First: Data about Ti is missing.

But apart from that you clearly get results (with any reasonable Ti) and with h=1. So what is the requirement that should enable you to determine h, i.e. the temperature dependence of the heat transfer coefficient?
Do you mean to insert h as a function of the surface temperature, like h=f(Theta(rb,t)) where f is known?

First: Data about Ti is missing.

But apart from that you clearly get results (with any reasonable Ti) and with h=1. So what is the requirement that should enable you to determine h, i.e. the temperature dependence of the heat transfer coefficient?
Do you mean to insert h as a function of the surface temperature, like h=f(Theta(rb,t)) where f is known?

Could we have the data? So we could see what is going on?

Variables := ExcelTools:-Import("C:\\Users\\Documents\\Maple\\MAPLE - MARCH 2011\\InputVariables.xls", "InputVariables", "D5:D25");

@marram When defining phi at the end you make it a function of (n,t,x), but no t appears in the expression. Also in Gterm there appears lambda(j)*t. But what is lambda?

@marram When defining phi at the end you make it a function of (n,t,x), but no t appears in the expression. Also in Gterm there appears lambda(j)*t. But what is lambda?

It works for me in Maple 15 and Maple 16. The elementwise operations of a function f on a container A (list,set Vector, Matrix, Array) works by appending a tilde onto f:   f~(A);

This was introduced in Maple 13.

It works for me in Maple 15 and Maple 16. The elementwise operations of a function f on a container A (list,set Vector, Matrix, Array) works by appending a tilde onto f:   f~(A);

This was introduced in Maple 13.

@marram You write vterm := int(D[2](Gterm)(n,1,x,t-tau),tau=0..t);

but Gterm is a function of 5 variables j,n,x,y,t.

@marram You write vterm := int(D[2](Gterm)(n,1,x,t-tau),tau=0..t);

but Gterm is a function of 5 variables j,n,x,y,t.

@marram I have edited my original answer quite a lot since it was not correct. I looked up the paper you referred to and now know what the authors call the characteristic equation in this context. Sorry about the confusion!

Be aware that in the Maple version they used in 1997 the LinearAlgebra package was not yet introduced. It came with Maple 6. The old package linalg should not be used these days.

@marram I have edited my original answer quite a lot since it was not correct. I looked up the paper you referred to and now know what the authors call the characteristic equation in this context. Sorry about the confusion!

Be aware that in the Maple version they used in 1997 the LinearAlgebra package was not yet introduced. It came with Maple 6. The old package linalg should not be used these days.

@J4James Since Nc appears in the boundary condition at 0 you need to include that in the eval part, like this:

dsolve(eval({eq2,theta(9)=0,D(theta)(0)=-Nc*(1-theta(0))},{beta=1,Ec=0.5,Q=-0.5,Nc=5,pr=3,f(eta)=F(eta),diff(f(eta),eta,eta)=F2(eta)}),numeric,known=[F,F2]):

@J4James Since Nc appears in the boundary condition at 0 you need to include that in the eval part, like this:

dsolve(eval({eq2,theta(9)=0,D(theta)(0)=-Nc*(1-theta(0))},{beta=1,Ec=0.5,Q=-0.5,Nc=5,pr=3,f(eta)=F(eta),diff(f(eta),eta,eta)=F2(eta)}),numeric,known=[F,F2]):

It would be helpful if you gave us the system.

@J4James This is not simple. I tried and have uploaded a worksheet which does get something looking like fig. 2a in the paper. Since infinity = 9 it is not surprising that there is a substantial difference. However, the general shape is good. As could be expected it is tricky at the leftmost end.

MaplePrimes12-09-06B.mw