figure_1_2.m

The Finite Difference Time Domain Method for Electromagnetics With MATLAB Simulations Atef Elshe

% create exact function and its derivative
N_exact = 301; % number of sample points for exact function
x_exact = linspace(0,6*pi,N_exact);
f_exact = sin(x_exact).*exp(-0.3*x_exact);
f_derivative_exact = cos(x_exact).*exp(-0.3*x_exact) ...
                    -0.3*sin(x_exact).*exp(-0.3*x_exact);

% plot exact function
figure(1);
plot(x_exact,f_exact,'k-','linewidth',1.5);
set (gca,'FontSize',12,'fontweight','demi');
axis([0 6*pi -1 1]); grid on;
xlabel('$x$','Interpreter','latex','FontSize',16);
ylabel('$f(x)$','Interpreter','latex','FontSize',16);
                
% create exact function for pi/5 sampling period 
% and its finite difference derivatives
N_a = 31; % number of points for pi/5 sampling period 
x_a = linspace(0,6*pi,N_a);
f_a = sin(x_a).*exp(-0.3*x_a);
f_derivative_a = cos(x_a).*exp(-0.3*x_a) ...
                    -0.3*sin(x_a).*exp(-0.3*x_a);

dx_a = pi/5;
f_derivative_forward_a  = zeros(1,N_a);
f_derivative_backward_a = zeros(1,N_a);
f_derivative_central_a  = zeros(1,N_a);
f_derivative_forward_a(1:N_a-1) = ...
        (f_a(2:N_a)-f_a(1:N_a-1))/dx_a;
f_derivative_backward_a(2:N_a) = ...
        (f_a(2:N_a)-f_a(1:N_a-1))/dx_a;
f_derivative_central_a(2:N_a-1) = ...
        (f_a(3:N_a)-f_a(1:N_a-2))/(2*dx_a);


% create exact function for pi/10 sampling period 
% and its finite difference derivatives
N_b = 61; % number of points for pi/10 sampling period 
x_b = linspace(0,6*pi,N_b);
f_b = sin(x_b).*exp(-0.3*x_b);
f_derivative_b = cos(x_b).*exp(-0.3*x_b) ...
                    -0.3*sin(x_b).*exp(-0.3*x_b);
dx_b = pi/10;
f_derivative_forward_b  = zeros(1,N_b);
f_derivative_backward_b = zeros(1,N_b);
f_derivative_central_b  = zeros(1,N_b);
f_derivative_forward_b(1:N_b-1) = ...
        (f_b(2:N_b)-f_b(1:N_b-1))/dx_b;
f_derivative_backward_b(2:N_b) = ...
        (f_b(2:N_b)-f_b(1:N_b-1))/dx_b;
f_derivative_central_b(2:N_b-1) = ...
        (f_b(3:N_b)-f_b(1:N_b-2))/(2*dx_b);
    
% plot exact derivative of the function and its finite difference 
% derivatives using pi/5 sampling period
figure(2);
plot(x_exact,f_derivative_exact,'k', ...
    x_a(1:N_a-1),f_derivative_forward_a(1:N_a-1),'b--', ...
    x_a(2:N_a),f_derivative_backward_a(2:N_a),'r-.', ...
    x_a(2:N_a-1),f_derivative_central_a(2:N_a-1),':ms', ...
    'MarkerSize',4, 'linewidth',1.5);
set (gca,'FontSize',12,'fontweight','demi');
axis([0 6*pi -1 1]);
grid on;
legend('exact', 'forward difference',...
    'backward difference','central difference');
xlabel('$x$','Interpreter','latex','FontSize',16);
ylabel('$f''(x)$','Interpreter','latex','FontSize',16);
text(pi,0.6,'$\Delta x = \pi/5$','Interpreter', ...
    'latex','fontsize',16, 'BackgroundColor','w','EdgeColor','k');

% plot error for finite difference derivatives 
% using pi/5 sampling period
error_forward_a  = f_derivative_a - f_derivative_forward_a;
error_backward_a = f_derivative_a - f_derivative_backward_a;
error_central_a  = f_derivative_a - f_derivative_central_a;

figure(3);
plot(x_a(1:N_a-1),error_forward_a(1:N_a-1),'b--', ...
    x_a(2:N_a),error_backward_a(2:N_a),'r-.', ...
    x_a(2:N_a-1),error_central_a(2:N_a-1),':ms', ...
    'MarkerSize',4, 'linewidth',1.5);

set (gca,'FontSize',12,'fontweight','demi');
axis([0 6*pi -0.2 0.2]);
grid on;
legend('forward difference', 'backward difference', ...
    'central difference');
xlabel('$x$','Interpreter','latex','FontSize',16);
ylabel('error $[f''(x)]$','Interpreter','latex','FontSize',16);
text(pi,0.15,'$\Delta x = \pi/5$','Interpreter', ...
    'latex','fontsize',16, 'BackgroundColor','w','EdgeColor','k');

% plot error for finite difference derivatives 
% using pi/10 sampling period
error_forward_b  = f_derivative_b - f_derivative_forward_b;
error_backward_b = f_derivative_b - f_derivative_backward_b;
error_central_b  = f_derivative_b - f_derivative_central_b;

figure(4);
plot(x_b(1:N_b-1),error_forward_b(1:N_b-1),'b--', ...
    x_b(2:N_b),error_backward_b(2:N_b),'r-.', ...
    x_b(2:N_b-1),error_central_b(2:N_b-1),':ms', ...
    'MarkerSize',4, 'linewidth',1.5);

set (gca,'FontSize',12,'fontweight','demi');
axis([0 6*pi -0.2 0.2]);
grid on;
legend('forward difference','backward difference', ...
    'central difference');
xlabel('$x$','Interpreter','latex','FontSize',16);
ylabel('error $[f''(x)]$','Interpreter','latex','FontSize',16);
text(pi,0.15,'$\Delta x = \pi/10$','Interpreter', ...
    'latex','fontsize',16,'BackgroundColor','w','EdgeColor','k');