Review on the applications of 2nd order linear P.D.E

Document Type : Original Article

Authors

1 New Cairo academy, 5th settlement, Egypt.

2 Department of Mathematics and Physical Science, New Cairo Academy, Higher Institute of Engineering and Technology, New Cairo City, Egypt.

10.21608/iugrc.2021.243392

Abstract

PDEs are very important in dynamics, elasticity, heat transfer, electromagnetic theory, and quantum mechanics by adding a few statistics to PDE it can be used in weather forecasting, prediction of crime places, disasters, how universe behave ……. Etc.
second order linear PDEs can be classified according to the characteristic equation into 3 types hyperbolic, parabolic and elliptic.
Hyperbolic equations have two distinct families of (real) characteristic curves, parabolic equations have a single family of characteristic curves, and the elliptic equations have none. All the three types of equations can be reduced to its first canonical form finding the general solution or the second canonical form similar to 3 basic PDE models.
Hyperbolic equations reduce to a form coinciding with the wave equation in the leading terms, the parabolic equations reduce to a form modeled by the heat equation, and the Laplace’s equation models the canonical form of elliptic equations. Thus, the wave, heat and Laplace’s equations serve as basic canonical models for all second order linear PDEs.

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