DSolve bvlim error due to Boundary condition at infinity

Question

I am solving an Ode using Mathematica that has boundary condition at infinity which is why Mathematica is showing error message. Please help to tackle this problem.

DSolve[t''[e] + Pr (1 - Exp[-Ke ])/K t'[e] == 0 && t[0] == 1 && t[Infinity] == 0, t, e]

The error message is

DSolve::bvlim: For some branches of the general solution, unable to compute the limit at the given points. Some of the solutions may be lost.

Answer 1

If you denote x-independent constant factor as

a = Pr (1 - Exp[-K \[Eta]])/\[Eta]

then

FullSimplify[DSolve[{t''[x] + a t'[x] == 0, t[0] == 1}, t[x], x]]

{{t[x] -> (a + C[1] - E^(-a x) C[1])/a}}

and t[Infinity] == 0 is satisfied if a>0 and C[1] = -a so answer is

E^(-a x)

You can solve these quickly by setting the limit:

Limit[DSolveValue[{t''[x] + a t'[x] == 0, t[0] == 1, t[M] == 0}, t[x], x], M -> \[Infinity], Assumptions -> a > 0]    

E^(-a x)

Limit[DSolveValue[{t''[x] + a t'[x] == 0, t[0] == 1, t[M] == 0}, t[x], x], M -> \[Infinity], Assumptions -> a < 0]    

1

Answer 2

There is a hint in the error message that the limit cannot be computed. When you have symbolic coefficients, it is possible, even likely, that the existence of the limit depends on them. If we specify the coefficient of t'[e] is positive, we get a result:

Assuming[Pr (1 - Exp[-K η])/η > 0,
 DSolve[t''[e] + Pr (1 - Exp[-K η])/η t'[e] == 0 && 
   t[0] == 1 && t[Infinity] == 0, t, e]
 ]

(*  {{t -> Function[{e}, E^((e (-1 + E^(-K η)) Pr)/η)]}}  *)

If it is not positive, then there is no solution:

Assuming[Pr (1 - Exp[-K η])/η < 0,
 DSolve[t''[e] + Pr (1 - Exp[-K η])/η t'[e] == 0 && 
   t[0] == 1 && t[Infinity] == 0, t, e]
 ]

(*  {}  *)

Likewise for the boundary case, if the coefficient is zero, which is probably not the case anyway.

Porbably one knows something about the coefficients from the context of the problem in which it arises. If the critical value of the coefficient is not obvious (greater/less than zero in this case), one might inspect the general solution. My eyes are weak so I used a big variable:

t[XXXXXX] /. DSolve[t''[e] + Pr (1 - Exp[-K η])/η t'[e] == 0 && t[0] == 1, t, e] // Simplify
(*
  {((-1 + E^(K η)) Pr - E^(K η) (-1 + 
       E^(((-1 + E^(-K η)) Pr XXXXXX)/η)) η C[1])/((-1 + E^(K η)) Pr)}
*)

The variable XXXXXX appears only in the exponential so the sign of its coefficient will determine whether the limit exists or not. Vitaliy's advice to replace a complicated coefficient with a simple symbol is good, too.