@Preben Alsholm Alright thanks so much for your clarifications! It was really really helpful.

I'll check with my professor regarding using taylor expansion for the exponential terms and see what he says first before proceeding on

@Preben Alsholm The reason i chose 20 as a arbituary constant for eta=infinity is due to another student working on a similar project setting eta=20 as the value for inifinity too hence using eta=10 should most certainly not be an issue.

I have a few question though

pds:-plot(g*sqrt(x), x = 0 ..3 , eta = .0001, color = red);

for x=0..3 it just shows the plot range of the plot isn't it? so if i use x=0..10 it should not affect the computation of results other than the shape of the plot am i right?

and how did you replace the exponential using 1+x? does it mean that you replace the exponential terms in 

subBC1 := -phi(x, 0)*exp(g(x,0)*sqrt(x)/(1+varepsilon*g(x,0)*sqrt(x)));
subBC2 := alpha*phi(x, 0)*sqrt(x)*exp(g(x,0)*sqrt(x)/(1+varepsilon*g(x,0)*sqrt(x)));

to 

subBC1 := -phi(x, 0)*(1+x);

subBC2 := alpha*phi(x, 0)*sqrt(x)*(1+x);

I'll be meeting my professor later on so I want to clear out these issues with him 

@Preben Alsholm I there any way I would be able to plot the graph of g*sqrt(x) vs x while starting at x value of lets say 0.0001? I was thinking the error could be due to a singularity error at x=0 that is causing the error but i'm not too sure.

sorry if im asking stupid questions cos i'm really bad at coding.

@Preben Alsholm 

I dont know if this helps but this are the codes i used which was basically yours, that gave me the results

restart; Digits := 10; sys_ode := 2*(diff(T(eta), eta, eta, eta))+T(eta)*(diff(T(eta), eta, eta)) = 0; ics := T(0) = 0, (D(T))(0) = 1, (D(T))(20) = 0; sol1 := dsolve({ics, sys_ode}, numeric, output = operator); q, w := op(subs(sol1, [T, D(T)])); PDE1 := eval([2*(diff(g(x, eta), eta, eta))/P+q(eta)*(diff(g(x, eta), eta))-g(x, eta)*w(eta) = 2*x*w(eta)*(diff(g(x, eta), x)), 2*(diff(phi(x, eta), eta, eta))/S+q(eta)*(diff(phi(x, eta), eta)) = 2*x*w(eta)*(diff(phi(x, eta), x))]); subBC1 := -phi(x, 0)*exp(g(x, 0)*sqrt(x)/(1+varepsilon*g(x, 0)*sqrt(x))); subBC2 := alpha*phi(x, 0)*sqrt(x)*exp(g(x, 0)*sqrt(x)/(1+varepsilon*g(x, 0)*sqrt(x))); BC := {g(0, eta) = 0, g(x, 20) = 0, phi(0, eta) = 1, phi(x, 20) = 1, (D[2](g))(x, 0) = subBC1, (D[2](phi))(x, 0) = subBC2}; P := 1; S := 1; varepsilon := 0; alpha := 0; pds := pdsolve(PDE1, BC, numeric, spacestep = .25)

@Preben Alsholm Hey thanks for the prompt reply

I definitely dont think i copied it wrongly though since this question was based on a research done by my professor. I managed to generate a solution though by imposing alpha and epsilon = 0, S and P = 1, with subBC1 and subBC2 still included in the boundary conditions.

managed to generate :pds := module() ... end module (I assume there is a result with this expression though i might be wrong)

But when i tried to plot the graph the "Error, (in pdsolve/numeric/plot) unable to compute solution for x>HFloat(0.0):
Newton iteration is not converging" error appears

@Preben Alsholm sorry to trouble but I've encountered with further problems on this question. I've adopted your codes and have changed to boundary conditions to 

subBC1 := -phi(x, 0)*exp(g(x, 0)*sqrt(x)/(1+varepsilon*g(x, 0)*sqrt(x)));

subBC2 := alpha*phi(x, 0)*sqrt(x)*exp(g(x, 0)*sqrt(x)/(1+varepsilon*g(x, 0)*sqrt(x)))

BCtest := {g(0, eta) = 0, g(x, 20) = 0, phi(0, eta) = 1, phi(x, 20) = 1, (D[2](g))(x, 0) = subBC1, (D[2](phi))(x, 0) = subBC2}

with

P := 1; S := 1

varepsilon := 0

alpha := 0

pds := pdsolve(PDE1, BCtest, numeric, spacestep = .25)

and managed to get a solution

However i needed to plot the graph of 

which is theta vs x and phi vs x

where

but when i tried using the codes

p1 := pds:-plot(g*sqrt(x), x = 0 .. 20, eta = 0, numpoints = 100, color = red)

this error appears: 

Error, (in pdsolve/numeric/plot) unable to compute solution for x>HFloat(0.0):
Newton iteration is not converging

Is it because of a singularity developing at x=0? or what does that error actually mean

@Preben Alsholm Thanks so much! you have no idea how long I was stuck on this for. I will try to clear up the boundary conditions for the problem first then and see how i can proceed on.