## Abbildungen - Kapitel 6

 ``` [x,y] = meshgrid(0:0.3:3,0.1:0.3:3); u = ones(size(x)); v = y; h = quiver(x,y,u,v,1.75); axis([0 3 0.1 3]); hold on; f = inline('x','t','x'); x0 = 0.2; T = 3; [t,x] = ode45(f,[0,T],[x0]); plot(t,x,'k-','linewidth', 3); x1 = x0; t1 = 0; h = 0.5 for i=1:T/h t2 = t1+h; x2 = x1 + h.*f(t1,x1); plot([t1,t2],[x1,x2],'k--o','linewidth', 3); x1 = x2; t1 = t2; end; ``` ``` [x,y] = meshgrid(0:0.3:3,0.1:0.3:3); u = ones(size(x)); v = y; h = quiver(x,y,u,v,1.75); hold on; f = inline('x','t','x'); x0 = 0.2; T = 3; [t,x] = ode45(f,[0,T],[x0]); plot(t,x,'k-','linewidth', 3); x1 = x0; t1 = 0; h = 0.5 for i=1:T/h t2 = t1+h; k = f(t1,x1); x2 = x1+h./2*(k+f(t1+h,x1+h.*k)) plot([t1,t2],[x1,x2],'k--o','linewidth', 3, 'MarkerFaceColor', 'k'); x1 = x2; t1 = t2; end; ``` ``` sigma = 10; beta=8/3; rho=28; dgl = inline('[10*(x(2)-x(1)); 28*x(1)-x(2)-x(1)*x(3); x(1)*x(2)-8/3*x(3)]','t','x'); opt = odeset('RelTol',1e-5,'refine',1); x01 = [1;1;1]; [t1,x1] = ode45(dgl,[0,40],x01,opt); h = plot(t1,x1(:,1),'k'); set(gca,'plotboxaspectratio',[1 0.2 1]) ``` ``` dt = t1(2:size(t1))-t1(1:size(t1)-1); h2 = plot(t1(1:size(t1)-1),dt,'-k'); ``` ``` lambda = -100; T = 0.3; ttex = [0:0.001:T]; exakt = exp(lambda.*ttex); h1 = 0.02; tt = [0:0.02:T]; exeu1 = (1+h1*lambda).^(tt./h1); impeu1 = 1./((1-h1*lambda).^(tt./h1)); plot(ttex,exakt,'k-',tt,exeu1,'k:o',tt,impeu1,'k--o',... 'linewidth',2,'MarkerFaceColor','k'); axis([0,T,-1.2,1.2]); ```

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