How to plot this Equation in Runge-Kutta 4th-5th Order?

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Question:How to plot this Equation in Runge-Kutta 4th-5th Order?

Posted: Saha 70 Product: Maple

September 02 2022

I am trying to solve this equation in RK Method 4th-5th Order but the following error keeps happening.Please help me to solve this equations.

(1)

ODE1 := (diff(f(x), x, x, x))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f(x)]))+(1/2)(`cosα`)*eta*(diff(f(x), x, x))+(1/2)(`sinα`)*f(x)*(diff(f(x), x, x))-M(diff(f(x), x)-1)/(1-phi+phi*rho[s]/rho[f(x)]) = 0:

(diff(diff(diff(f(x), x), x), x))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f(x)]))+(1/2)*cos(alpha)*eta*(diff(diff(f(x), x), x))+(1/2)*sin(alpha)*f(x)*(diff(diff(f(x), x), x))-M(diff(f(x), x)-1)/(1-phi+phi*rho[s]/rho[f(x)]) = 0

(2)

ODE2 := K[nf]*(diff(theta(x), x, x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))(eta*`cosα`+f*`sinα`)*(diff(theta(x), x)) = 0:

K[nf]*(diff(diff(theta(x), x), x))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))*(cos(alpha)*eta+sin(alpha)*f(x))*(diff(theta(x), x)) = 0

(3)

bcs := f(0) = 0, (D(f))(0) = 0, (D(f))(25) = 1;

(4)

a1 := [phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, M = 1, sin(alpha) = 0, cos(alpha) = 1];

[phi = .1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, K[nf] = .6842, Pr = 6.2, M = 1, sin(alpha) = (1/2)*3^(1/2), cos(alpha) = 1/2, K[f] = .613]

(5)

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*eta*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/4)*eta*(diff(diff(f(x), x), x))+(1/4)*3^(1/2)*f(x)*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

1.301348831*(diff(diff(diff(f(x), x), x), x))/(.9+520.0/rho[f(x)])+(1/2)*eta*(diff(diff(f(x), x), x))-(1)(diff(f(x), x)-1)/(.9+520.0/rho[f(x)]) = 0

(6)

c1 := dsolve({b1, bcs}, numeric); c1(0); c2 := dsolve({b2, bcs}, numeric); c2(0); c3 := dsolve({b3, bcs}, numeric); c3(0); c4 := dsolve({b4, bcs}, numeric); c4(0); c5 := dsolve({b5, bcs1}, numeric); c5(0)