MATH 308, Summer Examples in MATLAB

Transcription

1 MATH 308, Summer 2016 Examples in MATLAB % % %% % %% % Plotting a function % % %% % %% % %% % %% % %% % %% % %% % figure(1) clear t % Anonymous Functions u7=@(t) 0198*exp(-08590*t) *exp(-29142*t) + (1/45)*sin(5*t); uu7=@(t) (1/45)*sin(5*t); uuu7=@(t) 0198*exp(-08590*t) *exp(-29142*t) ; fplot(u7,[0,4*pi]); fplot(uu7,[0,4*pi], - ); fplot(uuu7,[0,4*pi], - ); title([ Function u(t)= 0198*exp(-08590t)+ + (1/45)*sin(5*t) ]);

2 figure(2) % Inline Functions syms x fplot(xˆ2,[-3 0], b ) fplot(x,[0 3], b ) figure(4) f t * y ˆ2; dirfield2(f,-2:02:2,-2:02:2) title([ Direction Field of dy/dt=f(x,y)= tyˆ2 ]); hold off figure(2 ) clear all for c=1:3 u=@(t) tˆ2 + c ; fplot(u,[0,3]); title([ Function u(t,c)= tˆ2 + c ]);

3 % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %%%% % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % Plotting a Direction Field % % %% % %% % %% % %% % %% % %% %%%% % %%% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% figure(3) %% Z =S(T,Y) [T,Y]=meshgrid(-2:02:3,-1:02:2); S=exp(-T)-2*Y; %% Using an inline defined function quiver(t,y, ones(size(s)),s), axis tight figure(4) %% Using an anonymous defined function f t*y ˆ2; dirfield2(f,-2:02:2,-2:02:2) function dirfield2(f,tval,yval) % dirfield(f, t1:dt:t2, y1:dy:y2) % plot direction field for first order ODE y = f(t,y) % using t-values from t1 to t2 with spacing of dt % using y-values from y1 to t2 with spacing of dy % % f is anonymous function, % or the name of an m-file with quotes

4 % % Example: y = ty 2 % Show direction field for t in [-2,2], y in [-2,2], use % spacing of 2 for both t and y: % % f t*y ˆ2; % dirfield2(f,-2:02:2,-2:02:2) [tm,ym]=meshgrid(tval,yval); dt = tval(2) - tval(1); dy = yval(2) - yval(1); fv = vectorize(f); if isa(f, function_handle ) fv = eval(fv); yp=feval(fv,tm,ym); s = 1/max(1/dt,abs(yp)/dy)*035; h = ishold; quiver(tval,yval,s,s*yp,0, r ); ; quiver(tval,yval,-s,-s*yp,0, r ); if h else hold off axis([tval(1)-dt/2,tval()+dt/2,yval(1)-dy/2,yval()+dy/2]) % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %%%% % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % %% % %% % Finding the zeros of a function % % %% % %% % %% % %% % %% % %% %% % %%% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% clear all lb = 0; % Set a lower bound for the function ub = 45; % Set an upper bound for the function N=900 ; % Number of reference points x = NaN*ones(N,1); % Initializes x starting_points=linspace(lb,ub,n); for i=1:n % Look for the zeros in the function s current window but we are interested only for the zeros of f on [0, 40] x(i)=fzero(@(x) 5*sin(19*x)+21*sin(91*x), starting_points(i)); x_unique=x(diff(x)>1e-12) x_final=zeros(size(x_unique)); x_final(1)= x_unique(1); for i=2:size(x_unique) if abs(x_unique(i) - x_unique(i-1)) < x_final(i)=0; else x_final(i)=x_unique(i); x_final y_final=zeros(size(x_unique)); fun 5*sin(19*x)+21*sin(91*x);

5 fplot(fun,[0,45]); plot(x_final, y_final, * ) title([ Zeros of f(x)=5*sin(19x)+21sin(91x), ]);

6 %%% %%%%%% %%% Plotting a Direction Field %%% %%%%%% %%% %%Define the function [(-1/2)*Y(1) + Y(2); -Y(1) - (1/2)*Y(2)]; %To generate the phase portrait, we need to compute the derivatives y1 %and y2 at t on a grid over the range of values for y1 and y2 we are %interested in We will plot the derivatives as a vector at each %(y1, y2) which will show us the initial direction from each point %We will examine the solutions over the range -2 < y1 < 8, and % -2 < y2 < 2 for y2, and create a grid of points y1 = linspace(-3,3,10); y2 = linspace(-3,3,10); % y3= linspace(-3,3,10); % creates two matrices one for all the x-values on the grid, and one for % all the y-values on the grid Note that x and y are matrices of the same % size and shape, in this case 20 rows and 20 columns [x,y] = meshgrid(y1,y2); %[x,y,z] = meshgrid(y1,y2, y3); size(x); size(y); u = zeros(size(x)); v = zeros(size(x)); t=0; for i = 1:numel(x) Yprime = f(t,[x(i); y(i)]); u(i) = Yprime(1); v(i) = Yprime(2); quiver(x,y,u,v, r ); figure(gcf) %quiver3(x,y,z,u,v,w, r ); figure(gcf) xlabel( x_1 ); ylabel( x_2 ); %zlabel( x_3 ) title([ Phase Space x_1-x_2 ]);

7 % % %% % %% % %% Phase portraits of a system of 2D or 3D-ODEs %% % %% % %% % %%%% % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %%% % %% % %% % % %% % %% % Y1 = (-1/2) Y1 + Y2 %% %% % %% % %% % %%% %%% % %%% % %% % %% % %% % Y2 = - Y1 - (1/2)Y2 %% % %% % %% % %% % %% % %% % %%% % %% % %% % %% % Defining the velocity vector field %% % %% % %% % %% % %% [(-1/2)*Y(1) + Y(2); -Y(1) - (1/2)*Y(2)]; y1 = linspace(-3,3,10); y2 = linspace(-3,3,10); %y3 = linspace(-3,3,10); % creates two matrices one for all the x-values on the grid, and one for % all the y-values on the grid Note that x and y are matrices of the same % size and shape, in this case 20 rows and 20 columns [x,y] = meshgrid(y1,y2); %[x,y,z] = meshgrid(y1,y2, y3); size(x); size(y); %size(z); u = zeros(size(x)); v = zeros(size(x)); %w = zeros(size(z)); % we can use a single loop over each element to compute the derivatives at % each point (y1, y2) t=0; for i = 1:numel(x) Yprime = f(t,[x(i); y(i)]); %Yprime = f(t,[x(i); y(i); z(i)]); u(i) = Yprime(1); v(i) = Yprime(2); %w(i) = Yprime(3); quiver(x,y,u,v, r ); figure(gcf) %quiver3(x,y,z,u,v,w, r ); figure(gcf) xlabel( x_1 ) ylabel( x_2 ) %zlabel( x_3 )

8 for xinit = -3:1:3 ; for yinit = -3:1:3 ; %for zinit = -2:1:2 ; [ts,ys] = ode45(f,[-10,50],[xinit;yinit]); %[ts,ys] = ode45(f,[-10,50],[xinit;yinit;zinit]); plot(ys(:,1),ys(:,2)) %plot3(ys(:,1),ys(:,2),ys(:,3)) plot(ys(1,1),ys(1,2), bo ) %plot3(ys(1,1),ys(1,2),ys(1,3), bo ) plot(ys(,1),ys(,2), ks ) %plot3(ys(,1),ys(,2),ys(,3), ks ) %axis([ ]); % axis([ ]); %axis([ ]); title([ Phase Space x_1 - x_2 ]); %title([ Phase Space x_1 - x_2 - x_3 ]); set(gca, Color, blue ); set(gcf, color, [0 0 1]) hold off

9 % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% % %% %% % % % % % % % % % % % % % % Mechanical Oscillations % % % % % % % % % % % % % % % % % % % % % % % % % % % % Examples % % % % % % % % % % % % % % % % % % % % % % % % Example 55 % % % % % % % % % % % % % % % % figure(1) clear t R delta omega R=05270; delta= 28198; omega=6; u1=@(t) R*cos(omega*t - delta); fplot(u1,[0,2*pi]); title([ Function u(t)= R*cos(omega*t - delta) ]); % % % % % % % % % % Example 56 % % % % % % % % % % % % % % % % figure(2) clear t u2=@(t) *exp(-24792*t) *exp( *t) ; fplot(u2,[0,2*pi]); title([ Function u(t)= *exp(-24792*t) *exp( *t) ]); axis([0 2*pi ])

10 % % % % % % % % % % Example 57 % % % % % % % % % % % % % % % % figure(3) clear t -(1/2)*exp(-6*t) - 2*t*exp(-6*t) ; fplot(u3,[0,2*pi]); title([ Function u(t)= -(1/2)exp(-6t) - 2texp(-6t); critical axis([0 2*pi ]) damping ]);

11 % % % % % % % % % % Example 58 % % % % % % % % % % % % % % % % figure(4) clear t u4=@(t) 0502*exp(-5*t/2)*cos( (sqrt(119)/2)*t ) ; fplot(u4,[0,pi]); title([ Function u(t)= -0502eˆ{-5t/2 }cos(54*t - 323) ]);