ABSTRACT
Mathematical models comprise a set of equations whose solution describes the behavior of the related physical problem. Mathematical models that are described by differential equations that include functions of several variables and their partial derivatives are used to formulate problems in science and engineering. The realization that entirely different physical problems may have identical or similar mathematical formulations is a stimulus for new advances in science. Mathematical models are created taking into account the phenomena present in the process and by applying laws and theories that describe the mass conservation, momentum, and energy [1]. In general, a mathematical model is a simplified description of real problems. However, building mathematical models requires a thorough understanding of the problem to be solved. This demand produces nonlinear models, and the significant progress concerning the classic solution of partial differential equations (PDE) made over the last two centuries only solves a small part of the simplest real problems. Finding their solutions by analytical procedures such as Laplace and Fourier methods is either impossible or impracticable since usually analytic solution are in the form of an integral or a series, requiring numerical evaluations. Thus most applications require numerical solutions, and efficient numerical algorithms are needed to represent an approximation to the equations describing the mathematical model. For that reason, it remains highly desirable to know some aspects of the behavior of the solution and the classical conventional classification of partial differential equations is extremely useful. For second-order linear partial differential equations, the theory defines three distinct kinds of partial differential equations: elliptic, hyperbolic and parabolic. We shall adopt a less rigorous classification based on the general properties of their solutions. For an elliptic equation, we can think of the Laplace equation. An example of a hyperbolic equation is the wave equation, and the diffusion equation is representative of a parabolic equation [2]. Many other problems have solutions exhibiting behaviors that change from one region to another [3].
