Eulode
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This document contains code to solve initial value problems (IVPs) using the Euler method and calculate the error. It defines a function called eulode that uses Euler's method to solve an IVP from t0 to tf with initial value y0 and step size h. It also defines a function called error_eulode that calculates the error between the exact and numerical solutions for varying step sizes h. The code generates log-log plots of the step size h versus the error to observe the convergence.
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Eulode
- 1. AML 702 Lab 8 ODE-IVPs 1 Program-1 function yfinal=eulode(t0,tf,y0,h) t(1)=t0; y(1)=y0; yexact = exp(-tf^2/2); c = t0; i = 0; while(c<tf) i = i+1; dydt=-t(i)*y(i); y(i+1)=y(i) + dydt*h; t(i+1)=t(i)+h; c = c+h; end yfinal = y(i+1) end .......................................................................... function yf = error_eulode(t0,tf,y0,h) yexact = exp(-tf^2/2); n = length(h); for i = 1:n yfinal = eulode(t0,tf,y0,h(i)) err(i) = abs(yfinal - yexact) end plot(log(h),log(err),'-o'); hold on; plot(log(h),log(h),'r'); xlabel('log(h)'); ylabel('log(err)'); end ............................................................................................................................................................... >>i=0:1:14 i= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >> h=2.^-i h= 1.0000 0.5000 0.2500 0.1250 0.0625 0.0313 0.0156 0.0078 0.0039 0.0020 0.0010 0.0005 0.0002 0.0001 0.0001 >> error_eulode(0,1,1,h)
- 2. yfinal = 0.6065 err = 0.3935 0.1435 0.0600 0.0275 0.0132 0.0065 0.0032 0.0016 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000