Chapter 26, Problem 1P

Given

$\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$

(a) Estimate the step-size required to maintain stability using the explicit Euler method.

(b) If $y\left(0\right)=0$, use the implicit Euler to obtain a solution from $t=0\text{ to 2}$ using a step size of 0.1.

To determine

To calculate: The step-size required to maintain stability of differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$ using the explicit Euler method.

Answer to Problem 1P

Solution:

The step-size required to maintain stability of the given differential equation is $h<1\times {10}^{-5}$.

Explanation of Solution

Given Information:

Differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$.

Formula used:

The stability of formula depends upon step size h and step size must satisfy the condition, $\left|1-ah\right|<1$.

Calculation:

Consider the differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$.

Now, it is known that if $\frac{dy}{dx}=-ax$ is a first order differential equation, then the solutionis $y=y_0{e}^{-ax}$.

So, using Euler’s method, $y_{i+1}=y_i+\frac{d{y}_i}{dx}h$.

Thus,

$\begin{array}{c}{y}_{i+1}=y_i+\frac{d{y}_i}{dx}h\\ =y_i-a{y}_{i}h\\ =y_i\left(1-ah\right)\end{array}$

The stability of formula depends upon step size h and step size must satisfy the condition, $\left|1-ah\right|<1$.

Now, the first order differential equation given is,

$\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$

The step size required to maintain the stability is,

$h<\frac{2}{200,000}$

Hence, $h<1\times {10}^{-5}$.

To determine

To calculate: The solution of the differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$ from $t=0\text{to}2$ using the implicit Euler method if $y\left(0\right)=0$ and step size is 0.1.

Answer to Problem 1P

Solution:

The solution of the given differential equation is:

$\begin{array}{cc}x& y\\ 0& 0\\ 0.1& 0.904788\\ 0.2& 0.818731\\ 0.3& 0.740818\\ 0.4& 0.67032\\ 0.5& 0.606531\\ 0.6& 0.548812\\ 0.7& 0.496585\\ 0.8& 0.449329\\ 0.9& 0.40657\\ 1& 0.36788\\ 1.1& 0.332871\\ 1.2& 0.301194\\ 1.3& 0.272532\\ 1.4& 0.246597\\ 1.5& 0.22313\\ 1.6& 0.201897\\ 1.7& 0.182684\\ 1.8& 0.165299\\ 1.9& 0.149569\\ 2& 0.135335\end{array}$

Explanation of Solution

Given Information:

The differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$, $t=0\text{to}2$. Initial value, $y\left(0\right)=0$ and step size is 0.1.

Formula used:

The implicit Euler’s formula is,

$y_{i+1}=y_i+\frac{d{y}_{i+1}}{dx}h$

Calculation:

Consider the differential equation, $\frac{dy}{dx}=-200,000y+200,000e^{-x}-e^{-x}$.

The implicit Euler’s formula is,

$y_{i+1}=y_i+\frac{d{y}_{i+1}}{dx}h$

Implicit formula for the given differential equation can be written as,

$y_{i+1}=y_i+\left(-200,000y_{i+1}+200,000e^{-x_{i+1}}-e^{-x_{i+1}}\right)h$

Simplify further,

$\begin{array}{l}{y}_{i+1}=\frac{y_i+\left(200,000e^{-x_{i+1}}-e^{-x_{i+1}}\right)h}{1+200,000h}\\ =\frac{y_i+\left(199999e^{-x_{i+1}}\right)h}{1+200,000h}\end{array}$

Substitute $h=0.1$ in above equation,

$\begin{array}{c}{y}_{i+1}=\frac{y_i+\left(199999e^{-x_{i+1}}\right)h}{1+200,000h}\\ =\frac{y_i+\left(199999e^{-x_{i+1}}\right)0.1}{1+200,000\left(0.1\right)}\\ =\frac{y_i+\left(19999.9e^{-x_{i+1}}\right)}{20001}\end{array}$

Thus, $y_{i+1}=\frac{y_i+\left(19999.9e^{-x_{i+1}}\right)}{20001}$

Substitute $i=0\text{ and }{x}_1=0.1$ in above equationas shown below,

$y_1=\frac{y_0+\left(199999e^{-x_1}\right)\times 0.1}{1+200,000\times 0.1}$

As $y\left(0\right)=0$

$\begin{array}{c}{y}_1=\frac{0+\left(19999.9e^{-0.1}\right)}{20001}\\ =0.904788\end{array}$

Use excel to find all the iteration with step size $h=0.1$ from $x=0\text{ to 2}$ as below,

Step 1. First put value of x in the excel as shown below,

Step 2. Now name the column B as y and go to column B2 and put value 0.

Step 3. Now, go to column B3 and write the formula as,

=(B2+(19999.9*(EXP(-A3))))/20001

Then, Press enter and drag the column up to the $x=2$.

Thus, all the iterations are as shown below,