“Helmholtz Differential Equation Polar Coordinates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Helmholtz Differential Equation Polar Coordinates

释义

Helmholtz Differential Equation Polar Coordinates

In 2-D Polar Coordinates, attempt Separation of Variables by writing

(1)

then the Helmholtz Differential Equation becomes

(2)

Divide both sides by

(3)

The solution to the second part of (3) must be periodic, so the differential equation is

(4)

which has solutions

Plug (4) back into (3)

(6)

This is an Euler Differential Equation with

and

. The roots are

.So for

and the solution is

(7)

But since

blows up at

, the only possible physical solution is

. When

, so

(8)

But since

blows up at

, the only possible physical solution is

. The solution for

isthen

(9)

for

, 1, ...and the general solution is

(10)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 502-504, 1953.

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