The governing equation for conduction through a continuous medium is a second order linear partial differential equation. This expression is, in effect, an assertion of the conservation of energy. It states that the heat conducted into a locale, plus the heat generated at that locale, is equivalent to the increase in energy stored by temperature at that locale.
$$ \nabla\cdot(k\nabla T)+\dot g=\rho c\frac{\partial T}{\partial t} $$
Cartesian Coordinate System Expansion
$$ \frac\partial{\partial x}(k\frac{\partial T}{\partial x})+\frac\partial{\partial x}(k\frac{\partial T}{\partial x})+\frac\partial{\partial x}(k\frac{\partial T}{\partial x})+\dot g=\rho c\frac{\partial T}{\partial t} $$
Cylindrical Coordinate System Expansion
$$ \frac 1r\frac\partial{\partial r}(kr\frac{\partial T}{\partial r})+ \frac 1{r^2}\frac\partial{\partial\phi}(k\frac{\partial T}{\partial\phi})+ \frac\partial{\partial z}(k\frac{\partial T}{\partial z})+\dot g=\rho c\frac{\partial T}{\partial t} $$
Spherical Coordinate System Expansion
$$ \frac 1{r^2}\frac\partial{\partial r}(kr^2\frac{\partial T}{\partial r})+ \frac 1{r^2\sin^2\theta}\frac\partial{\partial\phi}(k\frac{\partial T}{\partial\phi})+ \frac 1{r^2\sin\theta}\frac\partial{\partial\theta}(k\frac{\partial T}{\partial\theta})+\dot g=\rho c\frac{\partial T}{\partial t} $$
A number of assumptions may be used to simplify this equation. When steady-state conduction is reached, $\frac{\partial T}{\partial t}$ goes to zero, so the right hand side goes to zero. When no heat is internally generated, $\dot g$ also goes to zero. Finally, if the conductivity is constant, $k$ may be factored out of partial differentiation, and so the general form of the left-most term may be converted use the Laplacian operator: $\nabla\cdot(k\nabla T)=k\nabla\cdot\nabla T=k\nabla^2 T$.
Assuming steady state, constant conductivity, and single dimensionality, the following differential equation is obtained for cartesian coordinates. Given two boundary conditions such as temperatures at particular locations, or heat transfer into particular locations (which implies the gradient of temperature by the equation governing heat transfer by conduction), it is straightforward to solve the differential equation when constant heat generation is assumed.
$$ k\frac{d^2T}{d^2x}+\dot g=0 $$
Assuming steady state, constant conductivity, and two dimensionality, the following partial differential equation is obtained for cartesian coordinates.
$$ k(\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2})+\dot g=0 $$
Obtaining a solution to this equation is more tenuous, and there are analytical and numerical approaches to finding a temperature field that satisfies the partial differential equation.