restart; with(Student[NumericalAnalysis]); with(PDEtools, casesplit, declare); x := .5; a := 1; b := .5; d := 1; Q := 2; omega := .4; h1 := 1+a*cos(x); h2 := -d-b*cos(x+omega); F := Q-1-d; de := diff(alpha*f(y), y, y, y, y)+G*(diff(theta(y), y, y))+B*(diff(phi(y), y, y))+6*beta*(diff(f(y), y, y))*(diff(f(y), y, y, y))^2+3*beta*(diff(f(y), y, y, y, y))*(diff(f(y), y, y))^2 = 0, diff(theta(y), y, y)+Nb*(diff(theta(y), y))*(diff(phi(y), y))+Nt*(diff(theta(y), y))^2 = 0, diff(phi(y), y, y)+Nb*(diff(theta(y), y, y))/Nt = 0, f(h1) = (1/2)*F, f(h2) = -(1/2)*F, (D(f))(h1) = -1, theta(h2) = 1, phi(h2) = 1, (D(f))(h2) = -1, theta(h1) = 0, phi(h1) = 0; d1 := subs(Nb = 7, Nt = 1, G = 2, B = 2, beta = .2, alpha = 2, [de]); d2 := subs(Nb = 7, Nt = 3, G = 2, B = 2, beta = .2, alpha = 2, [de]); d3 := subs(Nb = 7, Nt = 5, G = 2, B = 2, beta = .2, alpha = 2, [de]); de1 := dsolve(d1, numeric); de2 := dsolve(d2, numeric); de3 := dsolve(d3, numeric); with(plots); p1 := odeplot(de1, [y, diff(f(y), y)], linestyle = 1, color = black, legend = "Nt=1"); p2 := odeplot(de2, [y, diff(f(y), y)], linestyle = 2, color = red, legend = "Nt=3"); p3 := odeplot(de3, [y, diff(f(y), y)], linestyle = 4, color = blue, legend = "Nt=5"); plots[display]({p1, p2, p3}); p4 := odeplot(de1, [y, theta(y)], linestyle = 1, color = black, legend = "Nt=1"); p5 := odeplot(de2, [y, theta(y)], linestyle = 2, color = red, legend = "Nt=3"); p6 := odeplot(de3, [y, theta(y)], linestyle = 4, color = blue, legend = "Nt=5"); plots[display]({p4, p5, p6}); p7 := odeplot(de1, [y, phi(y)], linestyle = 3, color = black, legend = "Nt=1."); p8 := odeplot(de2, [y, phi(y)], linestyle = 1, color = red, legend = "Nt =3"); p9 := odeplot(de3, [y, phi(y)], linestyle = 2, color = blue, legend = "Nt=5"); plots[display]({p7, p8, p9});

P1 := simplify((alpha+3*beta*(diff(f(y), y, y))^2)*(diff(f(y), y, y, y))+G*theta(y)+B*phi(y)); P2 := subs(f(y), theta(y), phi(y), [P1]);P3 := int(P2, x = 0 .. 1); plot(P3, Q = -3 .. 3);

1st problem,

#` how we'll substitute f(y),theta`(y) and phi(y) in P1 that i got from above solution

Noticew that after substituting P2 must remain function of x and Q.

2nd problem,

#P1 must be the function of x and Q only. while, we already substitute x=0.5 and Q. How P1 make the function of x and Q?