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PROPAGATION OF NON-UNIFORM STRING


## The string is made up of two different mass density strings
## seperated in the middle. Choose the right one to be ligther
## than the left. So, since velocity is inversely propotional
## to the mass density, than the rigth one is faster also.
## r1=c1*dt/dx and r2=c1*dt/dx
function ynext=propagate_two_parts(ynow,yprev,r1,r2)
## initiallization for, take y as previos one
ynext=ynow;
## we have two parts, hence two loops are needed
## the first part is between x=0 and
## let y(now)/2 for better visiulation.
## floor(x) returns the largest integer not greater than x
## length(x) determines the number of column
## or rows in matrix or vector
##I started the 'for' from x=0 but didn't work
##then started from x=1
## but in this case the
## string_two_parts didn't work
## let x0=i0=2
for i=2:floor(length(ynow)/2)
ynext(i) = 2*(1-r1^2)*ynow(i)-yprev(i)+r1^2*(ynow(i+1)+ynow(i-1));
endfor
## the second part is between x0=(L1+L2) /2 and L1+L2 where
## L1 and L2 are string lenghts respectively
## I started the 2nd for from N2steps
## and finished it by 'ynow' but it
## didn't work. So i tried the following
for i=floor(length(ynow)/2)+1:floor(length(ynow))-1
ynext(i) = 2*(1-r2^2)*ynow(i)-yprev(i)+r2^2*(ynow(i+1)+ynow(i-1));
endfor
endfunction

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NEWTON-RAPSON METHOD-8th degree Legendre polynomial

## Newton-Rapson Method to the smallest non negative root
## of the 8th degree Legendre Polynomial
## P8(x)=(1/128)(6435x^8-12012x^6+6930x^4-1260x^2+35)
## where -1<=x<=1.
## for the smallest non negative root, we can ignore
## all the terms except the last two by truncated
## the function to be zero and find
## x=0.167 as the initial smallest non negative
## root.
##Constants and initializations
x=[]; ## Empty array for the iterated x roots
x(1)=0.16700000; ## Initial guess to begin the iteration for the
## smallest non-negative root.
L8=[]; ## Empty array for the Legendre polynomial
L8p=[]; ## Empty array for the derivative of the Legendre polynomial
for i=1:100
##The value of the function at x
L8(i)=(1/128)*(6435*x(i)^8-12012*x(i)^6+6930*x(i)^4-1260*x(i)^2+35);
##The value of the derivative of the function at x
L8p(i)=(1/128)*(6435*8*x(i)^7-12012*6*x(i)^5+6930*4*x(i)^3-1260*2*x(i));
x(i+1)=x(i)-L8(i)/L8p(i); ## the iteration
endfor
## For plot let's define a new variable…

Second Harmonic Generation

gnuplot> set xrange [-180:180] gnuplot> set yrange [-180:180] gnuplot> set pm3d gnuplot> set hidden3d  gnuplot> set title 'SHG' gnuplot> splot sin(cos(x*pi/180))*sin(cos(x*pi/180))/(cos(x*pi/180)*cos(x*pi/180))*sin(cos(y*pi/180))*sin(cos(y*pi/180))/(cos(y*pi/180)*cos(y*pi/180)) title 'N=1
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