Solution of Differential and Integral Equations

Abstract

Differential and integral equations result from the mathematical modeling of processes. For most complex phenomena, the number of independent variables (e.g., spacial coordinates and time) is more than one, and the full description of the process will require more than one dependent variable (e.g., velocity, temperature, displacement, stress). The governing equations for such processes are expressible in the form of partial differential equations (PDEs). These equations indicate the dependence of the process on the plethora of independent variables. For many physical phenomena, PDEs emerge from a continuum viewpoint. The continuum equations derive from statistical averaging of microscopic phenomena. For example, consider a physical process such as the movement of fluid in a container. We can look at the forces acting on a small element of this fluid. This representative elementary volume, REV, should be small enough to allow us to say we are observing the flow at a point, but large enough to make a statistical average of the microscopic events meaningful. When we write statements pertaining to the conservation of mass, momentum and energy for this element, the result will be partial differential equations. Examples include Navier-Stokes equations, Laplace’s equation, Poisson equation, etc.