The biharmonic equation

Abstract

An important common theme in the developments presented in connection with Laplace’s equation, the diffusion equation and the wave equation is that they are all of the second-order and represent the fundamental equations which govern elliptic, parabolic and hyperbolic partial differential equations, respectively. A further general observation in previous expositions is that as the phenomena that are being modelled becomes either more complex or encompasses more complicated fundamental processes, the partial differential equations which describe such phenomena are expected to acquire a higher order. This was evident in the description of advection-diffusion phenomena governing the transport of chemicals in porous media. In the presence of only advective phenomena the transport process can be described by a first-order partial differential equation; when diffusive processes are taken into consideration, the transport process can be described by a second-order partial differential equation. The biharmonic equation is one such partial differential equation which arises as a result of modelling more complex phenomena encountered in problems in science and engineering. The term biharmonic is indicative of the fact that the function describing the processes satisfies Laplace’s equation twice explicitly. The exact first usage of the biharmonic equation is not entirely clear since every harmonic function which satisfies Laplace’s equation is also a biharmonic function.