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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An error bound for the Euler method

The question is from the chapter about numerical integration in Stoll. enter image description here The problem says that we can find a bound when we use the areas of rectangles to approximate an ...
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Calculating brownian motion in Eulers method

Say for example we have brownian motion term added to an expression $W(t)$ and our $t=2$, what would $W(2)$ be? I' ve tried searching the internet but I can't seem to find a way that gives a formula ...
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Euler's method produces mirrored solution?

I have the differential equation $$ 0.22y'' + y' + 10000y = 2000\pi \cos (2000 \pi t) $$ from this I get the Euler's method system of equations: $$ t_{n+1} = t_n + 0.001 $$ $$ x_{n+1} = x_n + 0.001\...
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What is the error of $x_i$ at a fixed time $t_i$ when using Euler's method for the IVP $\frac{dx}{dt}=x$ with $x(0)=1$?

So we have Euler's method: $x_{i+1}=x_i+hf(t_i,x_i)$ where $x'=f(t,x)$ and $h$ is a small step size. Applying this method to $x'=x$ gives $x_{i+1}=x_i+hx_i=(1+h)x_i$. Solving this difference equation ...
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33 views

Uniqueness of solution of differential equation

Consider the differential equation $ y'(x)=f(x,y(x)) $ with initial condition $y(x_0)=y_0$. According to some theorem, if $f$ is continuous, then there exists at least one solution that satisfies the ...
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43 views

System of differential equation, Euler's method

Does anyone know if there is any calculator or how can I input this system of differential equation in wolfram? $$\frac{dy_1}{dt}=0.279\left[-0.125y_1+2y_2-1.25y_C+1.25e\right]$$ $$\frac{dy_2}{dt}=-0....
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63 views

Hamilton equations-Symplectic Euler method

We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta ...
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Is it possible for me to model mathematically the revolution to sharpen a pencil using complex number?

I was planning to do research on complex number since I find the topic intriguing, How can I conduct a research by modelling mathematically to calculate the required number of rotation for a pencil in ...
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32 views

What does the order of an integration method means

My question is how do you define the order of an integration method. I know, that eulers method $$y_{k+1}=y(t)+h\cdot f(y,t)$$ And I also know that it is a first-order integration method. but i don't ...
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118 views

Taylor polynomials in Horner Form

Professional high school math teacher/ amateur mathematician here looking for feedback on my attempt to to derive the general case of a Maclaurin polynomial using finite difference methods. Here's my ...
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Verify order of convergence of the Euler--Maruyama schemenumerically

I have a question about the order of convergence of the Euler-Maruyama scheme and how one verifes this numerically. I have read that the Euler-Maruyama scheme is mean-squared convergent of order 1/2 ...
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38 views

Improved Euler's Method 2nd order ODE ,not sure if this is right

SO i have this differential equation and actually i am not sure if i have solved it right with Euler's improved method : $z'' = f_{z} - C_{z}*|z'|*z'$ $ z' = u$ $u' = z'' = f_{z}-C_{z}*|u|*u$ Improved ...
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40 views

Results better without adaptive $h$ than with adaptive $h$ when using RKF45?

I'm trying to solve a fairly simple ODE, $$-y'=t^{-2} +4(t-6)e^{-2(t-6)^2} ,~~~ y(1)=1~\text{ for }~t\in[0,10].$$ Via the Runge-Kutta-Fehlberg method. I don't understand why, but the solutions i get ...
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25 views

Are the sequences $\frac{ih}{2}((1-ih)^{k}-(1+ih)^{k})$ and $\frac{1}{(1+h^{2})^{k}}\frac{ih}{2}((1-ih)^{k}-(1+ih)^{k})$ bounded?

I want to understand the explicit and the implicit Euler method better. Assume I have the initial value problem $y''+y=0$, $y(0)=0$, $y'(0)=1$, which is of course solved by $y=\sin(x)$, and I convert ...
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66 views

Convert ODE to first-order form

I am struggling to understand reducing higher order DE to first-order form. Say I have an equation of motion, $$m \frac{d^2}{dt^2} \left( \begin{array}{c} x \\ y \end{array} \right) = -\frac{GmM}{{(...
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Use Euler's Formula to prove the following equation $ y = Ae^{\sqrt{\lambda}ix} + Be^{-\sqrt{\lambda}ix} $

I'm having a difficult time understanding how one would take $ y = Ae^{\sqrt{\lambda}ix} + Be^{-\sqrt{\lambda}ix} $ and end up with $ y = A \cos{\sqrt{\lambda}x} + B\sin{\sqrt{\lambda}x} $ knowing ...
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53 views

What conditions does Euler midpoint and the classical fourth-order Runge-Kutta method; is absolutely stable

Question: Given the differential equation and its initial condition as $$dy/dx =-20xy^2 \quad\text{and} \quad y(1)=1$$ respectively, determine, under what conditions each of: (i) Euler’s method; ...
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106 views

As step size $h$ decreases, which method is more efficient? Euler-midpoint method and the classical fourth-order Runge-Kutta method?

I came up with the following problem while using Euler-midpoint method and the classical fourth-order Runge-Kutta method to solve ordinary differential equations. As step size $h$ decreases, ...
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55 views

Derive local truncation error for Improved Euler

I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, ...
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49 views

Back in time Euler–Maruyama method

Given a stochastic differential equation $$ dX(t) = a(X(t), t) \, dt + b(X(t),t) \, dW(t), \qquad (1) $$ we can solve it forward in time with Euler–Maruyama scheme for a finite time step $\Delta t$: $$...
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Using Euler's method to compute the frequency of a nonlinear pendulum

In my studies of numerical methods I have come across the following exercise: We consider the following second-order ODE $$\ddot{\theta}+\sin(\theta) = 0 $$ and we reduce it to a two-dimensional ...
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95 views

Lower bound for $(1+x)^n$

I am trying to prove the following bound for $n \in N$ and $x \in [-1,0)$, but x closer to 0: $ e^{nx} - \frac{1}{2}nx^{2}e^{(n-1)x} \leq (1+x)^n $ I have: $ e^{nx} - \frac{1}{2}nx^{2}e^{(n-1)x} = (1 -...
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119 views

Euler's method to approximate a differential equation $\frac{dy}{dx} = x - y$

Question: Use Euler's method to find approximate values for the solution of the initial value $-$ problem $$\frac{dy}{dx} = x-y$$ $$y(0)=1$$ on the interval $[0,1]$ using five steps of size $h = 0.2$. ...
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63 views

Higher-order corrections for Euler's method

I would like to preface my question by confessing that I come from a Physics background, so I apologize for any abuse of notation. Given a 1st order ODE $$ y' = f(x, y) $$ we can use Euler's Method to ...
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111 views

Finding global error in Modified Euler method

I am doing some questions on modified Euler method (=Heun, expl. trapezium) and would appreciate some help with this one. The differential equation is $$\frac{dy}{dx}+2y=e^{3t}, ~~0\leq t\leq 1,~~ y(0)...
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Definition of a priori and a posteriori Error Estimates for General PDE

My question will be written rather loosely because I can only imagine any answer will be very general. Given some PDE : $$ u'(t)= -\nabla \phi(u(t)),~~u(0)=u_0, $$ what do authors mean when they ...
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$y'$ is being eliminated as I solve using Eulers formula what to do?

Investigate the extremals of the functional $\int((y^2 + (x^2)\cdot y'))dx$ with limits $0$ to $1$ under the conditions $y(0) = 0$, $y(1) = A$.
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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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133 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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526 views

1st order Runge-Kutta method and Euler's method

I was asked to work out a differential equation using the Euler method and then followed by the Runge-Kutta method. Based on the theory I have come across it says that the Euler method agrees with the ...
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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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99 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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79 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{d}...
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48 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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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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56 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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113 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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139 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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86 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'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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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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88 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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181 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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148 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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51 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 ...