Questions tagged [eulers-method]

Euler's method is a numerical method to solve first-order first-degree differential equations with a given initial value.

Filter by
Sorted by
Tagged with
6
votes
2answers
9k views

Local vs global truncation error

I was reading about local and global truncation error, and, I must be honest, I'm not really getting the idea of the two and what's the difference. Lets focus on the forward Euler method in ...
4
votes
1answer
2k views

A nice way to do Euler's method on a calculator?

As part of the calculator paper for IB (International Baccalaureate), we may be asked to do Euler's method, for say, 10 iterations. While it is feasible to do with a calculator (slightly easier if ...
3
votes
1answer
183 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 ...
2
votes
1answer
528 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 ...
2
votes
1answer
28 views

How do I calculate an approximation of the solution of the begining values at the point t = 1 with Euler method?

I have already posted question where I was asking about sketching Euler method. The explicit Euler method for numerically solving the begining values of differential equation $x′=f(t,x),x(t_0)=x_0$ ...
2
votes
1answer
371 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 ...
2
votes
1answer
141 views

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 ...
2
votes
0answers
72 views

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 $0\leq x\leq 1,-\infty <y<\infty$. ...
2
votes
1answer
89 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\...
1
vote
1answer
103 views

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 ...
1
vote
1answer
28 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 ...
1
vote
1answer
65 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-(...
1
vote
1answer
605 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 ...
1
vote
1answer
209 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[...
1
vote
2answers
152 views

Euler's method.

So for my assignment I have to code a program to solve first order ODE's using Euler's Method. My program works, it returns the right values. (I checked using an online calculator). However, solution ...
1
vote
1answer
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 ...
1
vote
2answers
70 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}=...
1
vote
1answer
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 ...
1
vote
1answer
807 views

Finding correct step-size for the Euler method

I don't understand how to find the correct step-size $h$ for the Euler method. My script says the following: One method consists in computing the numerical solution for an arbitrary $h$ and then $...
1
vote
1answer
84 views

Understanding definition of explicit Euler method

I'm quite new to ODE and the IVP (initial value problem), so I've some doubts related to these topics. I'm reading a draft provided by my professor which talks about the "polygon" or explicit Euler ...
1
vote
1answer
388 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 ...
1
vote
1answer
287 views

How to rewrite a 2nd order nonlinear ODE into an initial value problem of first order?

I am trying to rewrite the initial value problem $x''(t) = -sin(x(t))$ with initial values $x(0)= \pi/2 $ and $x'(0) = 0 $ into and initial value problem of first order ODE system. Later I will have ...
1
vote
1answer
49 views

Numerical Solution for ODE with a squared term inhibiting use of Euler's Method

I have this ODE: $$A(x)\ddot{x}+B(x)\dot{x}^2+C(x) = 0$$ $A,B,C$ are functions of $x(t)$. Usually I'd solve this numerically using Euler's method but in this case the $\dot{x}^2$ is giving me ...
1
vote
1answer
132 views

Eulers formula with an infinite series?

Alright so this is a real life problem and not just a homework thing. Ive borrowed money from a family member $16323 \rm dkk$ to be exact. Im borrowing this money for $211$ days and im borrowing it ...
1
vote
1answer
130 views

What is meant by Adams Bashforth being a “boot strap” method?

People seem to say that the Adams-Bashforth method requires some "boot strapping" because it needs two initial conditions: $y_{n+1}=y_n+\frac{\Delta t}{2}[3f(t_n,y_n) - f(t_{n-1}, y_{n-1})]$ I ...
1
vote
1answer
99 views

How to understand where I am going wrong in euler method

I have a question from my book and it says essentially, consider the IVP $x^{\bullet}=-x$ with $x(0)=1$, what is the exact value of x(1), then using Eulers method with step size1 , estimate $x(1)$ ...
1
vote
1answer
37 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 ...
1
vote
0answers
35 views

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 ...
1
vote
0answers
132 views

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(...
1
vote
0answers
19 views

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 ...
1
vote
0answers
37 views

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 ...
1
vote
1answer
118 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 ...
1
vote
1answer
47 views

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 ...
1
vote
2answers
625 views

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 ...
1
vote
0answers
177 views

Stability analysis for eulers method for second order differential equation

I need an example for stability analysis of Euler's method for second order ODE.. thanks alot
1
vote
1answer
70 views

Using Eulers method to solve equation of motion of aircraft

I am told I can solve the following differential equation by utilising Euler's method. I can solve simple differential equations with the method, I am however, having some trouble wrapping my head ...
1
vote
1answer
345 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{...
0
votes
2answers
99 views

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)...
0
votes
1answer
57 views

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?
0
votes
1answer
270 views

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 ...
0
votes
1answer
879 views

Error Bound for Euler's Method

I'm studying the Euler Method trough the book "Numerical Analysis", but I didn't understand an example where we have to calculate the error of this method... First of all we have a Corollary which ...
0
votes
1answer
28 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 ...
0
votes
1answer
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 ...
0
votes
1answer
43 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 ...
0
votes
1answer
45 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: ...
0
votes
2answers
43 views

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-...
0
votes
1answer
224 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 ...
0
votes
1answer
181 views

Using Euler's relation to transform to cosine

input is $f_2(t) = Acos(w_0 t + \phi)$ output $y_2(t)$ is $\frac{A}{2} e^{j \phi} H(jw_0) e^{jw_0 t} + \frac{A}{2} e^{-j \phi} H(-jw_0) e^{jw_0 t} $ okay so I need the step in between, how were the ...
0
votes
1answer
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 ...
0
votes
1answer
32 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 ...