Adjoint symmetry and conservation law of nonlinear diffusion equations with convection and source terms

A complete adjoint symmetry classification of the nonlinear diffusion equations with convection and source terms is performed and all adjoint symmetries are expressed in a unified form X=φ(x,t)∂uX=φ(x,t)∂u, where φ(x,t)φ(x,t) satisfies a linear partial differential equation. Moreover, we find that all the adjoint symmetries are conservation law multipliers of the equation under study. We also show that the adjoint symmetries are just the substitutions of nonlinear self-adjointness and vice versa. Finally, a general conservation law formula associated with the symmetry and adjoint symmetry is given and two illustrated examples are considered.