Partial Differential Equations in Semi-infinite Domains

Abstract

Mathematical modeling of mass or heat transfer in solids involves Fick’fs law of mass transfer or Fourier’s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition ition at x = 0. The dependent variable has to be finite at distances far ( x = ∞) from the origin. Both parabolic and elliptic partial differential equations will be discussed in this chapter. The Laplace transform technique will be used for parabolic partial differential equations. A similarity solution technique will be used for parabolic, elliptic and nonlinear partial differential equations.