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STIFFNESS OF BENDING STRINGS


## A 'realistic', 'non-elastic' string, which responses to any
## bending and has stifness. This script takes in the previous
## and the present profiles and iterates to find
## the profile in the next time step. The ratio 'r' is not 1
## like in the 'non-realistic' string since the speed of the wave
## always less than the speed of the string it should be less than 1
## for best and most stable solution
##constants
dx=1e-2 ## Spatial increment (m)
L=2 ## Length of the string (m)
M=L/dx ## Dimensionless partition
r=0.25 ## Famous dimensionless ratio
E=1e-4 ## Dimensionless stiffnes
x=-1:dx:1;
l=length(x);
x0=0.5;
k=1e2;
## Set up the initial profile
y=initial_profile(x,x0,k);
plot(x,y)
pause
## Impose the time boundary condition
ynow=y;
yprev=y;
ynow(1)=ynow(2)=0
Nsteps=2000;
for n=3:Nsteps
ynow(Nsteps-1)=ynow(Nsteps-2)=0;
ynext=propagate_stiff(ynow,yprev,r);
plot(x,ynext,';;')
axis([-1.05,1.05,-1.1,1.1])
pause(0);
endfor

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NEWTON-RAPSON METHOD FOR HEAT FLOW

##Constants and initializations
a=5.67E-8; ## Stefan-Boltzman constant[Watt/meter^2Kelvin^4]
e=0.8; ## Rod surface emissivity [Dimensionless]
h=20; ## Heat transfer coefficient of air flow [W/m^2-K]
Tinf=Ts=25; ## Temperature of air and the walls of the close[Celcius]
D=0.1; ## Diameter of the rod[meter]
I2R=100; ## Electric power dissipated in rod (Ohmic Heat)[W]
T=[]; ## Temperature of the rod[*C]
T(1)=25; ## Initial guess of the temperature of the rod[*C]
Q=[]; ## Heat function [W]
Qp=[]; ## First derivative of Q wrt T [W/C*].
for i=1:100
Q(i)=pi*D*(h*(T(i)-Tinf)+e*a*(T(i)^4-Ts^4))-I2R;
Qp(i)=pi*D*(h+4*e*a*T(i)^3);
T(i+1)=T(i)-Q(i)/Qp(i); ## Newton-Rapson Method
endfor
printf('The steady state temperature is %f\n',T(i+1))
save -text HeatFlowTemp.dat
## The plot
t=1:100; ##temperature
for n=1:100
H(n)=pi*D*(h*(t(n)-Tinf)+e*a*(t(n)^4-Ts^4))-I2R;
endfor
plot(t,H)
xlabel('T(Celcius)');
ylabel('Q(Watt)');
legend('Q(T)');
title('Heat flow vs Temperatu…