Solution Of Parabolic Partial Differential Equations By Finite Difference Methods

Abstract

This paper is concerned with finding the solutions to a particular type of partial differential equations. It will be left up to the engineers and physicists to derive the actual equations which describe a physical situation. This paper will concern itself only with the solutions to parabolic differential equations. Linear partial differential equations of degree two are frequently classified as either elliptic, hyperbolic, or parabolic. Of these three, this paper is concerned only with the last. It would seem necessary to define, in exact terms, what is to be considered a linear PDS of degree two. If the equation is reducable to the form [FORMULA] where u is the dependent variable and the x’s the independent variables, then the equation is linear and of degree two. Further, if all the A's except one are -1 or all except one +1, and the exception, eg Ak, Is zero and if only Bk is not zero, then the equation is considered parabolic. This is the definition presented by Caranahan (1). Some examples of the parabolic PDE as defined above are [FORMULA] Equations of this type arise principally in the study of heat flow and related occurrances.