{y}(t) = - t (y(t))^2, y(t_0) =y_0 > 0$

We consider the two IVP $$\dot{y}(t) = - \lambda y(t), y(t_0) =y_0$$ $$\dot{y}(t) = - t (y(t))^2, y(t_0) =y_0 > 0$$ We're asked to execute one step with step-size $h$ with the Explicit and ...
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28 views

“Preliminary Shooting” using a single step of Euler's method

Here is my problem: Consider the boundary-value problem \begin{align*} y''=y^{3}+x, \qquad y(a) = \alpha, \qquad y(b)=\beta, \qquad a \leq x \leq b \end{align*} To use the shooting method to solve ...
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36 views

Error upper bound using Euler's Method

Determine an upper bound on the error made using Euler's method with step size $h$ to find an approximate value of the solution to the initial-value problem: $\frac{dy}{dt} = t - y^4$, $y(0) = 0$ at ...
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how to use improved Eulers method ODE?

ok, So I have this question: Gompertz used the following model do describe environmental pressure on the rate of growth of different groups of a given animal species. $$\frac{\mathrm{d} x}{\mathrm{...
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23 views

Error for Euler's method for higher order ODE

Consider the IVP $$\frac{dy}{dt}(t)=f(t,y(t)), \quad t\in [a,b], \quad y(a)=\alpha \in \mathbb{R}.$$ One can show that Euler's method, i.e., the scheme $$y_{i+1}=y_i+hf(t_i.y_i), \quad y_0=\alpha, \...
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35 views

How can I use Euler's Method in order to find the value of a constant k given the differential equation?

So, I am trying to attempt part b of this problem using my answer from part a. I just want to confirm if I did part a correctly and how I can do part b?
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Help with using the Euler midpoint equation

I am going to use the Euler midpoint equation to find the position of a block in a spring system at the time $0.05$s. The movement equation is as follows $$\frac{d^2x}{dt^2} + kx/m = 0$$ and doesn't ...
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37 views

is Euler's method stable for this problem?

Consider the IVP \begin{align*} y''=-y \end{align*} for $t \geq 0$, and $y(0)=1$, $y'(0)=2$. I have rewritten this differential equation as a system of first-order ODE's such that \begin{align*} u'=...
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56 views

Determining step size for Euler's method.

I have completed part(a) to find the $y$-Lipschitz constant $L=3$, I'm just not sure how to start part (b), determining a step size to guarantee a global error less than $10^{-3}$. Any help would be ...
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59 views

Exact solution and Euler method approximation for a first order differential equation

Having trouble wrapping my brain around this question.. don't know where to start. A. Consider the differential equation $f'(x) = (x + 1)f(x)$ with $f(0) =1$. What is the exact formula for $f(x)$? ...
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33 views

Writing a second order ODE as a system of first order ODEs and applying one step of Euler's method

I have been struggling with this problem for awhile now and I just can't seem to get the hang of it. The problem: Write the problem as a system of the first order and perform a step with Euler's ...
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Solving kinematic ODEs

I am making a particle filter implementation using the kinematic equations as my state transition function, that is to say: $$ \dot{p} = v\\ \dot{v} = a $$ where $a$ is assumed to be given. My first ...
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Euler Method for Dimensionless System

Consider the following system of couple ODEs $$ \begin{align} \frac{\partial n_A}{\partial \tau}&= \tilde{f}(n_B)-\delta_An_A\\ \frac{\partial n_B}{\partial \tau}&= \tilde{g}(n_A)-\delta_Bn_B. ...
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Euler's method for concave and monotonic increasing functions?

Will Euler's method (implicit or explicit) always approximate a function on or above the real solution, if the function is concave and monotonous increasing? It looks like that in my examples (all ...
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34 views

Determining graphed solution to Euler's method is incorrect?

A sample differential equations exam shows a y-t graph for dy/dt = f(y) with the initial condition y(0) = 2, wherein the plot was was determined using Euler's method. The sample answer states where ...
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62 views

Compute approximation of ODE using one step of explicit/implicit Euler method

I'm given the IVP: $$u^{(3)}(t) + u'(t) = tu(t)$$ $$u''(2) = 2$$ $$u(2) = 0$$ $$u'(2) = 1$$ and am asked to approximate the solution for $t=2.5$ using one step of the explicit Euler-method and one ...
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75 views

Understanding proof of Euler method having consistency of order 1

In my current lecture we derived that the Euler method has consistency of order 1. At one point in the proof it reads: If $f \in C^1(D)$ on a compact set $D$ around the graph of $u$, we can bound ...
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80 views

Numerical integration of differential equation on the surface of a sphere

I'm trying to simulate the motion of a particle (a position vector) that is constrained to living on the surface of a unit sphere. Each time-step, $\Delta t$, the particle moves in some direction on ...
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44 views

Solve initial value problem

I have an exercise that I do not understand. We have to solve an initial value problem: $$ \begin{array}{ccl} y'(t) &=& f(t,y(t)) \\ y(a) &=& y_0 \end{array} $$ We have to derive ...
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Euler's method to solve a differential equation

Apply Euler's method on the initial value problem $y'(t)=y(t)$ with $y(0)=1$ (in the interval $[0,1]$) and equidistant grid $I_h$, $h=\frac1n$. Give the approximation $y_h$ explicitly. This question ...
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76 views

ODE Taylor series expansion using midpoint Euler method

I am trying to solve part (c) of the question I attached as a picture as preparation for an exam. I can see how to get the result in the correct form using a taylor series expansion, however I don't ...
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Euler's method example: Where this value came from?

So, i've found an example that suits what i want to do, and i understood the majority of it, but i didn't really figured out where the last value, of the last line came from. I wanna know where did ...
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240 views

Example of a engineering problem solved using the Euler's method

I'm new to this forum, mechanical engineering and to matlab as well and this will be my first true academic work at university. I'm only at the 1st semester of my course and really need some help. So,...
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115 views

What's the minimum step size that can be used in Euler's method before it becomes unreliable?

In particular, if Euler's method is implemented on a computer, what's the minimum step size that can be used before rounding errors cause the Euler approximations to become completely unreliable? I ...
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71 views

Consistency improved Euler method

I have the butcher tablaeu for the improved Euler method \begin{array} {cc|c} 0 & 0 & 0 \\ 1 &0 & 1 \\ \hline \frac{1}{2} &\frac{1}{2} \end{array} I need to show that this ...
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246 views

Deriving Implicit Euler Method Update Rule to be used in iterations

Starting with the approximation $$y'(t) = \frac{y(t) - y(t - h)}h$$ arriving at an update rule of the form: $$y(t + h) = y(t) + hf(t + h, y(t + h))$$ Derive the implicit Euler update rule for this ...
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57 views

Trapezoidal iteration method for solving differential equation

I am learning on how to use various numerical methods to approximate solutions to differential equations. We are using R, to actually iterate these functions, but I am having difficulties wrapping my ...
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47 views

Calculation using Euler's method

Given y' = 1 - 2x - 3y, starting condition y(4)=5 and h = 1/2 I am asked to estimate by hand the value for y(5). My question is, if my staring value are as follows: ...
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41 views

ODE with Euler Method

I have to solve the following ODE: $y'(t)=y(t)+t$, $y(0)=0$ with Eulers Method in two steps, where $h=0.1$. I tried the following: $y'(0)=y(0)+0=0$ and then I get $y(0.1)=y(0)+h*y'(0)=0$ but then ...
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100 views

Euler's method on IVP, finding the global error.

I have the following system: $y'=y+e^x$ $y(0)=0$ The problem asks for applying Euler's method and then finding an expression for the global error. Finally, supposing that $$\lim_{h->0} \frac{1-(...
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What's the power of quaternion $[e^{a} * (cos(r) + \frac{sin(r)}{r}(bi+cj+dk)) ]^ 2$?

Euler's identity extended into quaternions is: $q = a + bi + cj + dk$ with a,b,c,d real numbers for the below: $\sqrt{b^2+ c^2 + d^2} = r > 0$, and $\frac{bi+cj+dk}{r}$ = $\sqrt{-1}$, ...
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Using Quaternion Extension of Eulers Formula what is $e^{qw} * e^{qw} = e^{qw2}$?

Knowing that for quaternions Euler's identity is: $q = a + bi + cj + dk$ with a,b,c,d real numbers $\sqrt{b^2+ c^2 + d^2} = r > 0$ $e^q = e^{a + r\sqrt{-1}} = e^ae^{r\sqrt{-1}} = e^a(...
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Implicit and explicit Euler Method for expression of stochastic solutions in time T

I have expression for implicit iteration (backward Euler Method): $$X_{n+1} = X_{n}+\delta tX_{n+1}+\sqrt{\delta t}X_{n+1}Z_n,$$ where $n=0,...,N-1; Z_n iid\sim N(0,1)$ and explicit iteration (...
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180 views

implementing Euler method in matlab for second order ODE

I have to use the Euler method for the differential equation : $$\begin{cases} x^{\prime}=y \\ y^{\prime}=-\frac{k}{m}x-\frac{\beta}{m}x^{3} \end{cases}$$ with $k=4, \beta =-0.04 , m=1$ in matlab. We ...
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Homework Problem Help with Euler's Formula

I have a question with my homework. I'm pretty sure I have to use Euler's formula to solve it, but I'm kind of stuck on how to use the formula to solve this problem. I started to draw the problem on ...
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108 views

Forward Euler Method Given Two Step Sizes

I am attempting to compute an approximation of the solution with the forward Euler method in $[0,1]$ with step lengths $h_{1}= 0.2$, $h_{2}= 0.1$ given the initial value problem below $$\frac{dy}{dz}=...
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93 views

Euler's method for different differential equations

I have equation: \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}t} = -0.04\sqrt{y} \end{align*} How would I find the expression for Euler's method? I know the general expression is: $$y_n=y_{n-1}+h\...
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Local stability analysis for a differential equation

I am trying to perform a local stability analysis for the differential equation $$dy/dt = tcos(y)+e^{-t}$$ I want to determine the step size I will need at a particular time t and corresponding ...
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2k views

How to determine the step size using Euler's Method?

Consider the initial value problem $x' = x+e^{-x}$ , $x(0)= 0$. This problem can’t be solved analytically. Using the Euler method, compute $x$ at $t = 1$, correct to three decimal places. How ...
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655 views

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 ...
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349 views

Implicit Euler and trapezoidal method

Fixed point iteration is sometimes used for solving the implicit Euler method $$y_{n+1}^\mathrm{(I)}= y_n + hf(y_{n+1}^\mathrm{(I)})$$ with the iteration procedure starting with explicit Euler $$y_{...
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378 views

Why does the Euler method go bad when the time step $T$ is decreased?

Consider below differential eqn $$\dfrac{dy}{dt} = y$$ In discrete form, with a time step $T$, using forward euler, it becomes $$\dfrac{y[n+1]-y[n]}{T} = y[n]$$ Solving the difference eqn we get $$y[...
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262 views

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 ...