% plot_Bessel_function.m

%

% This MATLAB file plots the first Bessel functions of order 0 and 1.

% Then, it plots the Bessel function of order 0.5 and 1.5.

% Both Bessel functions of the first and second kind are plotted.

%

% K. Beers

% MIT ChE 7/31/2002

xmin = 1e-3;

xmax = 10;

resolution = 200;

x = linspace(xmin,xmax,resolution);

y_zero = 0*x;

% First, compute the Bessel functions.

% Bessel functions of the first kind

J_0 = besselj(0,x);

J_1 = besselj(1,x);

J_0x5 = besselj(0.5,x);

J_1x5 = besselj(1.5,x);

% Bessel functions of the second kind

Y_0 = bessely(0,x);

Y_1 = bessely(1,x);

Y_0x5 = bessely(0.5,x);

Y_1x5 = bessely(1.5,x);

% Now, plot the Bessel functions.

figure;

% set common axis limits

xplot_min = 0;

xplot_max = xmax;

yplot_min = -1.1;

yplot_max = +1.1;

% Bessel functions of the first kind

% integer orders

subplot(2,2,1);

plot(x,J_0);

hold on;

plot(x,J_1,'--');

plot(x,y_zero,':');

legend('J_0(x)', 'J_1(x)');

xlabel('x');

ylabel('J_n(x)');

axis([xplot_min xplot_max yplot_min yplot_max]);

% non-integer orders

subplot(2,2,2);

plot(x,J_0x5);

hold on;

plot(x,J_1x5,'--');

plot(x,y_zero,':');

legend('J_{0.5}(x)','J_{1.5}(x)');

xlabel('x');

ylabel('J_\nu(x)');

axis([xplot_min xplot_max yplot_min yplot_max]);

% Bessel functions of the second kind

% integer orders

subplot(2,2,3);

plot(x,Y_0);

hold on;

plot(x,Y_1,'--');

plot(x,y_zero,':');

legend('Y_0(x)', 'Y_1(x)');

xlabel('x');

ylabel('Y_n(x)');

axis([xplot_min xplot_max yplot_min yplot_max]);

% non-integer orders

subplot(2,2,4);

plot(x,Y_0x5);

hold on;

plot(x,Y_1x5,'--');

plot(x,y_zero,':');

legend('Y_{0.5}(x)','Y_{1.5}(x)');

xlabel('x');

ylabel('Y_\nu(x)');

axis([xplot_min xplot_max yplot_min yplot_max]);

gtext('Bessel functions of the first and second kinds');