Advanced Mathematics For Engineering and Science
As the product of a close collaboration between two mathematicians and an engineer, this pragmatic text provides the know-how and know-why to understanding mathematics and its many applications.
TABLE OF CONTENTS Advanced Mathematics For Engineering and Science
Chapter 6: Partial Differential Equations II
6.1 INTRODUCTION
In Chapter 5, we have used the method of separation of variables and transform methods to solve partial differential equations. In this chapter, we introduce additional methods which can be used to solve P.D.E. s. We consider also non-linear equations.
P.D.E. s can be classified in three types, hyperbolic, elliptic, and parabolic, and for each type we discuss one method. Here, the hyperbolic equations are solved by the method of characteristics, a method which has been encountered in Chapter 5. The method of Green s function which was employed in Chapter 1 to solve non-homogeneous O.D.E. s is now extended to solve elliptic P.D.E. s, and a similarity variable is introduced to solve parabolic equations.
The equations we considered in Chapter 5, namely the wave equation, the diffusion (heat) equation, and Laplace s equation are all important equations in classical physics. In this chapter, we seek the solution of Schr dinger s equation which is one of the most important equations in quantum mechanics.
6.2 METHOD OF CHARACTERISTICS
In Chapter 5, we have used the method of characteristics to solve first order partial differential equations and the wave equation (d Alembert s solution). We recall that the characteristics are the curves along which information is propagated. Discontinuous initial data are carried along the characteristics. It is also shown that hyperbolic equations have real characteristics. We now describe Riemann s method of solving hyperbolic equations.
The canonical form of a linear hyperbolic equation is
| (6.2 1a,b) |  |
We assume that the prescribed conditions are...
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