Partial differential equations of the second-order

Abstract

Partial differential equations of the second- and higher-order are encountered in a number of areas of importance to engineering and mathematical physics. As the processes that are accounted for in the development of the governing differential equations increase in complexity, the differential equations themselves tend to acquire a higher-order. The theory of second-order partial differential equations has found extensive applications in the study of problems in fluid mechanics, flow in porous media, heat conduction in solids, diffusive transport of chemicals in porous media, wave propagation in strings and membranes, and in mechanics of solids. Higher-order partial differential equations have been developed particularly in relation to studies in mechanics of solids. These include the formulation of the biharmonic equation for the solution of problems in plane elasticity and in the flow of viscous fluids, the fourth-order partial differential equations associated with the flexural mechanics of prismatic plates, the sixth-order partial differential equations used to describe the mechanics of shell structures and other fourth-order partial differential equations which describe the mechanics of plastic solids. In this chapter we shall restrict attention to the examination of second-order partial differential equations.