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Introduction and Fundamentals for Mathematics and Continuum Mechanics 29

1.6.5 Green Function
If L is a linear operator, a partial (or ordinary) differential equation may be written as:

L u(x,y,z) = f(x,y,z)
As L is a linear operator, it satisfies the following conditions for linear operator:
For two functions f 1 (x,y,z) and f 2 (x,y,z), and their corresponding solutions u 1 (x,y,z)
and u 2 (x,y,z), and two scalars a 1 and a 2 , the solution to L u(x,y,z) = a 1 f 1 (x,y,z) + a 2 f 2 (x,y,z)
will be a 1 u 1 (x,y,z) + a 2 u 2 (x,y,z). This is an important feature. If it is known the solution
u* to a unit source at point (x 0 , y 0 , z 0 ), that is f(x,y,z) = d(x − x 0 )d(y − y 0 )d(z − z 0 ), then one
can integrate the solution u* to obtain the solution to any source z(x 0 , y 0 , z 0 ).
Green functions are widely used in the Boundary Element Method. Interested read-
ers may find more detailed description in Qin (2007).