Questions tagged [eulers-method]
Euler's method is a numerical method to solve first-order first-degree differential equations with a given initial value.
187
questions
-4
votes
1
answer
43
views
numerical methods and chaos, eulers method in excel
this is my first post. I am trying to figure out this "exploration" for my high level college math class and I think trying to code in excel is messing me up.
"1. Consider the “simple” ...
0
votes
0
answers
22
views
Stability of Forward Euler in nonlinear ODE
I have the following ode: $y^{'}= \frac{k}{\sqrt{y}}$ where k is a positive value.
Applying the Forward Euler method gives the following: $v^{n}=\frac{\Delta t \ k}{\sqrt{v^{n-1}}}+v^{n-1}$
I'm ...
0
votes
0
answers
19
views
How to integrate arbitrary discrete-time linear and angular body fixed velocity to world space?
I have body fixed angular velocity values and linear acceleration values streaming in to my application. at some interval $\delta t$.
I need to get a world position from these, assuming the start ...
- 133
0
votes
0
answers
28
views
Absolute stability and numerical robustness.
I am having problem understanding the definition of absolute stability. One definition of absolute stability I have heard is
"A numerical solution $ w_n $ to a problem is absolute stable for a ...
2
votes
0
answers
176
views
Backward Euler method with spatial derivative terms
Consider the following differential equation:
$$\frac{\partial u(x,t)}{\partial t} = - \frac{\partial u(x,t)}{\partial x} + u(x,t) \tag{1}$$
with $u(x,0)=f(x)$. The solution of $(1)$, using MOC, is $e^...
- 338
0
votes
0
answers
72
views
Global truncation error of backward Euler method
It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like
$$ \frac{\exp(LT) -1}{L} \frac{Mh}{2},$$
with $L$...
- 3,252
1
vote
0
answers
51
views
Euler method coupled stochastic differential equations with time-derivative
Given a complex variable $\alpha(t)$ following this stochastic differential equation $\dot{\alpha}(t)=-\kappa\alpha+\sqrt{\kappa n_{\rm th}} \eta(t)$, where $\eta(t)$ describes a Winer process, we ...
- 99
0
votes
0
answers
41
views
Time Discretisation of a System of PDEs
Suppose we have an arbitrary system of PDEs
$$ \partial_t u - D_1 \Delta u + a(v) u = f(t)$$
$$\partial_t v - D_2 \Delta v + b(u) v = g(t) $$
We want to discretise the system in time. We use the ...
- 75
0
votes
1
answer
42
views
Trying to solve exercise about Euler method
Help to solve this exercise with Euler Method, my problem here problem is that the solutions I calculated are not close to the real values and they diverge. Thanks!
$$y'=e^y $$
with conditions $$ 0\...
- 13
0
votes
1
answer
98
views
Proving that the semi-implicit Euler method is symplectic
I'm having trouble understanding the following proof that the semi-implicit Euler method
$$\begin{cases} p^{k+1} = p^k-\Delta t \frac{\partial H}{\partial q}(p^{k+1},q^k) \\ q^{k+1} = q^k + \Delta t \...
- 1
4
votes
0
answers
137
views
Convergence of Euler scheme for ODEs
Consider the initial value problem for the ODE
\begin{align}
\frac{dy}{dt}&=f(y), \\
y(0)&=y_0,
\end{align}
where $f$ is a Lipschitz continuous function on $\mathbb{R}.$ Since $f$ is globally ...
- 506
0
votes
1
answer
78
views
Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method
Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method, knowing that $y(0)=1$, $t=[0,1]$, $h=0.25$
I know the local truncation error (LTE) introduced by the Euler method is given ...
0
votes
1
answer
39
views
euler method results have a huge error
I have The following IVP:
$$y'=-20y+20 \cos t-\sin t ,~~~ 0≤t≤2,~~~ y(0)=0;$$
With $N = 5$, $h = 0.4$, and $t_i=0.4i$, for $i = 0,1,2,3,4,5$.
I am performing the steps as following:
1st:
$y1$=$y0+...
- 11
0
votes
1
answer
90
views
Why are the costate equations solved backwards in time?
I'm trying to find an optimal control to a simple nonlinear SIR model. I am trying to undersand the Pontryagin minimum principle but I don't understand why the costate equations must be solved ...
- 148
0
votes
0
answers
50
views
Euler's method for differential equations (estimation)
Is it true that if a curve is increasing, Euler's method will always underestimate an actual solution?
So if a curve is either increasing and concave down, or increasing and concave up, we can simply ...
0
votes
0
answers
37
views
Understanding the stability of eulers method for a nonlinear system of differential equations and the relationship with the Jacobian.
So suppose I have the system of nonlinear differential equations:
$y'(t)=F(y(t))$
It looks like one equates the drive function to a local taylor approximation about $y(t_n)=y_n$ e.g
$F(y(t)) \approx F(...
- 653
1
vote
0
answers
47
views
The proof for the Euler decomposition of the VaR
For a portfolio-wide profit/loss variable $X= \sum_{i=1}^{n}w_iX_i$ the value-at-risk of $X$ at confidence level $\alpha$ (usually close to 1) is defined as the $\alpha$-quantile of $-X$:
\begin{...
- 191
1
vote
3
answers
272
views
Problem with numerical solving 2nd order nonlinear differential equation
I have this 2nd order nonlinear ODE:
$$ \frac{I_0}{C} \left[ e^{\frac{1}{2nU_T} \left(U_{in}(t) - U_{out}(t) - RCU'_{out}(t) - \frac{R}{R_z} U_{out}(t) -LCU''_{out}(t) - \frac{L}{R_z} U'_{out}(t) \...
- 53
0
votes
0
answers
22
views
Can I verify the weak order of convergence of a numerical scheme for SDEs by checking convergence in distribution?
let $X_{t=0}^T$ be a continuous-time stochastic process defined by a certain stochastic differential equation
$$
X_t=a(t,X_t)dt+b(y,X_t)dW_t
$$
where W_t is a Wiener process.
Let $\tilde X_{i=0}^{N}$ ...
- 237
0
votes
1
answer
62
views
For what $h$ is Euler's explicit method stable?
For which value of the discretization step $h$ the Euler explicit method applied to the differential equation $y'(x)+\frac{1}{4}y(x)=x$ with the initial condition $y(1)=1$ is stable?
I wrote the ...
- 53
0
votes
1
answer
41
views
Explanation to Euler's Method
I am trying to figure out Euler's method.
The initial value problem is :
$$P'(t)=0.7P(t)(1-\frac{P(t)}{750})-20, P(0)=30$$
The time step is set to $Δt=7$ days
For the algorithm we have:
$f(t,P)=0.7P(...
2
votes
1
answer
186
views
Doubts on the solution of a differential equation
I've encountered myself with a differential equation which I'm not sure that I've solved correctly.
The situation is the following: I have a set of parametric equations, which I'll call $\vec{r}$
\...
0
votes
0
answers
50
views
Implicit Euler's method for differential equations
So on my assignment, I have this first-order differential equations system:
R' = f(t,R,T)
J' = g(t,R,T)
R(0) = R0, J(0) = J0
( R(t), J(t) depend on t).
because the ...
0
votes
1
answer
49
views
Explicit/Implicit Euler method. Show that $ \lim_{k \rightarrow \infty} ||y_k||_2^2 = \infty $ and $ \lim_{k \rightarrow \infty} ||y_k||_2^2 = 0 $
We consider the following initial value problem:
$ y'(t) = \begin{pmatrix}
-y_{2}(t) \\
y_1(t)\\
\end{pmatrix} $ with $ y(0) = \begin{pmatrix}
a \\
0\\
\end{pmatrix} ...
- 55
0
votes
0
answers
65
views
Solve SIR model using Euler's method in C#
The SIR model is:
$$ \begin{aligned} \dot S &= −\beta I S\\ \dot I &= \beta I S − \gamma I \\ \dot R &= \gamma I \end{aligned} $$
I know how to solve SIR with Euler's method in C#. I don't ...
- 41
0
votes
0
answers
45
views
Euler method for ODE with noisy derivative
I am interested in numerically integrating a noisy differential equation:
$\frac{dx}{dt} = f(x,t) + \epsilon(t)$ where $\epsilon(t) \sim \mathcal{N}(\mu, \sigma^2)$. Is this a RODE or SODE?
How does ...
- 221
1
vote
2
answers
44
views
How does Euler's method work when step size and wanted values are different?
I have a question that goes as such:
Use Euler’s method with $dt= 0.1$, to estimate $p$ for these values of $t$, using initial condition $(0,5)$.
$t= 2,4,6,8,10,12,14,16$
$\frac{Dp}{dt}= 0.05(p-4)(...
- 11
0
votes
1
answer
258
views
Local truncation error vs Global truncation error
I know that global truncation error is proportional to $h^p$ while local truncation error is proportional to $h^{(p+1)}$, where $h$ is the step size.
But where does this relationship come from and how ...
- 341
0
votes
1
answer
43
views
Approximating second order differential equation with Euler's method
My equation is $x''=\frac{kx}{m}$, to apply Euler's method it is seperated into a system of equation consists of two first order ODE:
$\frac{dx}{dt}=v$
$\frac{dv}{dt}=-\frac{kx}{m}$
Lets say that the ...
- 341
0
votes
0
answers
65
views
How to find local truncation error in Forward Euler Method without knowing the actual value
In my high school project, I am solving differential equations using the forward Euler numerical method and this is because the equations were too hard to solve analytically.
To find the local error ...
- 21
0
votes
1
answer
20
views
Forward Euler and 1st-order linear ODE with exponential solution
Let us consider the simple ODE $y'=y$ with $y(0)=1$ on the interval $[0,1]$. The solution is obviously $y(t)=\mathrm{e}^t$. Now, consider a Forward Euler approximation of the solution to the ODE. The ...
- 1,191
0
votes
1
answer
114
views
Estimating pi with Euler's method and Runge-Kutta 4
I'm trying to estimate pi by solving the IVP
$y'' + y = 0$
where
$y(0) = 1, y'(0) = 0$
numerically by defining $\frac{\pi}{2}$ as the first value on t such that $y(t) = 0$
I'm trying to solve this ...
- 5
1
vote
1
answer
87
views
Where can I find "detailed" error analysis of modified Euler's methods?
I'm studying the local truncation error of each Heun's, Midpoint, and Ralston's methods. For Heun's method, I found a material in here. However, I don't get how the following is derived.
$$
f(t+h, y(t+...
- 97
1
vote
2
answers
70
views
Euler method without defined function
Before I begin, I must say that reading and quickly understanding the definitions of mathematical theorems is not my strongest suit.
We had a test where one of the subjects was Eulers methods. The ...
- 11
0
votes
1
answer
102
views
How is the global truncation error and stability criterion of the forward Euler method consistent with each other?
The forward Euler method can be used to solve ODE's with some initial value by using the update rule
$$y_{k+1} = y_k + hf(t_k, y_k),$$
where $0\leq k \leq n-1$ and $h$ is the step size $\dfrac{t_n-t_0}...
- 13
0
votes
0
answers
172
views
Logistic equation - How to go from continuous form to discrete form?
Logistic equation in continuous form:
$\frac{\mathrm{d} y}{\mathrm{d} t} = ry(1 - ay)$ (Autonomous Differential Equations and Population Dynamics, equation 6 in Boyce Diprima's book, eleventh Edition)...
- 153
1
vote
1
answer
396
views
Backwards Euler Method's error converges to step size (Python)
I am trying to programmatically solve the ODE,
$$\displaystyle Y'( t) =-Y( t) +\frac{1}{1+t^{2}} +\tan^{-1}( t)$$
with initial condition $Y(0)=0$. I know the analytical solution is $\displaystyle Y( t)...
- 13
1
vote
0
answers
66
views
Using Euler's integration to solve a problem that is not written as a derivative
I am brand new to solving differential equations. I am taking a modeling course that is asking me to use Euler's method to solve for the following:
$y = \displaystyle \int_{0}^{2} (x^3) dx$
However, ...
0
votes
1
answer
87
views
Implicit Euler to solve this $y(x_1)$
$$y_{n+1} = y_n +hf(h,y_{n+1})$$
How do I use implicit Euler to solve this $y(x_1)$?
$$y' = \frac{-x}{y^2}$$
$$y(0) = 1$$
$$h=0.1 $$
$$x_1=0.1$$
I have got this far:
$$y_1 = y_0 + hf(h,y_1) = y_0-\...
- 21
0
votes
1
answer
28
views
how does this example of runge kutta translate to calculating forces
I saw this example of runge kutta online, I use the unity game engine, and am working on making softbody physics (squishy things like a sponge).
It was working, until you run it for like a minute or ...
- 197
0
votes
0
answers
46
views
Can you derive the limit definition of an integral with an Euler's method construction?
If you were to try to approximate the value of $f(x+3\Delta x)$, you could use Euler's method to make 3 linear approximations.
$$f(x + \Delta x) \approx f(x) +\frac{df}{dx}(x)*\Delta x $$
$$f(x + 2\...
- 157
0
votes
1
answer
630
views
Rotation matrix decomposition into mixed local/global Euler angles
I am trying to decompose a rotation matrix into Euler angles.
The standard ways appear to work fine to find the rotations around the moving/local axes (example 1) OR to find the rotations around the ...
- 3
2
votes
1
answer
62
views
Confusion about Finite Differences Method Notation
I would need a confirmation that I am understanding one of the steps in the lecture notes about numerical methods right. We have an ODE with $y\in C^1(t)$:
\begin{align}
y'(t) &= f(t, y(t))\\
y(...
- 21
3
votes
1
answer
530
views
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 ...
- 139
0
votes
1
answer
56
views
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{\...
11
votes
1
answer
593
views
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 ...
user999691
0
votes
0
answers
115
views
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 ...
- 13
0
votes
2
answers
739
views
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 ...
- 874
3
votes
0
answers
100
views
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 ...
- 1,535
1
vote
0
answers
122
views
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 ...
- 19