There are two solutions for the heat equation. The first one involves \'sine\' a

Question

There are two solutions for the heat equation. The first one involves 'sine' and is not in the problem. The second one is the equation in the problem. Both have been worked out below. Feel free to work them out individually as needed and then please explain how this problem can be represented computationally.

Below is the solution to the problem. Both equations have been solved and are correct.

Now that we have the solution and explanation from two different equations for the heat equation how can the information turn into a computer program?

How can this turn into a computer program? Think about differences (not derivatives) and using an intermediate equation. The goal is to have an array that gives temperature at different positions, and will need to have an equation that updates that array for each step. Please explain how this can turn into a computer program with the given data.

Consider a uniform rod of material whose temperature varies only along its length, in the x direction. By considering the heat flowing from both directions into a small segment of length derive the heat equation, X, 2 where K-Ktop, c is the specific heat of the material, and is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that K is independent of temperature, show that a solution of the heat equation is -2/4Kt vt where TO is a constant background temperature and A is any constant. Sketch (or use a computer to plot) this solution as a function of x, for several values of t. Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.

Explanation / Answer

// this is a code in c++ modify the constants as per your requirement

#include <iostream>
#include<math.h>
using namespace std;

int main()
{
// cout << "Hello World!" << endl;
double T[1000];// temperature array where T[i] corresponds to value of Temperature at x = i/400
double T0 = 0;// assign random constant values
double A = 4;// assign desired A
double k = 1;//assign desired k
double t;
double e = 2.71828;
for(t=1;t<100;t++)
{
for(int x =1;x<1000;x++)
{
  
double x0 = x/400;
T[x] = T0 + (A/pow(t,0.5))*(pow(e,-x0*x0/4*k*t));
cout<<T[x]<<" ";
}
cout<<endl;
}
return 0;
}

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