A new method for deriving analytical solutions of partial differential equations — Algebraically explicit analytical solutions of two-buoyancy natural convection in porous media

Analytical solutions of governing equations of various phenomena have their irreplaceable theoretical meanings. In addition, they can also be the benchmark solutions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equation set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given. Supported by the National Natural Science Foundation of China (Grant No. 50576097) and the National Basic Research Development Program of China (Grant No. 2007CB206902)