Chapter 31, Problem 1P

Repeat Example 31.1, but for $T\left(0,t\right)=75\text{ and }T\left(10,t\right)=150$ and a uniform heat source of 15.

To determine

To calculate: The solution to the Poisson’s equation $\frac{d^{2}T}{d{x}^2}=-f\left(x\right)$ for the boundary condition $T\left(0,t\right)=75$ and $T\left(10,t\right)=150$.

$$

Answer to Problem 1P

Solution: Solution to the Poisson’s equation for provided boundary equation is $T=-7.5x^2+82.5x+75$.

Explanation of Solution

Given Information:

Poisson’s equation:

$\frac{d^{2}T}{d{x}^2}=-f\left(x\right)$

Here, function $f\left(x\right)$ is the heat source along the rod.

Boundary conditions are as $T\left(0,t\right)=75$ and $T\left(10,t\right)=150$

The value of the uniform heat source is $15$.

Formula used:

If a differential is in the form $\frac{d^{2}T}{d{x}^2}=-f\left(x\right)$, then the general equation to the differential equation is $T=a{x}^2+bx+c$.

Calculation:

Consider the problem statement, the expression for the boundary condition is:

$\frac{d^{2}T}{d{x}^2}=-15$

The Boundary condition is provided as follows:

$T\left(0,t\right)=75$ and $T\left(10,t\right)=150$

The general solution to the differential equation is:

$T=a{x}^2+bx+c$

Differentiate it with respect to $x$.

$\begin{array}{c}\frac{dT}{dx}=2ax+b\\ \frac{d^{2}T}{d{x}^2}=2a\end{array}$

Compares the equations with each other.

$\begin{array}{c}2a=-15\\ a=\frac{-15}{2}\\ =-7.5\end{array}$

Apply the boundary condition $T\left(0,t\right)$ also substitute $-7.5$ for $a$.

$\begin{array}{c}75=-7.5{\left(0\right)}^2+b\left(0\right)+c\\ c=75\end{array}$

Apply the boundary condition $T\left(0,t\right)$ also substitute $-7.5$ for $a$, and $75$ for $c$.

$\begin{array}{c}150=-7.5{\left(10\right)}^2+b\left(10\right)+75\\ 150+750-75=10b\\ b=\frac{825}{10}\\ =82.5\end{array}$

Substitute $-7.5$ for $a$, $82.5$ for $b$ and $75$ for $c$.

$T=-7.5x^2+82.5x+75$

Use the following MATLAB command to execute the code and plot the temperature as a function of position along the rod.

x=0:0.001:15;

%define the range of the length

y = -7.5*x.^2+82.5*x+75;

%define the function in form of matlab code

plot(x, y)

%write the varibale into plot command in the same sequence you want to plot

xlabel('Length')

%Label axes

ylabel('Temperature')

Now, execute the program by pressing run button. Following plot is obtained that shows the plot of temperature verses distance.