Matlab Cat 2

MATLAB

1)Bessel fist kind
X = 0:0.1:20;
J = zeros(5,201);
for i=0:4
J(i+1,:) = besselj(i,X);
end
figure('Position',[0 0 1024 768])
plot(X,J,'LineWidth',1.5)
axis([0 20 -.5 1])
grid on;
legend('J_0','J_1','J_2','J_3','J_4','Location','Best')
title('Bessel Functions of the First Kind for n = 0,1,2,3,4')
xlabel('X')
ylabel('J_n(X)')

2)Bessel second kind
X = 0:0.1:20;
Y = zeros(5,201);
for i=0:4
Y(i+1,:) = bessely(i,X);
end
figure('Position',[0 0 1024 768])
plot(X,Y,'LineWidth',1.5)
axis([-0.1 20.2 -2 0.6])
grid on;
legend('Y_0','Y_1','Y_2','Y_3','Y_4','Location','Best')
title('Bessel Functions of the Second Kind for n = 0,1,2,3,4')
xlabel('X')
ylabel('Y_n(X)')

3)Fourier Series

clear all
clc
syms x k L U n
f = input('Enter the function: ');
L = input('Enter the lower limit: ');
U = input('Enter the upper limit: ');
l = (U-L)/2;
n = input('Enter the number of terms required: ');
ak = @(f,x,k) int(f*cos(k*pi*x/l)/l,x,L,U);
bk = @(f,x,k) int(f*sin(k*pi*x/l)/l,x,L,U);
fs = @(f,x,n) ak(f,x,0)/2 + ...
symsum(ak(f,x,k)*cos(k*pi*x/l) + bk(f,x,k)*sin(k*pi*x/l),k,1,n);
pretty(fs(f,x,n))
fst = ak(f,x,0)/2;
for i=1:n
fst = fst + ak(f,x,i)*cos(i*pi*x/l) + bk(f,x,i)*sin(i*pi*x/l);
disp(['Harmonics upto:', num2str(i)]);
disp(fst);
h=ezplot(f,[L U]);
set(h,'LineWidth',2)
hold on;
h = ezplot(fst,[L U]);
set(h,'LineStyle','-','Color',[i/n 1/i 1/n],'LineWidth',2);
title(['Partial sums up to n=',num2str(i)])
end

4)Harmonic analysis
clear all
clc
syms t
n = input('Enter the number of data points n : ');
x_0 = input('Enter the starting value of x : ');
s = input('Enter the length of the spacing between successive values of x :
');
count = input('Enter 0 if the unit of x is degrees, otherwise enter a nonzero
number: ');
y = input('Enter the y values (as a row vector): ');
n1 = input('Enter the number of harmonic of the series n1 : ');
x = x_0 + (0:n-1)*s;
if(count == 0)
x=x*pi/180;
s=s*pi/180;
end
l=0.5*(x(n)+s-x(1)); % l=pi if it is degree
a_0 = (2/n)*sum(y);
for i=1:n1
yc=y.*cos(i*pi*x/l);
ys=y.*sin(i*pi*x/l);
a(i)=(2/n)*sum(yc);
b(i)=(2/n)*sum(ys);
end
F_s=a_0/2;
figure('Position',[1 1 1024 768])
for i=1:n1
subplot(n1,1,i)
plot(x,y,'r*');
hold on
F_s = F_s+a(i).*cos(i*pi*t/l)+b(i).*sin(i*pi*t/l);
ezplot(F_s, [x(1) x(n)])
title(['Harmonics up to: ', num2str(i)])
end
disp('Fourier series : ')
vpa(F_s,4)

5)second order difference equation with constant coeff
clear all
clc
syms n k1 k2
a = input('Enter the coefficient of y(n+2): ');
b = input('Enter the coefficient of y(n+1): ');
c = input('Enter the coefficient of y(n): ');
g = input('Enter the non-homogeneous part: ');
r = roots([a b c]);
if imag(r)~=0
rho = sqrt(real(r(1))^2 + imag(r(1))^2);
theta = atan(abs(imag(r(1)))/real(r(1)));
y1 = (rho^n)*cos(n*theta);
y2 = (rho^n)*sin(n*theta);
elseif r(1)==r(2)
y1 = r(1)^n;
y2 = n*r(1)^n;
else
y1 = r(1)^n;
y2 = r(2)^n;
end
Co = det([y1, y2;subs(y1,n,n+1), subs(y2,n,n+1)]); %Casoratian of the solutions
y_c = k1*y1 + k2*y2;
disp('Complementary Solution is: ');
disp(y_c);
if(g ~= 0)
y11 = subs(y1,n,n+1);
y21 = subs(y2,n,n+1);
36
Co1 = subs(Co,n,n+1);
u1 = simplify(symsum(-g*y21/Co1,n,0,n-1));
u2 = simplify(symsum(g*y11/Co1,n,0,n-1));
y_p = simplify(u1*y1+u2*y2);
y = y_c + y_p;
else
y = y_c;
end
check = input('If the given problem has initial conditions then enter 1 else enter 0: ');
if (check == 1)
yval1 = input('Enter the initial condition at n = 0: ');
yval2 = input('Enter the initial condition at n = 1: ');
cond1 = strcat(char(subs(y,n,0)),'=',num2str(yval1));
cond2 = strcat(char(subs(y,n,1)),'=',num2str(yval2));
[k1,k2] = solve(cond1,cond2);
y = subs(y);
end
disp('Complete Solution is: ')
disp(collect(collect(y,y1),y2))
if(check ~= 0)
nrange = linspace(0,10,40);
Y = subs(y,n,nrange);
stem(nrange,Y);
set(gca,'XTick',linspace(0,10,11))
xlabel('n');
ylabel('y(n)');
end

6)SOLUTION OF FIRST AND SECOND ORDER DIFFERENCE EQUATIONS USING Z – TRANSFORM

clc
clear all
syms n z Yz g
yn = sym('y(n)');
yn1 = sym('y(n+1)');
yn2 = sym('y(n+2)');
a = input('The Coefficient of y(n+2) = ');
b = input('The Coefficient of y(n+1) = ');
c = input('The Coefficient of y(n) = ');
nh = input('The Non Homogeneous part ');
eq = a*yn2 + b*yn1 + c*yn - nh;
Zeq = ztrans(eq, n, z);
if (a==0)
d = input('The initial value at 0 is ');
Zeq = subs(Zeq,{'ztrans(y(n), n, z)', 'y(0)'}, {Yz, d});
else
d = input('The initial value at 0 is ');
e = input('The initial value at 1 is ');
Zeq = subs(Zeq,{'ztrans(y(n), n, z)', 'y(0)', 'y(1)'}, {Yz, d, e});
end
eq = collect(Zeq, Yz);
Y = solve(eq, Yz);
Y = iztrans(Y, z, n);
Y = simple(Y)
m = 0:20;
y = double(subs(Y,n,m));
stem(y)
title('Difference equation');
xlabel('n'); ylabel('y(n)');


forms

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<td><input type="text" name="address" value="" size="25"></td>
</tr>

<tr>
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<td><input type="text" name="city" value="" size="25"></td>
</tr>

<tr>
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<td><input type="text" name="state" value="" size="25"></td>
</tr>

<tr>
<td>Zip:</td>
<td><input type="text" name="zip" value="" size="25"></td>
</tr>

<tr>
<td>Magazine:</td>
<td><select name = "newspaper">
<option value="1" selected = "selected">The Times Of India</option>
<option value="2" selected = "selected">Hindustan Times</option>
</select>
</td>
</tr>

<tr>
<td>Subscription:</td>
<td><input type ="radio" name="1yr" value="1" checked ="checked">1 year
<input type ="radio" name="1yr" value="1" >2 year</td>
</tr>
</table>
</br>

Additional Comments: </br>
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<input type="submit" name="submitbtn" value ="submit"/>


</body>
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frames

<!DOCTYPE html>
<html>
<title>Frames</title>
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    <frameset cols="20%,80%">
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link to frame

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<html>
<title>group</title>
<body>
<a href="http://www.tutorialspoint.com" target="right">click here!</a>
</body>
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