# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...