% Solve u'(t)=lambda u, with u(0)=1
% Exact solution is u(t)=exp(lambda t)
% Use theta method with different values of theta

uexact=@(t,lambda) exp(lambda*t);

% first run with N=10 and lambda=1
N=10; lambda=1;
figure(1), clf
fplot(@(t) uexact(t,lambda),[0,1],'b','linewidth',2), hold on
% explicit Euler
[tn,Uexp]=theta_method1(lambda,N,0);
% implicit Euler
[tn,Uimp]=theta_method1(lambda,N,1);
% Crank Nicolson
[tn,Ucn]=theta_method1(lambda,N,0.5);
plot(tn,Uexp,'r',tn,Uimp,'g',tn,Ucn,'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('u','fontsize',20)
legend('exact solution','explicit Euler','implicit Euler','Crank Nicolson','location','NorthWest','fontsize',20)
title(['Solution with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)
% exact solution at grid points
Ue=uexact(tn,lambda);
figure(2),clf
plot(tn,(Uexp-Ue),'r',tn,(Uimp-Ue),'g',tn,(Ucn-Ue),'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('error','fontsize',20)
legend('error in explicit Euler','error in implicit Euler','error in Crank Nicolson','location','NorthWest','fontsize',20)
title(['Error with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)


% repeat but with N=100 and lambda=1
N=100; lambda=1;
figure(3), clf
fplot(@(t) uexact(t,lambda),[0,1],'b','linewidth',2), hold on
% explicit Euler
[tn,Uexp]=theta_method1(lambda,N,0);
% implicit Euler
[tn,Uimp]=theta_method1(lambda,N,1);
% Crank Nicolson
[tn,Ucn]=theta_method1(lambda,N,0.5);
plot(tn,Uexp,'r',tn,Uimp,'g',tn,Ucn,'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('u','fontsize',20)
legend('exact solution','explicit Euler','implicit Euler','Crank Nicolson','location','NorthWest','fontsize',20)
title(['Solution with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)
% exact solution at grid points
Ue=uexact(tn,lambda);
figure(4),clf
plot(tn,(Uexp-Ue),'r',tn,(Uimp-Ue),'g',tn,(Ucn-Ue),'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('error','fontsize',20)
legend('error in explicit Euler','error in implicit Euler','error in Crank Nicolson','location','NorthWest','fontsize',20)
title(['Error with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)


% final run with N=10 and lambda=-25
N=10; lambda=-25;
figure(5), clf
fplot(@(t) uexact(t,lambda),[0,1],'b','linewidth',2), hold on
% explicit Euler
[tn,Uexp]=theta_method1(lambda,N,0);
% implicit Euler
[tn,Uimp]=theta_method1(lambda,N,1);
% Crank Nicolson
[tn,Ucn]=theta_method1(lambda,N,0.5);
plot(tn,Uexp,'r',tn,Uimp,'g',tn,Ucn,'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('u','fontsize',20)
legend('exact solution','explicit Euler','implicit Euler','Crank Nicolson','location','NorthWest','fontsize',20)
title(['Solution with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)
% exact solution at grid points
Ue=uexact(tn,lambda);
figure(6),clf
plot(tn,(Uexp-Ue),'r',tn,(Uimp-Ue),'g',tn,(Ucn-Ue),'k','linewidth',2)
xlabel('t','fontsize',20), ylabel('error','fontsize',20)
legend('error in explicit Euler','error in implicit Euler','error in Crank Nicolson','location','NorthWest','fontsize',20)
title(['Error with N=',num2str(N),' and lambda=',num2str(lambda)],'fontsize',20)