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19_nonlinear_ivp/.ipynb_checkpoints/lecture_23-checkpoint.ipynb

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+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

19_nonlinear_ivp/eulode.m

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+function [t,y] = eulode(dydt,tspan,y0,h,varargin)
+% eulode: Euler ODE solver
+%   [t,y] = eulode(dydt,tspan,y0,h,p1,p2,...):
+%           uses Euler's method to integrate an ODE
+% input:
+%   dydt = name of the M-file that evaluates the ODE
+%   tspan = [ti, tf] where ti and tf = initial and
+%   final values of independent variable
+%   y0 = initial value of dependent variable
+%   h = step size
+%   p1,p2,... = additional parameters used by dydt
+% output:
+%   t = vector of independent variable
+%   y = vector of solution for dependent variable
+if nargin<4,error('at least 4 input arguments required'),end
+ti = tspan(1);tf = tspan(2);
+if ~(tf>ti),error('upper limit must be greater than lower'),end
+t = (ti:h:tf)'; n = length(t);
+% if necessary, add an additional value of t
+% so that range goes from t = ti to tf
+if t(n)<tf
+  t(n+1) = tf;
+  n = n+1;
+end
+y=zeros(n,length(y0));
+y(1,:)=y0;
+for i = 1:n-1 %implement Euler's method
+  y(i+1,:) = y(i,:) + dydt(t(i),y(i,:),varargin{:})'*(t(i+1)-t(i));
+end

19_nonlinear_ivp/predprey.m

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+function yp=predprey(t,y,a,b,c,d)
+	% predator-prey model (Lotka-Volterra equations)
+	yp=zeros(2,1);
+	yp(1)=a*y(1)-b*y(1)*y(2);
+	yp(2)=-c*y(2)+d*y(1)*y(2);
+end

19_nonlinear_ivp/rk4sys.m

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+function [tp,yp] = rk4sys(dydt,tspan,y0,h,varargin)
+% rk4sys: fourth-order Runge-Kutta for a system of ODEs
+%   [t,y] = rk4sys(dydt,tspan,y0,h,p1,p2,...): integrates
+%           a system of ODEs with fourth-order RK method
+% input:
+%   dydt = name of the M-file that evaluates the ODEs
+%   tspan = [ti, tf]; initial and final times with output
+%           generated at interval of h, or
+%   = [t0 t1 ... tf]; specific times where solution output
+%   y0 = initial values of dependent variables
+%   h = step size
+%   p1,p2,... = additional parameters used by dydt
+% output:
+%   tp = vector of independent variable
+%   yp = vector of solution for dependent variables
+if nargin<4,error('at least 4 input arguments required'), end
+if any(diff(tspan)<=0),error('tspan not ascending order'), end
+n = length(tspan);
+ti = tspan(1);tf = tspan(n);
+if n == 2
+  t = (ti:h:tf)'; n = length(t);
+  if t(n)<tf
+    t(n+1) = tf;
+    n = n+1;
+  end
+else
+  t = tspan;
+end
+tt = ti; y(1,:) = y0;
+np = 1; tp(np) = tt; yp(np,:) = y(1,:);
+i=1;
+while(1)
+  tend = t(np+1);
+  hh = t(np+1) - t(np);
+  if hh>h,hh = h;end
+  while(1)
+    if tt+hh>tend,hh = tend-tt;end
+    k1 = dydt(tt,y(i,:),varargin{:})';
+    ymid = y(i,:) + k1.*hh./2;
+    k2 = dydt(tt+hh/2,ymid,varargin{:})';
+    ymid = y(i,:) + k2*hh/2;
+    k3 = dydt(tt+hh/2,ymid,varargin{:})';
+    yend = y(i,:) + k3*hh;
+    k4 = dydt(tt+hh,yend,varargin{:})';
+    phi = (k1+2*(k2+k3)+k4)/6;
+    y(i+1,:) = y(i,:) + phi*hh;
+    tt = tt+hh;
+    i=i+1;
+    if tt>=tend,break,end
+  end
+  np = np+1; tp(np) = tt; yp(np,:) = y(i,:);
+  if tt>=tf,break,end
+end