Let us think of a situation where a fluid (temperature given by function $T_f$) is passing over a surface (of length $l$) whose temperature is an unknown function $T(x,y,0)$. The fluid inlet temperature is known as $T_{fi}$. The heat transfer between the surface and the fluid is governed by a Robin condition as:
$$\frac{\partial T(x,y,0)}{\partial z} = c\bigg(\underbrace{\color{red}{e^{-b_cy/l}\left[T_{fi} + \frac{b_c}{l}\int_0^y e^{b_cs/l}T(x,s,0)\mathrm{d}s\right]}}_{f(y)} - T(x,y,0)\bigg) \tag 1$$
$c,b_c$ are known constants.
Is it a pre-requisite in a Robin condition that the colored term $f(y)$, in $(1)$ be a constant i.e. free of $y$ dependence ?
In my analysis I have found that when $(1)$ is expressed as: $$c^{-1}\frac{\partial T(x,y,0)}{\partial z} = \frac{1}{2}\bigg[T_{fi}(1+e^{-b_c}) + \frac{e^{-b_c}b_c}{l}\int_0^l e^{\frac{b_c s}{l}}T(x,s,z) \mathrm{d}s\bigg] - T(x,y,0) \tag 2$$
In $(2)$, I have just replaced the term $f(y)$ with an average quantity defined as $$\frac{T_{fi}+f(l)}{2}$$ This step just removes the $e^{\frac{-b_c y}{l}}$.
When I use $(2)$ to solve my governing equation (a laplacian), it yields a solution but albeit at the cost of the assumption I just mentioned ?
This leads me to ask whether the presence of a constant is necessary in a Robin condition ?