How to solve this solution in R-K Method Fourth Order?

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Question:How to solve this solution in R-K Method Fourth Order?

Posted: Saha 70 Product: Maple

June 19 2022

Please Help! To Clear this Error and Plot this series in R-K Method

(1)

eq1 := (diff(f(x), `$`(x, 3)))/((1-phi)^2.5*(1-phi+phi*rho[s]/rho[f]))+(1/2)(`cosα`)*eta*(diff(f(x), `$`(x, 2)))+(1/2)(`sinα`)*f(x)*(diff(f(x), `$`(x, 2)))-M(diff(f(x), `$`(x, 1))-1)/(1-phi+phi*rho[s]/rho[f]) = 0;

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

(2)

eq2 := K[nf]*(diff(Theta(x), `$`(x, 2)))/K[f]+(1/2)*Pr*(1-phi+phi*rho[s]*c[s]/(rho[f]*c[f]))(eta*`cosα`+f*`sinα`)*(diff(Theta(x), `$`(x, 1))) = 0;

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

(3)

bcs := eta = 1, rho[s] = 5200, rho[f] = 997.1, c[s] = 670, c[f] = 4179, K[s] = 6, K[f] = .613, K[nf] = K[f]*(K[s]+2*K[f]-2*phi*(K[f]-K[s]))/(K[s]+2*K[f]+phi*(K[f]-K[s])):

(4)

a1 := [cos(beta) = 1, sin(beta) = 0, M = 1, Pr = 6.2, phi = .1];

(5)

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

(6)

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(.1)(eta*`cosα`+f*`sinα`)+(.1)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(1-(0.)(eta*`cosα`+f*`sinα`)+(0.)(eta*`cosα`+f*`sinα`)*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0

(7)

Error, (in dsolve/numeric/process_input) invalid argument: {eta = 1, K[f] = .613, K[nf] = K[f]*(K[s]+2*K[f]-2*phi*(K[f]-K[s]))/(K[s]+2*K[f]+phi*(K[f]-K[s])), K[s] = 6, c[f] = 4179, c[s] = 670, rho[f] = 997.1, rho[s] = 5200, K[nf]*(diff(diff(Theta(x), x), x))/K[f]+3.100000000*(.9+.1*rho[s](eta*`cosα`+f*`sinα`)*c[s](eta*`cosα`+f*`sinα`)/(rho[f](eta*`cosα`+f*`sinα`)*c[f](eta*`cosα`+f*`sinα`)))*(diff(Theta(x), x)) = 0}

(8)

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Question: How to solve this solution in R-K Method Fourth Order?

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