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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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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Proof verification: Baby Rudin Chapter 7 Exercise 25 (Euler Method)

Problem 7.25 in Rudin's Principles of Mathematical Analysis goes as follows: Suppose $\phi$ is a continuous bounded real function in the strip defined by $0\leq x\leq 1,-\infty <y<\infty$. ...
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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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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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38 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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29 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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41 views

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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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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80 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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42 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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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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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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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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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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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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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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67 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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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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557 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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351 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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191 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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202 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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181 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 ...
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Using Euler's method to estimate a value of y(1.1) if y(1,0) = 0.

I am currently attempting some past paper exam questions and have come across a question on Euler's method that I am unsure on how to solve. This is the question; MCQ on Euler's Method: I have had ...
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337 views

Implementing backward Euler with a nonlinear system

I'm trying to implement a backward Euler method in a situation when I have a nonlinear system of equations, and I'm having some trouble. My set up is as follows: I have $\vec{f}(\vec{u}, t) = \begin{...
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Finding a Function Given a Piece

Suppose you're given function on a domain ([1,2] for instance) that exactly resembles $\frac{1}{x}$. Is this enough to know (assuming the function doesn't have a weird behavior, such as piecewise or ...
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445 views

Modified Euler Absolute Stability Proof

Given the modified Euler method: $u_{n+1} = u_n + hf(u_n + \frac{h}{2}f(u_n))$ applied to the test equation $y' = f(y) = \lambda y$, how do you prove that no imaginary value $h\lambda$ is contained in ...
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Euler's method: plotting total error (round-off included) as a function of stepsize

I'm trying to show that when the stepsize is too small, round-off error accumulates making the total error too big, hence doing the euler method with a smaller stepsize yields better results but only ...
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113 views

Euler method with infinite gradient at initial value

The title itself is self explanatory - I am trying to numerically solve an ODE with an initial value that has an infinite gradient. It seemed problematic to me and I am not certain as to how I should ...
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Question on Implicit Euler Method Proof

I'm working on a proof for the local truncation error of the implicit Euler method. I've been given a little hint to get started, but I'm stuck on a line that involves the chain rule. Can some please ...
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380 views

Stability Condition of Forward Euler Method

I've been searching all over the internet, but cannot find an answer to the following question: Given some ODE $y' = f(y, x)$ with initial condition $y(0) = 0$, where $f$ is some real function in two ...
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Max vertices in Odd Degree Graph

If I have a graph whose vertices all have odd degree greater than 1, what is the maximum possible number of vertices if the graph has at most 14 edges? My thought for this is basically that your best ...
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Deriving Eulers method

I am wondering if it is possible to derive Eulers method without doing it graphically. I am trying to give a detailed derivation of Eulers method, and how the formula is found $$y_i=y_{i-1}+hf(x_{i-...
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218 views

Crank-Nicolson Scheme equivalent to a forward and backward Euler method

I am trying to show that, for the equation $$y'+\alpha y=0$$ alternating between a forward Euler method step for $y_{2n}$ and a backward Euler step for $y_{2n+1}$ with time-step $h$ is equivalent ...
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Using Euler derive the relationships between cosine and exponential function [closed]

$$\cos\phi=\frac{1}{2}(e^{j\phi} + e^{-j\phi}).$$ Please i was told to do this assignment but cant prove if anyone can help me out
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Using Complex exponential/Euler's Method to integrate, bizarre case

$ \int e^{-3t} cos(2-\sqrt 3 t) dt $ I have been asked to evaluate that using complex exponential/euler's method. I have done many similar questions but all of them had something like (cos3x), sin(5t)...
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Euler's Method Global Error: How to calculate $C_1$ if $error = C_1 h$

My textbook claims that, for small step size $h$, Euler's method has a global error which is at most proportional to $h$ such that error $= C_1h$. It is then claimed that $C_1$ depends on the initial ...
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Solving coupled system of ODE's

I'm reading a paper on system of ordinary differential equations, and I'm a bit confused with the following definition of a such: $ \dot{\phi} = -A^T\phi-Qx , \quad \dot{x} = \left\{ \begin{array}{ll}...
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Numerical Approximation Values Exceed Exact Value

Is it possible for the numerical approximation (Euler, Improved Euler,or Runge-Kutta) to both exceed and fall under the exact value of a first order differential equation?
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Order of predictor-corrector method.

Consider the following initial value problem $$\begin{cases} x'(t) = f(t,x(t)), \; t \in [t_0, T]\\ x(t_0) = \mu_0 \end{cases}$$ where $f \in \mathcal{C}\left([t_0, T] \times \mathbb{R}^n;\mathbb{R}^n ...
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Strange oscillations in Matlab

I have created a midpoint algorithm to solve 2nd order ODEs in Matlab. And now Im comparing my solver with built in - ode23s. I have used a harmonic motion described as a 2nd order ODE, the .m file ...
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634 views

Modified Euler Method: region of absolute stability

I am having trouble finding the region of absolute stability for modified Euler method: \begin{align} w^*_{i+1}&=w_i+hf(t_i,w_i) \\ w_{i+1}&=w_i+\frac{h}2\left[f(t_i,w_i)+f(t_{i+1},w^*_{i+1})\...
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1answer
517 views

Implicit Euler method for linear first order ODE's

We have the linear first order ODE $y'(t) = m(t) y(t) + n(t)$ where $m(t), n(t)$ are given functions. We need to show that after applying the implicit Euler method we receive a linear system of ...
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208 views

Euler backward method

The ODE that leads to Figure shown is $y' = 10y$. The exact solution satisfies $y(0) = 1$, and Euler’s method is applied with step size $h = 0.1$. What are the initial values $y(0)$ for the other two ...