# L-32 MCS 471 Fri 4 Nov 2022 : adamsbashforth3.jl

# Applies the method of undetermined coefficient to construct
# a 3-point Adams-Bashforth method using SymPy.

using SymPy

# declare the parameter h, the weights, and a free variable
h, c1, c2, c3, z = Sym("h, c1, c2, c3, z")
# if xn is 0, then xnminus1 must be -h, xnminus2 must be -2*h
(xn, xnminus1, xnminus2) = (0, -h, -2*h)
# we can now define the rule symbolically
rule(f) = c1*f(xn) + c2*f(xnminus1) + c3*f(xnminus2)
# the basis functions are
b1(x) = Sym("1")
b2(x) = x
b3(x) = x^2
# the right hand sides of the equations
rhs1 = SymPy.integrate(b1(z), (z, xn, xn+h))
rhs2 = SymPy.integrate(b2(z), (z, xn, xn+h))
rhs3 = SymPy.integrate(b3(z), (z, xn, xn+h))
# the rule must integrate the basis functions correctly
eq1 = rule(b1) - rhs1
eq2 = rule(b2) - rhs2
eq3 = rule(b3) - rhs3
println("The equations :")
println(eq1)
println(eq2)
println(eq3)
sol = SymPy.solve([eq1,eq2,eq3],[c1, c2, c3])
println("The solution :", sol)