# Question:problem solving coupled boundary value problem

## Question:problem solving coupled boundary value problem

Maple 13

I tried to solve two coupled non-linear ordinary differential equations that arises in case of natural convection from an isothermal vertical flat plate. The problem is a boundary value one. The boundary conditions for the problem are at eta=0, f '(0)=0, f(0)=0, g(0)=1 and at eta=infinity f '(eta)=0, g(eta)=0. f ' should go from zero to zero through positive values but after a certain value of eta f ' is going to negative values until the value of eta =infinity at which it is forced to go to zero. But g is going from 1 to 0 as is should do. (Ref: Convective Heat Transfer, Burmeister,Second ed., Page 515-516)So why is the problem coming?I shall be highly obliged if you can kindly solve my problem.

with plots

Eq[1] := diff(f(eta), `\$`(eta, 3))+3*f(eta)*(diff(f(eta), `\$`(eta, 2)))-2*(diff(f(eta), eta))^2+g(eta)

Eq[2] := (diff(g(eta), `\$`(eta, 2)))/Pr+3*f(eta)*(diff(g(eta), eta))

BCs := [f(0), (D(f))(0), (D(f))(25), g(0)-1, g(25)]

pars := {Pr = 0.1e-1}

for i to 2 do eq[i] := subs(pars, Eq[i]) end do

eqs := eq[1], eq[2]

vars := f(eta), g(eta)

bcs := op(subs(pars, BCs))

sol := dsolve({bcs, eqs}, {vars}, type = numeric, output = array([seq(i, i = 0 .. 25)]))

The problem is coming only for low values of Pr when after a certain value of eta (10 for the above problem) f ' is becoming negative and going to zero at eta=25 through negative values.

Kindly help me if you can, because it is a part of my thesis work.

Thanking you.

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