Aubrie Mccall

2022-09-06

Consider the following problem

${y}^{\prime}=\mathrm{ln}\mathrm{ln}(4+{y}^{2}),\phantom{\rule{1em}{0ex}}x\in [0,1],y(0)=1$

We can formulate the problem the approximate the solution

${y}_{n+1}={y}_{n}+h\mathrm{ln}\mathrm{ln}(4+{y}_{n}^{2}),\phantom{\rule{2em}{0ex}}n=0,1,...,N-1,\phantom{\rule{1em}{0ex}}{y}_{0}=0$

with mesh points ${x}_{n}=nh$. I am tasked with finding the truncation error. Now considering a Taylor series we find that

$|{T}_{n}|\le \frac{h}{2!}|(\mathrm{ln}\mathrm{ln}(4+{\xi}^{2}){)}^{\prime}|\phantom{\rule{1em}{0ex}}\xi \in ({x}_{n},{x}_{n+1})$

Now

$\mathrm{ln}\mathrm{ln}(4+{y}^{2}{)}^{\prime}=\frac{d}{dy}\mathrm{ln}\mathrm{ln}(4+{y}^{2})=\frac{2y\mathrm{ln}\mathrm{ln}(4+{y}^{2})}{\mathrm{ln}(4+{y}^{2})(4+{y}^{2})}\le \frac{2y}{4+{y}^{2}}\le 1/2$

$\therefore |{T}_{n}|\le \frac{h}{4}$

Is this the correct way to go about obtaining this?

${y}^{\prime}=\mathrm{ln}\mathrm{ln}(4+{y}^{2}),\phantom{\rule{1em}{0ex}}x\in [0,1],y(0)=1$

We can formulate the problem the approximate the solution

${y}_{n+1}={y}_{n}+h\mathrm{ln}\mathrm{ln}(4+{y}_{n}^{2}),\phantom{\rule{2em}{0ex}}n=0,1,...,N-1,\phantom{\rule{1em}{0ex}}{y}_{0}=0$

with mesh points ${x}_{n}=nh$. I am tasked with finding the truncation error. Now considering a Taylor series we find that

$|{T}_{n}|\le \frac{h}{2!}|(\mathrm{ln}\mathrm{ln}(4+{\xi}^{2}){)}^{\prime}|\phantom{\rule{1em}{0ex}}\xi \in ({x}_{n},{x}_{n+1})$

Now

$\mathrm{ln}\mathrm{ln}(4+{y}^{2}{)}^{\prime}=\frac{d}{dy}\mathrm{ln}\mathrm{ln}(4+{y}^{2})=\frac{2y\mathrm{ln}\mathrm{ln}(4+{y}^{2})}{\mathrm{ln}(4+{y}^{2})(4+{y}^{2})}\le \frac{2y}{4+{y}^{2}}\le 1/2$

$\therefore |{T}_{n}|\le \frac{h}{4}$

Is this the correct way to go about obtaining this?

bequejatz8d

Beginner2022-09-07Added 6 answers

I believe that the only mistake is in your application of the chain rule. I find it helpful to break the function into its composite parts and apply the chain rule one step at a time.

Let $f,g,h$ be the functions given by

$\begin{array}{rl}f(y)& =4+{y}^{2}\\ g(y)& =\mathrm{ln}(y)\\ h(y)& =g(y).\end{array}$

Then

$k(y)=h(g(f(y))=\mathrm{ln}(\mathrm{ln}(4+{y}^{2})$

is the function of interest and $k$ is defined and differentiable for all $y$. Moreover,

${k}^{\prime}(y)={h}^{\prime}(g(f(y)){g}^{\prime}(f(y)){f}^{\prime}(y)=\frac{1}{\mathrm{ln}(4+{y}^{2})}\frac{1}{4+{y}^{2}}2y.$

The derivative ${k}^{\prime}$ can be bounded as follows

$|{k}^{\prime}(y)|\le \frac{1}{\mathrm{ln}(4)}\frac{2|y|}{4+{y}^{2}},$

because ln is monotone increasing. You have already applied the helpful inequality

$|ab|\le \frac{1}{2}({a}^{2}+{b}^{2}),$

which in our case, where $a=2$ and $b=|y|$, allows for the estimate

$|{k}^{\prime}(y)|\le \frac{1}{2\mathrm{ln}(4)}.$

Breaking complicated functions into composite parts is especially useful when programming computers. One line per component produces a program which is easy for a human being to debug/verify.

Let $f,g,h$ be the functions given by

$\begin{array}{rl}f(y)& =4+{y}^{2}\\ g(y)& =\mathrm{ln}(y)\\ h(y)& =g(y).\end{array}$

Then

$k(y)=h(g(f(y))=\mathrm{ln}(\mathrm{ln}(4+{y}^{2})$

is the function of interest and $k$ is defined and differentiable for all $y$. Moreover,

${k}^{\prime}(y)={h}^{\prime}(g(f(y)){g}^{\prime}(f(y)){f}^{\prime}(y)=\frac{1}{\mathrm{ln}(4+{y}^{2})}\frac{1}{4+{y}^{2}}2y.$

The derivative ${k}^{\prime}$ can be bounded as follows

$|{k}^{\prime}(y)|\le \frac{1}{\mathrm{ln}(4)}\frac{2|y|}{4+{y}^{2}},$

because ln is monotone increasing. You have already applied the helpful inequality

$|ab|\le \frac{1}{2}({a}^{2}+{b}^{2}),$

which in our case, where $a=2$ and $b=|y|$, allows for the estimate

$|{k}^{\prime}(y)|\le \frac{1}{2\mathrm{ln}(4)}.$

Breaking complicated functions into composite parts is especially useful when programming computers. One line per component produces a program which is easy for a human being to debug/verify.

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?