The biharmonic equation
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 a
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The biharmonic equation 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 furt her 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. Many of the applications of the biharmonic equation stem from the consideration of more complex mechanical and physical processes involving solids and fluids. One of the earliest applications of the biharmonic equation deals with the classical theory of flexure of elastic plates developed, among others, by J. Bernoulli (1667-1748), Euler (1707-1831), Lagrange (1736-1813), Germain (1776-1831), Poisson (1781-1840), Navier (1785-1836), Cauchy (1789-1857) and Lame (1795-1870). Developments to the mathematical modelling of the theory of plates continued with contributions by Kirchhoff (1824-1887), Levy
A. P. S. Selvadurai, Partial Differential Equations in Mechanics 2 © Springer-Verlag Berlin Heidelberg 2000
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8. The biharmonic equation
(1838-1910), J.C. Maxwell (1831-1879) and Sir Horace Lamb (1849-1934). The mathematical theory of thin plates, generally attributed to Poisson and Kirchhoff, has been extensively applied to the stress analysis of structural plates composed of both metallic and non-metallic materials. The reduction of the analysis of the two-dimensional problem in the classical theory of elasticity to the solution of the biharmonic equation is due to Airy (1801-1892), who used the calculations in the design of a structural support system for an astronomical telescope. The work of Green (1793-1841), Sir G.G. Stokes (1819-1903), Kelvin (1824-1907), J.C. Maxwell (1831-1879), Boussinesq (1842-1929), Hertz (1857-1894), Michell (1863-1940), Love (18631940), Morera (1856-1909), Beltrami (1835-1900) and Galerkin (1871-1945) in
