Question

22.2 Solve the following problem over the interval from x = 0 to 1 using a step size of 0.25 where y(0)-1. Display all your results on the same graph. r dV = (1 + 4x) (a) Analytically. (b) Using Euler's method. (c) Using Heun's method without iteration. (d) Using Ralston's method. (e) Using the fourth-order RK method.

I want Matlab code.

Answer 1

%%%Matlab code

clc;
close all;
clear all;
f=@(x , y) (1+4*x)*sqrt(y);
y1(1)=1;
x(1)=0;
h=0.25;
t=0:h:1;
%%% Analytical method
[t1 y1]=ode45(f,[0 1],y1(1));

%%% Eular method
y2(1)=1;
for n=1:length(t)-1;
y2(n+1)=y2(n)+h*f(t(n),y2(n));
end

%%%% Hen's Method
y3(1)=1;
for n=1:length(t)-1
yp=h*f(t(n),y3(n));
y3(n+1)=y3(n)+h/2*(f(t(n),y3(n))+f(t(n),yp));
end

%%%% RK-Method
yr(1)=1;
for k=1:length(t)-1
  
k1=h*feval(f,t(k),yr(k));
k2=h*feval(f,t(k)+h/2,yr(k)+k1/2);
k3=h*feval(f,t(k)+h/2,yr(k)+k2/2);
k4=h*feval(f,t(k)+h,yr(k)+k3);
yr(k+1)=yr(k)+(k1+2*k2+2*k3+k4)/6;
end

%%% Ralstan method
yrl(1)=1;
for k=1:length(t)-1
k1=h*f(t(k),yrl(k));
k2= h*f(t(k) + 2 *h / 3, yrl(k) + 2* k1 / 3 );
yrl(k+1)=yrl(k)+1/4*(k1+3*k2);
end

figure;
plot(t1,y1);
hold on
plot(t,y2,'-*');
hold on
plot(t,y3,'-o');
hold on
plot(t,yr','-^')
hold on
plot(t,yrl);
grid on
legend('Exact solution','Eular method','Hen''s Method','RK-4 method','Ralston method');
xlabel('x');
ylabel('y');

OUTPUT:

Exact solution Eular method -Hen's Methood 6 RK-4 method Ralston method 5 2 0.3 0.6 0.8 0.1 0.2 0.4 0.5 0.7 0.9