APPENDIX A: A SHORT INTRODUCTION TO THE LAPLACE 

 TRANSFORM TECHNIQUE 



1 . The Laplace transform is a powerful technique for solving linear 

 partial differential equations. This technique allows the target partial dif- 

 ferential equation to be converted to an ordinary linear differential equation 

 in the transformed plane for solving one-dimensional problems in space. The 

 Laplace transform of a function y is denoted as L{y} and is defined by the 

 operation: 



L{y} = y = / y(x,t) e" st dt (ai) 



-/y(x,, 



o 



The over bar denotes the transformed function. The transform of a derivative 

 of a function with respect to time is 



{£} = 



= sy - y(x,0) (A2) 



This relationship may be derived by performing a partial integration of Equa- 

 tion Al. The term y(x,0) represents the initial conditions for the system. 

 Accordingly, the transform of the diffusion equation may be written (if, with 

 the convention y(x,0) = , that is, a shoreline which is initially parallel 

 to the x-axis) : 



& - f y = (A3) 



dx E 



The general solution of this homogeneous linear differential equation is 



y = Ae qX + Be" qX (A4) 



where 



2 s 

 q =- 



Al 



