I am modelling steady Joule heating within an annulus, cooled on the inner and outer edges.
I would therefore like to solve the axisymmetric heat equation with internal generation, i.e.
$$\nabla^2T + \frac{q_v}{k}=0 \label{eqn:'Heat'}$$
However, I would like to account for temperature in both the volumetric rate of generation $q_v = J^2\rho$ and conductivity $k$. Both electrical resistivity $\rho$ and $k$ have linear constitutive models (temperature coefficients $\alpha$ and $\beta$ respectively) referred to reference conditions $(\rho_0,T_0)$ and $(k_1,T_1)$ respectively, shown below:
$$\rho(T) = \rho_0(1 + \alpha(T-T_0))$$ $$k(T)=k_1(1+\beta(T-T_1))$$ Note that reference conditions for each property need not be the same, hence the different subscripts.
To simplify, the governing equation may be written:
$$\nabla^2T + \frac{A+BT}{C+DT} = 0; \quad A \equiv \frac{\rho_0J^2}{k_1}(1-\alpha T_0), \quad B \equiv \frac{\alpha\rho_0J^2}{k_1}, \quad C \equiv 1-\beta T_1, \quad D\equiv\beta$$
Thus, the equation is now slightly nonlinear (a modified Bessel equation).
I have been able to solve a version with $\beta = 0$; the resulting equation is linear and admits a solution that is a sum of Bessel functions (constants $c_1$ and $c_2$ depending on boundary conditions):
$$T_{\beta=0}(r) = c_1J_0(r\sqrt{B}) + c_2Y_0(r\sqrt{B}) - \frac{A}{B}$$
I would like to know whether there is an analytic solution more generally. I have attempted various variable substitutions (e.g. see below) but cannot find anything to remove the non-linearity.
By writing $u(r) \equiv C + DT$, Equation \ref{eqn:'Heat'} becomes (using prime notation for brevity):
$$u^{\prime\prime}+\frac{u^{\prime}}{r}+\frac{E}{u}+B = 0, \quad E \equiv AD-BC$$
Note that if there is no variation with temperature $E=0$ and we recover the canonical solution $$T_{E=0}(r) = c_1\log(r) + c_2 -\frac{q_vr^2}{4k}$$
Any insight or assistance would be much appreciated. Many thanks.