Concept explainers
Repeat Prob. 27.28, but for the following heat source:
(a)
To calculate: The temperature distribution by shooting method for a heated rod with a uniform heat source given by Poisson equation,
, where heat source
, also
Answer to Problem 29P
Solution: The table of the solution of the boundary value problem is,
Explanation of Solution
Given:
A differential equation,
, where heat source
, also
Formula used:
Linear-interpolation formula:
Calculation:
Consider the following Poisson equation,
Since,
. Then,
Now, change the above boundary value problem into equivalent initial-value problem. Then,
But
Thus,
Use the shooting method in the above system of first order linear differential equation
Suppose,
Then, the system of system of first order linear differential equation with initial condition is,
Now, solve the above system of differential equation.
Thus,
Integrate on both the sides of the above differential equation, to get
Where
Now, use the initial condition,
. Thus,
Put the value of
Thus,
Since,
. But
Therefore,
Integrate on both the sides of the above differential equation, to get
Where
is constant of integration.
Use the initial condition,
. Then,
Put the value of
, then
Now, evaluate the above for
. Thus,
But the above of
Then, put another guess. Suppose
Then, the system of system of first order linear differential equation with initial condition is,
Now, solve the above system of differential equation. Then,
Integrate on both the sides of the above differential equation
Thus,
Where
Now, use the initial condition,
Therefore,
Put the value of
Since,
. But
Thus,
Integrate on both the sides of the above differential equation. Then,
Where
Use the initial condition,
. Thus,
Put the value of
Now, evaluate the above for
. Thus,
Since, the first guess value
corresponds to
and the second-guess value
corresponds to
Now, use these values to compute the value of
that yields
Then, by linear interpolation formula,
Therefore, the right value of
which yields
Then, the equivalent initial value problem corresponding to the boundary value problem is,
Now, use the fourth order RK method with step size
The RK method for above system of first order linear differential equation with initial condition is,
Where
And
And
Where
Then, for
And
Also,
Thus,
And
In the similar way, find the remaining
. Then,
And
Therefore, the table of the solution of the boundary value problem is
Hence, the graph of the temperature distribution is
(b)
To calculate: The temperature distribution by finite difference method for a heated rod with a uniform heat source given by Poisson equation,
, where heat source
, also
Answer to Problem 29P
Solution:
The table of the solution of boundary value problem is
Explanation of Solution
Given:
A differential equation,
, where heat source
, also
Formula used:
(1) The finite difference method is:
(2) The Gauss-Seidel iterative method is:
Calculation:
Consider the following Poisson equation,
Since,
Thus,
The finite difference method is given by,
Now, substitute the value of second order derivative in the boundary value problem.
Then, the boundary value problem becomes,
Or
Since,
. Then,
For the first node,
For the second node,
For the third node,
For the fourth node,
Then, write the system of equations in matrix form
Since, the coefficient matrix is tridiagonal matrix, then use Gauss-Seidel iterative technique
The Gauss-Seidel iterative method is,
Now, evaluate
by above Gauss-Seidel method
Then,
And
Then, the table of the solution of boundary value problem is
Therefore, the graph of temperature distribution is
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