On functionally graded balls and cones

The heat equation, for both steady and unsteady situations, is considered when the material parameters are spherically symmetric functions of position. Explicit separated solutions are derived when the material parameters are exponential functions; the radial part of these solutions is given in terms of confluent hypergeometric functions or Whittaker functions. In the steady case, explicit solutions are found when the conductivity k(r) = exp (−βrq ), where β and q are parameters with q > 0. The behaviour near the tip of a spherically-graded cone is also investigated.