Recent questions in Euler's Method

Integral CalculusAnswered question

elynnea4xl 2023-02-24

Using Euler's formula, find the unknown: $\begin{array}{|cccc|}\hline Faces& ?& 5& 20\\ Vertices& 6& ?& 12\\ Edges& 12& 9& ?\\ \hline\end{array}$

Integral CalculusAnswered question

Carina Nash 2023-01-07

How to use DeMoivre's Theorem to find $(1+i)}^{20$ in standard form?

Integral CalculusAnswered question

mooltattawsmq8 2023-01-03

How does "e" (2.718) help apply to applications/implications in real life?

Integral CalculusAnswered question

John Noble 2022-12-23

According to Euler’s formula for any solid, the number of Faces (F) and vertices (V) added together is exactly two more than the number of edges (E).

Integral CalculusAnswered question

unecewelpGGi 2022-11-27

Use Euler methed to estimate the value of ( 2 loops), where $y$ is the solution of the initial value problem ${y}^{\prime}=1-2xy,h=0,1,y(1)=0.538$

Integral CalculusAnswered question

Justine Pennington 2022-11-22

I am going to program Eulers method in Octave to approximate gravity in 1-dimension. I understand the formula for Eulers method, which is equal to:

${y}_{i+1}={y}_{i}+(\mathrm{\u25b3}t\star f({t}_{i},{y}_{i}))$

What I don't understand is what my function f(t,y) is in this case. What do I have to insert into the formula to get my next y-point?

${y}_{i+1}={y}_{i}+(\mathrm{\u25b3}t\star f({t}_{i},{y}_{i}))$

What I don't understand is what my function f(t,y) is in this case. What do I have to insert into the formula to get my next y-point?

Integral CalculusAnswered question

Aleah Avery 2022-11-20

I am trying to calculate the global error bound for Euler's method, but I am having trouble. I am given the formula $|y({t}_{i})-{u}_{i}|\le \frac{1}{L}(\frac{hM}{2}+\frac{\delta}{h})({e}^{L({t}_{i}-a)}-1)+|{\delta}_{0}|{e}^{L({t}_{i}-a)}$ where ${u}_{i}$ is the Euler approxmation. I am also given $M,L,a,\delta ,{\delta}_{0}$, h. If I am not mistaken this will give me the error for each step, but how do I find the upper bound for the total error?

Integral CalculusAnswered question

Nico Patterson 2022-11-18

What is the reason behind the multiplication of the function's derivative with the step size (and the subsequent addition) in the numerical Euler method?

${y}_{n+1}={y}_{n}+hf({t}_{n},{y}_{n})$

I can't figure out why exactly this would work for generating a new value. How can scaling the output of the function $f$ and subsequently adding it to the prior value ${y}_{n}$ approximate a new value? Is there some formal or graphical explanation of this? The formula seems somewhat counterintuitive.

${y}_{n+1}={y}_{n}+hf({t}_{n},{y}_{n})$

I can't figure out why exactly this would work for generating a new value. How can scaling the output of the function $f$ and subsequently adding it to the prior value ${y}_{n}$ approximate a new value? Is there some formal or graphical explanation of this? The formula seems somewhat counterintuitive.

Integral CalculusAnswered question

kaltEvallwsr 2022-11-13

I have to write application which solves task presented below. I only know some c# so I will stick to it. It is some kind of homework but I am asking for help with understanding this and advice for implementation.

Take natural $\text{}n=1$ and real $\text{}a1$

differential problem

$\text{}{y}^{\prime}(x)=f(x,y(x)),y(1)=0$

where $\text{}f(x,y)=\sqrt{y+1}+x$

solve for $\text{}[1,a]$ with step $\text{}h=\frac{a-1}{n}$ with Euler and Heun methods.

Compare results with exact solution $\text{}y(x)={x}^{2}-1$ by finding maximum error of each method.

Take natural $\text{}n=1$ and real $\text{}a1$

differential problem

$\text{}{y}^{\prime}(x)=f(x,y(x)),y(1)=0$

where $\text{}f(x,y)=\sqrt{y+1}+x$

solve for $\text{}[1,a]$ with step $\text{}h=\frac{a-1}{n}$ with Euler and Heun methods.

Compare results with exact solution $\text{}y(x)={x}^{2}-1$ by finding maximum error of each method.

Integral CalculusAnswered question

evitagimm9h 2022-11-13

We are interested in solving the advection equation ${u}_{t}={u}_{x}$ where $0\le x<1$, $t\ge 0$

with periodic boundary conditions and $u(x,0)=f(x),f(x)=f(x+1)$ In the grid ${x}_{0},{x}_{1},\dots {x}_{N}$, ${t}_{0},{t}_{1},\dots {t}_{N}$.

using the forward euler method, or $\frac{v({x}_{j},{t}_{n+1})-v({x}_{j},{t}_{n})}{\mathrm{\Delta}t}=\frac{v({x}_{j+1},{t}_{n})-v({x}_{j-1},{t}_{n})}{2\mathrm{\Delta}x}$ where $v$ is the approximation of $u$ at the grid points.

What I don't understand:

We are essentially using $v({x}_{j-1},{t}_{n}),v({x}_{j},{t}_{n}),v({x}_{j+1},{t}_{n})$ to calculate $v({x}_{j},{t}_{n+1})$. Initially this is fine because $v({x}_{j},0)$ is known for all $j$, but how would we calculate $v({x}_{0},{t}_{1})$? we can't, since that would require us to know $v({x}_{-1},{t}_{0})$ which doesn't exist. And this issue propagates, each new ${t}_{n}$ we can calculate one less point than ${t}_{n-1}$.

What am I missing?

with periodic boundary conditions and $u(x,0)=f(x),f(x)=f(x+1)$ In the grid ${x}_{0},{x}_{1},\dots {x}_{N}$, ${t}_{0},{t}_{1},\dots {t}_{N}$.

using the forward euler method, or $\frac{v({x}_{j},{t}_{n+1})-v({x}_{j},{t}_{n})}{\mathrm{\Delta}t}=\frac{v({x}_{j+1},{t}_{n})-v({x}_{j-1},{t}_{n})}{2\mathrm{\Delta}x}$ where $v$ is the approximation of $u$ at the grid points.

What I don't understand:

We are essentially using $v({x}_{j-1},{t}_{n}),v({x}_{j},{t}_{n}),v({x}_{j+1},{t}_{n})$ to calculate $v({x}_{j},{t}_{n+1})$. Initially this is fine because $v({x}_{j},0)$ is known for all $j$, but how would we calculate $v({x}_{0},{t}_{1})$? we can't, since that would require us to know $v({x}_{-1},{t}_{0})$ which doesn't exist. And this issue propagates, each new ${t}_{n}$ we can calculate one less point than ${t}_{n-1}$.

What am I missing?

Integral CalculusAnswered question

Zackary Diaz 2022-11-13

Use the Euler Method to write down an iterative algorithm so that ${y}_{n+1}=y({t}_{n+1})$ can be determined from ${y}_{n}=y({t}_{n})$, where ${t}_{n}=n\mathrm{\u25b3}t$ is the size of the time step, for the following Ordinary Differential Equation $\frac{dy}{dt}={y}^{2}+\mathrm{sin}(t)$

I'm confused about this question because the question's I'm been doing upto this one is that I am given an initial $({x}_{0},{y}_{0})$ and then I draw up a table to find a $y$ value for a corresponding $x$ value.. so this question is very confusing as I have never came across a question such as this one.

I'm confused about this question because the question's I'm been doing upto this one is that I am given an initial $({x}_{0},{y}_{0})$ and then I draw up a table to find a $y$ value for a corresponding $x$ value.. so this question is very confusing as I have never came across a question such as this one.

Integral CalculusAnswered question

Siena Erickson 2022-11-12

Since we have a simple conversion method for converting from radians to degrees, $\frac{180}{\pi}$ or vice versa, could we apply this to Euler's Identity, ${e}^{i\pi}=-1$ and traditionally in radians, to produce an equation that is in degrees; one that may or may not be as simplistic or beautiful as the radian version, but yet is still quite mathematically true?

If not, why? If so, how did you derive it?

If not, why? If so, how did you derive it?

Integral CalculusAnswered question

Nola Aguilar 2022-11-11

I'm trying to solve this 1st order ODE numerically by bringing it into an explicit form, but I don't think it is valid because of the dependency on x_n in the final expression.

$\frac{dy}{dx}+x=0\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}=-x\phantom{\rule{0ex}{0ex}}$

Alpha is the angle from point with index (n) to point with index (n+1).

$tan(\alpha )=\frac{dy}{dx}\phantom{\rule{0ex}{0ex}}tan({\alpha}_{n})=-{x}_{n}\phantom{\rule{0ex}{0ex}}$

I call h the step size.

$tan({\alpha}_{n})=\frac{{y}_{n}-{y}_{n-1}}{{x}_{n}-{x}_{n-1}}\phantom{\rule{0ex}{0ex}}tan({\alpha}_{n})=\frac{{y}_{n}-{y}_{n-1}}{h}\phantom{\rule{0ex}{0ex}}$

When I rearrange this I obtain the following form.

${y}_{n}={y}_{n-1}+h\cdot tan({\alpha}_{n})\phantom{\rule{0ex}{0ex}}{y}_{n}={y}_{n-1}+h\cdot (-{x}_{n})\phantom{\rule{0ex}{0ex}}{y}_{n}={y}_{n-1}-h\cdot {x}_{n}$

Is this the final numerical solution of this 1st order ODE?

${y}_{n}={y}_{n-1}-h\cdot {x}_{n}\phantom{\rule{0ex}{0ex}}$

EDIT: I've brought it to a better form I think.

${y}_{1}={y}_{0}-h\cdot {x}_{1}\phantom{\rule{0ex}{0ex}}{y}_{2}={y}_{1}-h\cdot {x}_{2}={y}_{0}-h\cdot {x}_{1}-h\cdot {x}_{2}\phantom{\rule{0ex}{0ex}}{y}_{3}={y}_{2}-h\cdot {x}_{3}={y}_{0}-h\cdot {x}_{1}-h\cdot {x}_{2}-h\cdot {x}_{3}\phantom{\rule{0ex}{0ex}}$

So that the final solution is the following.${y}_{n}={y}_{0}-h\cdot ({x}_{1}+...+{x}_{n})\phantom{\rule{0ex}{0ex}}$

$\frac{dy}{dx}+x=0\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}=-x\phantom{\rule{0ex}{0ex}}$

Alpha is the angle from point with index (n) to point with index (n+1).

$tan(\alpha )=\frac{dy}{dx}\phantom{\rule{0ex}{0ex}}tan({\alpha}_{n})=-{x}_{n}\phantom{\rule{0ex}{0ex}}$

I call h the step size.

$tan({\alpha}_{n})=\frac{{y}_{n}-{y}_{n-1}}{{x}_{n}-{x}_{n-1}}\phantom{\rule{0ex}{0ex}}tan({\alpha}_{n})=\frac{{y}_{n}-{y}_{n-1}}{h}\phantom{\rule{0ex}{0ex}}$

When I rearrange this I obtain the following form.

${y}_{n}={y}_{n-1}+h\cdot tan({\alpha}_{n})\phantom{\rule{0ex}{0ex}}{y}_{n}={y}_{n-1}+h\cdot (-{x}_{n})\phantom{\rule{0ex}{0ex}}{y}_{n}={y}_{n-1}-h\cdot {x}_{n}$

Is this the final numerical solution of this 1st order ODE?

${y}_{n}={y}_{n-1}-h\cdot {x}_{n}\phantom{\rule{0ex}{0ex}}$

EDIT: I've brought it to a better form I think.

${y}_{1}={y}_{0}-h\cdot {x}_{1}\phantom{\rule{0ex}{0ex}}{y}_{2}={y}_{1}-h\cdot {x}_{2}={y}_{0}-h\cdot {x}_{1}-h\cdot {x}_{2}\phantom{\rule{0ex}{0ex}}{y}_{3}={y}_{2}-h\cdot {x}_{3}={y}_{0}-h\cdot {x}_{1}-h\cdot {x}_{2}-h\cdot {x}_{3}\phantom{\rule{0ex}{0ex}}$

So that the final solution is the following.${y}_{n}={y}_{0}-h\cdot ({x}_{1}+...+{x}_{n})\phantom{\rule{0ex}{0ex}}$

Integral CalculusAnswered question

charmbraqdy 2022-11-08

Hy. Can someone please explain me how can I resolve ${\int}_{0}^{\mathrm{\infty}}{e}^{-(sI-A)t}\phantom{\rule{thinmathspace}{0ex}}dt$ ? I must have the final result $(sI-A{)}^{-1}$. I think it has the euler form, for beta integral but I don't know how can I get to that result.

Integral CalculusAnswered question

Emmanuel Giles 2022-11-08

I need to find the following integral:

$\int \frac{1}{(1+x)\sqrt{1+x-{x}^{2}}}dx.$

I tried using Euler's formula and put $xt+1=\sqrt{1+x-{x}^{2}}$ and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

$\int \frac{1}{(1+x)\sqrt{1+x-{x}^{2}}}dx.$

I tried using Euler's formula and put $xt+1=\sqrt{1+x-{x}^{2}}$ and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

Integral CalculusAnswered question

Kenna Stanton 2022-11-06

$\frac{dx}{dt}=\omega \sqrt{A-{x}^{2}},$

where $A,\omega >0$ and $x={x}_{0}$ at $t=0$.

It is to be solved from $t=0$ to $t=50.0$. It has analytical solution

$x(t)=\sqrt{A}\mathrm{sin}(\omega t+\phi ),$

where $\mathrm{sin}(\phi )=\frac{{x}_{0}}{\sqrt{A}}$ if ${x}_{0}\le \sqrt{A}$.

The question I am trying to solve is the following

Rewrite the differential equation you have been given at the start of this document in the correct form for applying the Euler and Euler-Cauchy numerical schemes. Write down an appropriate 1 Euler method recursive scheme to solve this differential equation for the following values of the parameters and initial conditions:

$\omega =3.1$, $A=10.0$, ${x}_{0}=2.0$.

where $A,\omega >0$ and $x={x}_{0}$ at $t=0$.

It is to be solved from $t=0$ to $t=50.0$. It has analytical solution

$x(t)=\sqrt{A}\mathrm{sin}(\omega t+\phi ),$

where $\mathrm{sin}(\phi )=\frac{{x}_{0}}{\sqrt{A}}$ if ${x}_{0}\le \sqrt{A}$.

The question I am trying to solve is the following

Rewrite the differential equation you have been given at the start of this document in the correct form for applying the Euler and Euler-Cauchy numerical schemes. Write down an appropriate 1 Euler method recursive scheme to solve this differential equation for the following values of the parameters and initial conditions:

$\omega =3.1$, $A=10.0$, ${x}_{0}=2.0$.

Integral CalculusAnswered question

pin1ta4r3k7b 2022-11-04

$\int {e}^{-3t}cos(2-\sqrt{3}t)dt$

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) etc. This is the first time I have come across a question where its of the format (cos a+bx) and cannot understand how to deal with the extra term. I have been stuck for a while now and have thus decided to ask the community for help, please help me out.

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) etc. This is the first time I have come across a question where its of the format (cos a+bx) and cannot understand how to deal with the extra term. I have been stuck for a while now and have thus decided to ask the community for help, please help me out.

Integral CalculusAnswered question

Amy Bright 2022-11-03

I have an Euler method that has this form:

$\hat{I}({t}_{n+1})=\hat{I}({t}_{n})+h\beta \hat{I}({t}_{n})[1-\frac{\hat{I}({t}_{n})}{N}]$

which can also be written like

$\hat{I}({t}_{n+1})=\varphi (\hat{I}({t}_{n}))$

where $\varphi (x)$ is the iteration function down below:

$\varphi (x)=x+h\beta x(1-\frac{x}{N})$

I use $h=6$ in this method but if I use a $h$ which is a little bit bigger (for example $h=20$), I have an absolute instability error. I want to find the value of $h$ from which this absolute error is shown?

$\hat{I}({t}_{n+1})=\hat{I}({t}_{n})+h\beta \hat{I}({t}_{n})[1-\frac{\hat{I}({t}_{n})}{N}]$

which can also be written like

$\hat{I}({t}_{n+1})=\varphi (\hat{I}({t}_{n}))$

where $\varphi (x)$ is the iteration function down below:

$\varphi (x)=x+h\beta x(1-\frac{x}{N})$

I use $h=6$ in this method but if I use a $h$ which is a little bit bigger (for example $h=20$), I have an absolute instability error. I want to find the value of $h$ from which this absolute error is shown?

Integral CalculusAnswered question

blackdivcp 2022-10-31

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 approach this.

e.g. $\frac{dy}{dx}=\frac{y}{\sqrt{x}},y(0)=1$

(Obviously this can be solved analytically but I would like to know if there is any numerical method that tackles problems like this)

e.g. $\frac{dy}{dx}=\frac{y}{\sqrt{x}},y(0)=1$

(Obviously this can be solved analytically but I would like to know if there is any numerical method that tackles problems like this)

Integral CalculusAnswered question

ndevunidt 2022-10-23

I have this differential equation ${y}^{\prime}=\frac{y(\mathrm{sin}t)}{t}$, $y(0)=2$, and $h=\frac{1}{4}$. The first set of values, the inital, are (0,2). For the next iteration would it be ${y}_{1}=2+(\frac{1}{4})\left(\frac{2\cdot \mathrm{sin}(0)}{0}\right)$? I know you can't divide by zero, so what would should I do in this case?

When you are dealing with Euler's method problems, it means that you are focusing on the technique that is used to analyze differential equations. The idea is to use the local linearity that is also known as the linear approximation. Take a look at Euler's method equation where you implement small tangent lines over a short distance. Then continue with the differential equation rewriting. It will help you to approximate the solution when you have an initial-value challenge. While you’re at it, you can look at the various questions related to computational science and mathematics based on the provided solutions.