Question

Problem #3: The Ralston method is a second-order method that can be used to solve an initial-value, first-orde ordinary differential equation. The algorithm is given below: Vi#l=>: +($k+ş kz)h Where ky = f(ti,y:) * = f(mehr) You are asked to do the following: 3.1 Following that given in Inclass activity #10a, develop a MATLAB function to implement the algorithm for any given function, the time span, and the initial value. 3.2 Use your code to solve the following first-order ordinary differential equation from 1 = 0 to 1 = 1.5 with the initial condition y = 3 at t=0 with a time step of h = 0.3: dy =-1.5y + 7e-041 dt 3.3 Make a plot of your numerical solution (with symbols) of a time step of h=0.3 and compare it with the solution by the MATLAB built-in ODE solver, "ode 45". 3.4 Test the effect of the time step h on the accuracy of your numerical solutions.

Answer 1

%% 3.1
function [t,y] = ralstomMethod(f,tSpan,y0)
% f(t,y) = dy/dt
% tSpan = time span
% y0 = initial value

y(1) = y0;
t = tSpan';

n = length(tSpan)-1;

for i =1:n
h = t(i+1)-t(i);
k1 = f(t(i),y(i));
k2 = f(t(i)+3*h/4,y(i)+3*k1*h/4);
  
y(i+1) = y(i) + h*(k1+2*k2)/3;
end

end

%% 3.2
clc
clear all
y0 = 3;
tf = 1.5;
h = 0.3;
% time span
tSpan = 0:h:tf;
% differential function dy/dt = f(t,y)
f =@(t,y) -1.5*y+7*exp(-0.4*t);
% solution of differential equation using ralstomMethod method
[t,y] = ralstomMethod(f,tSpan,y0);

%% 3.3
plot(t,y);
xlabel('time (s)');
ylabel('y(t)');
grid on
hold on
% solution of differential equation using ode45 function
[t,y] =ode45(f,tSpan,y0);
plot(t,y,'-r');
legend('Ralstom Method','ode45 function');

%% 3.4
figure
grid on
hold on
colour = ['r','b','g','k'];
h = [0.3 0.25 0.1 0.05];
for i = 1:length(h)
[t,y]=ralstomMethod(f,0:h(i):tf,y0);
plot(t,y,colour(i));
end
hold off
legend('h=0.3','h=0.25','h=0.1','h=0.05');
xlabel('time (s)');
ylabel('y(t)');

3.8 Ralstom Method ode45 function 3.7 3.6 3.5 3.4 3.31 3.2 3.1 0.5 1.5 time (s)

3.8 h=0.3 -h=0.25 h=0.1 h=0.05 3.7 3.6 3.5 3.4 3.3 3.2 3.1 0.5 1.5 time (s)