“Green's Function--Poisson's Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Green's Function--Poisson's Equation

释义

Green's Function--Poisson's Equation

Poisson's Equation equation is

(1)

where

is often called a potential function and

a density function, so the differential operator in this case is

. As usual, we are looking for a Green's function

such that

(2)

But from Laplacian,

(3)

(4)

and the solution is

(5)

Expanding in the Spherical Harmonics gives

(6)

where

and

are Greater Than/Less Than Symbols. This expression simplifiesto

(7)

where

are Legendre Polynomials, and

. Equations(6) and (7) give the addition theorem for Legendre Polynomials.

In Cylindrical Coordinates, the Green's function is much more complicated,

(8)

where

and

are Modified Bessel Functions of the Firstand Second Kinds (Arfken 1985).

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 485-486, 905, and 912, 1985.

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