Adams-Moulton Method

https://www.mathworks.com/matlabcentral/fileexchange/63034-adams-bashforth-moulton-method

Numerical Methods (single step and multi step) for solving First Order Ordinary Differential Equations. 
Methods included: 
1. Euler's Method 
2. Heun's Method 
3. Fourth Order Runge Kutta methods 
4. Adams-Bashforth Method 
5. Adams-Moulton Method 
These methods are commonly used for solving IVP, a first order Initial Value Problem (IVP) is defined as a first order differential equation together with specified initial condition at t=t₀: 
y' = f(t,y) ; t0 ≤ t ≤ b with y(t₀) = y₀

There exist several methods for finding solutions of differential equations.

%% Adams-Bashworth-Moulton Predictor, corrector and other methods

%% initialization

%%

syms x y

% Range of x

xspan=[2,3];

% Step size

h=0.1

% Initial values

y0=3, x0=2

y(1)=y0;x(1)=x0;

% No. of steps

steps=(xspan(2)-xspan(1))/h;

%% Define f(x,y)

%%

f= @(x,y)x*(y-x);

%% Exact Solution

%%

[Xe,Yexact]=ode45(f,xspan,y0);

Xe=Xe(1:4:end);

Yexact=Yexact(1:4:end);

Exact_Sol=vpa([Xe Yexact],5)

%% Euler's Method

%%

for j=2:steps+1

x(j,1)=x(j-1)+h;

y(j,1)=y(j-1,1)+h*f(x(j-1,1),y(j-1,1));

end

euler=vpa([x y],5)

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

%% Heun's Method

%%

for j=2:steps+1

x(j,1)=x(j-1)+h;

k1(j-1,1)=h*f( x(j-1), y(j-1) );

k2(j-1,1)=h*f( x(j-1)+h, y(j-1)+k1(j-1));

y(j,1)=y(j-1)+0.5*(k1(j-1)+k2(j-1));

end

heuns=vpa([x y],5)

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

%% RK 4th order method

%%

for j=2:steps+1

x(j,1)=x(j-1)+h;

k1(j-1,1)=h*f( x(j-1), y(j-1) );

k2(j-1,1)=h*f( x(j-1)+h/2, y(j-1)+0.5*k1(j-1) );

k3(j-1,1)=h*f( x(j-1)+h/2, y(j-1)+0.5*k2(j-1) );

k4(j-1,1)=h*f( x(j-1)+h, y(j-1)+k3(j-1) );

y(j,1)=y(j-1)+(1/6)*(k1(j-1)+2*k2(j-1)+2*k3(j-1)+k4(j-1));

end

rk4=vpa([x y],5)

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

%% Adam-Bashworth predictor

%%

for k=5:steps+1

x(k,1)=x(k-1)+h;

y(k,1)=y(k-1) +(h/24)*( -9*f(x(k-4),y(k-4)) +37*f(x(k-3),y(k-3))...

-59*f(x(k-2),y(k-2)) +55*f(x(k-1),y(k-1)));

end

p=y;

Adam_Bashworth=vpa([x p],5)

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

%% Adam-Moulton corrector

%%

for k=5:steps+1

x(k,1)=x(k-1)+h;

y(k,1)=y(k-1) +(h/24)*( f(x(k-3),y(k-3)) -5*f(x(k-2),y(k-2))...

+19*f(x(k-1),y(k-1)) +9*f(x(k),p(k)));

end

Adam_Moulton=vpa([x y],5)

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

%%

% Created by- Bhartendu Thakur, Machine Learning & Computing

Adams-Bashworth-Moulton Predictor, corrector and other methods

initialization

syms x y

% Range of x

xspan=[2,3];

% Step size

h=0.1

h = 0.1000

% Initial values

y0=3, x0=2

y0 = 3

x0 = 2

y(1)=y0;x(1)=x0;

% No. of steps

steps=(xspan(2)-xspan(1))/h;

Define f(x,y)

f= @(x,y)x*(y-x);

Exact Solution

[Xe,Yexact]=ode45(f,xspan,y0);

Xe=Xe(1:4:end);

Yexact=Yexact(1:4:end);

Exact_Sol=vpa([Xe Yexact],5)

Exact_Sol = 

2.02.12.22.32.42.52.62.72.82.93.03.03.21643.47263.78144.16114.63665.24326.03067.06968.461810.354

Euler's Method

for j=2:steps+1

x(j,1)=x(j-1)+h;

y(j,1)=y(j-1,1)+h*f(x(j-1,1),y(j-1,1));

end

euler=vpa([x y],5)

euler = 

2.02.12.22.32.42.52.62.72.82.93.03.03.23.4313.70184.02424.41414.89265.48866.24167.20528.4537

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

mse = 2.5235

Heun's Method

for j=2:steps+1

x(j,1)=x(j-1)+h;

k1(j-1,1)=h*f( x(j-1), y(j-1) );

k2(j-1,1)=h*f( x(j-1)+h, y(j-1)+k1(j-1));

y(j,1)=y(j-1)+0.5*(k1(j-1)+k2(j-1));

end

heuns=vpa([x y],5)

heuns = 

2.02.12.22.32.42.52.62.72.82.93.03.03.21553.47013.77654.15224.62165.21865.99097.00638.361410.195

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

mse = 0.20447

RK 4th order method

for j=2:steps+1

x(j,1)=x(j-1)+h;

k1(j-1,1)=h*f( x(j-1), y(j-1) );

k2(j-1,1)=h*f( x(j-1)+h/2, y(j-1)+0.5*k1(j-1) );

k3(j-1,1)=h*f( x(j-1)+h/2, y(j-1)+0.5*k2(j-1) );

k4(j-1,1)=h*f( x(j-1)+h, y(j-1)+k3(j-1) );

y(j,1)=y(j-1)+(1/6)*(k1(j-1)+2*k2(j-1)+2*k3(j-1)+k4(j-1));

end

rk4=vpa([x y],5)

rk4 = 

2.02.12.22.32.42.52.62.72.82.93.03.03.21643.47263.78144.16114.63655.24316.03057.06948.461510.353

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

mse = 0.00066539

Adam-Bashworth predictor

for k=5:steps+1

x(k,1)=x(k-1)+h;

y(k,1)=y(k-1) +(h/24)*( -9*f(x(k-4),y(k-4)) +37*f(x(k-3),y(k-3))...

-59*f(x(k-2),y(k-2)) +55*f(x(k-1),y(k-1)));

end

p=y;

Adam_Bashworth=vpa([x p],5)

Adam_Bashworth = 

2.02.12.22.32.42.52.62.72.82.93.03.03.21643.47263.78144.16054.63485.23956.02387.05768.441310.319

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

mse = 0.042467

Adam-Moulton corrector

for k=5:steps+1

x(k,1)=x(k-1)+h;

y(k,1)=y(k-1) +(h/24)*( f(x(k-3),y(k-3)) -5*f(x(k-2),y(k-2))...

+19*f(x(k-1),y(k-1)) +9*f(x(k),p(k)));

end

Adam_Moulton=vpa([x y],5)

Adam_Moulton = 

2.02.12.22.32.42.52.62.72.82.93.03.03.21643.47263.78144.16114.63655.24286.02977.06758.457510.345

% Mean Square Error

mse = vpa(norm(Yexact-y,2),5)

mse = 0.0097564

% Created by- Bhartendu Thakur, Machine Learning & Computing