From \"Differential Equations With Matlab\", Hunt ODE 45 integrates the system o

Question

From "Differential Equations With Matlab", Hunt

ODE 45  integrates the system of differential equations y'=f(t,y) from t0 to tf with initial conditions y0. Each row in the solution array y corresponds to a value returned in column vector t.

The error function erf of x is erf(x)=2/squareroot pi integral^x_0 e^-t^2 dt. erf(x) erf(x) returns the error function evaluated for each elementof x. Solve nonstiff differential equations - medium order method [t, y]= ode45 (ode fun.tspan, y 0) [t, y]= ode45 (odefun.tspan, y0, options) [t, y, te, ye, ie]= ode45 (odefun.tspan, y0, options) sol = ode45(___) The function erf, discussed in Chapter 5 and in Problem 8 in this set, is the solution to the initial value problem dy/dt=2/squareroot pi e^-t^2, y(0)=0, so if we solve this initial value problem numerically we get approximate values for the built-in function erf. Use ode45, employing the accuracy options discussed in Chapter 7, to calculate values for erf(0.1), erf(0.2), ..., erf(l) having at least 10 correct digits. Present your results in a table. In a second column print the values of erf(x) for x = 0.1, 0.2, ..., 1, obtained by using the built-in function erf. Compare the two columns of values.

Explanation / Answer

matlab code :

clc;
clear all;
close all;
format long
tspan=0:0.1:1;
y0=0;

[t,y]=ode45(@(t,y) 2/sqrt(pi)*exp(-t^2),tspan,y0);
g=erf(tspan)';

Result:

t value using ode45 using erf(x)
0 0 0
0.100000000000000 0.112462907543300 0.112462916018285
0.200000000000000 0.222702581918710 0.222702589210478
0.300000000000000 0.328626754034792 0.328626759459127
0.400000000000000 0.428392351926769 0.428392355046668
0.500000000000000 0.520499877142764 0.520499877813047
0.600000000000000 0.603856092483087 0.603856090847926
0.700000000000000 0.677801197390138 0.677801193837418
0.800000000000000 0.742100969623544 0.742100964707661
0.900000000000000 0.796908218073471 0.796908212422832
1.000000000000000 0.842700793289598 0.842700792949715

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