function [t,T] = dao(alpha,delta,tf)
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% Course: APPM 2360
% Author: James Robinson
% Date: Febuaury 2007
%
% WARNING:  Do not change any of the code in this file while working on
% your lab assignment.
%
% Description:  This function numerically solves the delayed difference
% model for the Niņo 3 temperature anomaly used in the El Niņo lab (T'=T-T^3-alpha*T(t-delta)).  
% The solution is computed using Matlab's dde23 function.  The solution always starts at
% t=0, and the intial data from t=-delta to t=0 is given by the subfunction
% Thist.
%
% Input:
% alpha        Constant in equation
% delta        Time delay
% tf           The solver will produce a solution from t=0 to t=tf.
%
% Output:
% t         The vector of time values at which T is calculated.
% T         A vector of Niņo 3 temperature anomaly values.
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% Code for dao
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% Call dde23 to solve the equation with the given constants.
sol = dde23(@dTdt,delta,@Thist,[0 tf],[],alpha,delta);

% Set the outputs to the appropriate values from the solution structure.
t = sol.x;
T = sol.y;

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% Subfunctions
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function T = Thist(t,alpha,delta)
%This function provides intial values for T from T=-delta to T=0.
T = -2-0.1*sin(2*pi*delta*t);

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function Tp = dTdt(t,T,Z,alpha,delta)
% This function calculates the first derivative of T with respect to t.
Tp = T-T.^3-alpha*Z;

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