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The Maple code for the modified Euler-Heun adaptive method is:
f := (t, y) -> -2*y + exp(-2*(t - 6)^2); dsolve( {D(y)(t) = f( t, y(t) ), y(0) = 1} ); t[0] := 0.0; y[0] := 1.0; h := 0.1; eps[abs] := 0.001; for i from 1 do K[0] := f( t[i - 1], y[i - 1] ); yt := y[i - 1] + h*K[0]; K[1] := f( t[i - 1] + h, yt ); zt := y[i - 1] + h*(K[0] + K[1])/2; s := sqrt( eps[abs] / abs( yt - zt ) ); if s < 0.75 then h := max( h/2, 0.0125 ); elif s > 1.5 then h := min( 2*h, 0.4 ); end; t[i] := t[i - 1] + h; y[i] := y[i - 1] + h*K[0]; if t[i] > 10 then break; end if; end do: plots[pointplot]( [seq( [t[k], y[k]], k = 1..i )] );
The Maple code for the modified Runge-Kutta-Fehlberg method is:
f := (t, y) -> -2*y + exp(-2*(t - 6)^2); dsolve( {D(y)(t) = f( t, y(t) ), y(0) = 1} ); t[0] := 0.0; y[0] := 1.0; h := 0.1; eps[abs] := 0.001; for i from 1 do K[1] := f( t[i-1], y[i-1] ); K[2] := f( t[i-1] + (1/4)*h, y[i-1] + (1/4)*h*K[1] ); K[3] := f( t[i-1] + (3/8)*h, y[i-1] + (3/8)*h*( 1/4*K[1] + 3/4*K[2]) ); K[4] := f( t[i-1] + (12/13)*h, y[i-1] + (12/13)*h*( 161/169*K[1] - 600/169*K[2] + 608/169*K[3]) ); K[5] := f( t[i-1] + h, y[i-1] + h*( 8341/4104*K[1] - 32832/4104*K[2] + 29440/4104*K[3] - 845/4104*K[4]) ); K[6] := f( t[i-1] + (1/2)*h, y[i-1] + (1/2)*h*(-6080/10260*K[1] + 41040/10260*K[2] - 28352/10260*K[3] + 9295/10260*K[4] - 5643/10260*K[5]) ); yt := y[i - 1] + h*( 2375*K[1] + 11264*K[3] + 10985*K[4] - 4104*K[5] )/20520; zt := y[i - 1] + h*( 33440*K[1] + 146432*K[3] + 142805*K[4] - 50787*K[5] + 10260*K[6] )/282150; s := root[4]( (eps[abs]*h)/(2*abs(yt - zt)) ); if s < 0.75 then h := max( h/2, 0.025 ); continue; end if; t[i] := t[i - 1] + h; y[i] := yt; if t[i] >= 10 then break; end if; if s > 1.5 then h := min( 2*h, 1.6 ); end if; end do: plots[pointplot]( [seq( [t[k], y[k]], k = 1..i )] );
Copyright ©2006 by Douglas Wilhelm Harder. All rights reserved.