Assignment no: 8. Euler and Modified Euler equation to solve Differen-
tial Equations
Method:1 Solving Differential equation by NDSolve function
clear@ppDclear@ppD
pp = NDSolve@8y'@tD � t^2 + Hy@tDL^2 - 1, y@-2D � -2<, y, 8t, -2, 2<D88y ® InterpolatingFunction@88-2., 2.<<, <>D<<
plt@1D = Plot@Evaluate@y@tD �. ppD, 8t, -2, 2<, PlotStyle ® 8Orange, Thick<D
-2 -1 1 2
-2.0
-1.5
-1.0
-0.5
0.5
clear@modiefiedeulerDclear@modiefiedeulerD
modifiedeuler@f_, 8x_, x0_, xn_<, 8y_, y0_<, steps_D :=
Block@8xold = x0, yold = y0, sollist = 88x0, y0<<, x, y, h<, h = N@Hxn - x0L � stepsD;
Do@xnew = xold + h;
fold = f �. 8x ® xold, y ® yold<;
ynew = yold + Hh � 2L * HHfoldL + f �. 8x ® xold + h, y ® yold + h * fold<L;
sollist = Append@sollist, 8xnew, ynew<D;
xold = xnew;
yold = ynew, 8steps<D;
Return@sollistDD
solution = heun@x^2 + y^2 - 1, 8x, -2, 2<, 8y, -2<, 20D
98-2, -2<, 8-1.8, -1.04<, 8-1.6, -0.537726<, 8-1.4, -0.253993<, 8-1.2, -0.1073<,
8-1., -0.0621201<, 8-0.8, -0.0973579<, 8-0.6, -0.193606<, 8-0.4, -0.327991<,
8-0.2, -0.47472<, 9-2.77556 ´ 10-16
, -0.60954=, 80.2, -0.714329<, 80.4, -0.778617<,
80.6, -0.797869<, 80.8, -0.770441<, 81., -0.694706<, 81.2, -0.566662<,
81.4, -0.377375<, 81.6, -0.108672<, 81.8, 0.276739<, 82., 0.863166<=
In[77]:= plt@3D = ListPlot@%7,
PlotStyle ® Directive@RGBColor@0., 0.95, 0.09D, [email protected], DashedD,
Joined ® TrueD
Out[77]=
-2 -1 1 2
-2.0
-1.5
-1.0
-0.5
0.5
euler@f_, 8t_, t0_, tn_<, 8y_, y0_<, steps_D :=
euler@f_, 8t_, t0_, tn_<, 8y_, y0_<, steps_D :=
Block@8told = t0, yold = y0, sollist = 88t0, y0<<, t, y, h<, h = N@Htn - t0L � stepsD;
Do@tnew = told + h;
ynew = yold + h * Hf �. 8t ® told, y ® yold<L;
sollist = Append@sollist, 8tnew, ynew<D;
told = tnew;
yold = ynew, 8steps<D;
Return@sollistDD
solution1 = euler@t^2 + y^2 - 1, 8t, -2, 2<, 8y, -2<, 20D
98-2, -2<, 8-1.8, -0.6<, 8-1.6, -0.08<, 8-1.4, 0.23328<, 8-1.2, 0.436164<,
8-1., 0.562212<, 8-0.8, 0.625428<, 8-0.6, 0.63166<, 8-0.4, 0.583459<,
8-0.2, 0.483544<, 9-2.77556 ´ 10-16
, 0.338307=, 80.2, 0.161197<, 80.4, -0.0256058<,
80.6, -0.193475<, 80.8, -0.313988<, 81., -0.36627<, 81.2, -0.33944<,
81.4, -0.228396<, 81.6, -0.0259629<, 81.8, 0.286172<, 82., 0.750551<=
2 Assignment 8.nb
plt@2D = ListPlot@%19, PlotStyle ®
Directive@RGBColor@1., 0., 0.56D, [email protected], DashedD, Joined ® TrueD
-2 -1 1 2
-2.0
-1.5
-1.0
-0.5
0.5
In[78]:= plt11 = Show@8plt@1D, plt@2D<, plt@3DD
Out[78]=
-2 -1 1 2
-2.0
-1.5
-1.0
-0.5
0.5
From these three graphs it is evident that the modified euler’s method is perfect match to the
analytical solution obtained from Mathematica but there is significant error if soultion is obtained
from Euler’s method.
Assignment 8.nb 3
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