restart;
with(PDEtools):
with(plots):
with(LinearAlgebra):

A1 := 1:

# A2: Density coefficient
A2 := 1:

# A3: Thermal conductivity coefficient (Maxwell model)
A3 := 1:

# A4: Heat capacity coefficient  
A4 := 1:

# A5: Electrical conductivity coefficient (Maxwell model)
A5 := 1:
 

# Default parameter values (can be varied in studies)
M := 0:               # Magnetic field parameter
               # Unsteadiness parameter  
lambda_val := 0.5:      # Film thickness parameter (β²)
R := 0:               # Radiation parameter
A_star := 0.5:          # Heat source parameter
B_star := 0.5:          # Heat sink parameter
Ec := 0:              # Eckert number
Pr := 1:            # Prandtl number

OdeSys := A1 * diff(f(eta), eta, eta, eta) +
                     A2 * lambda_val * (f(eta) * diff(f(eta), eta, eta) -
                     diff(f(eta), eta)^2 - S * diff(f(eta), eta) -
                     (S * eta/2) * diff(f(eta), eta, eta)) -
                     M * A5 * diff(f(eta), eta) = 0,(A3 + (4/3)*R) * diff(theta(eta), eta, eta) -
                   Pr * A4 * lambda_val * ((S/2) * (3*theta(eta) + eta*diff(theta(eta), eta)) +
                   2*diff(f(eta), eta)*theta(eta) - f(eta)*diff(theta(eta), eta)) +
                   lambda_val * (B_star * theta(eta) + A_star * diff(f(eta), eta)) = 0;

# Boundary conditions
    Cond :=f(0) = 0, D(f)(0) = 1, theta(0) = 1, f(1) = S/2,  D(theta)(1) = 0:
#(D(D(f)))(1) = 0:

SVals := [1, 1.2, 1.4, 1.6,1.8]:



for j to numelems(SVals) do
  
        Ans[j] := dsolve(eval([OdeSys, Cond], S = SVals[j]), numeric,
                         output = listprocedure):
end do: