المساعد الشخصي الرقمي

مشاهدة النسخة كاملة : ساعدوني في إثبات معادلات لابلاس



**جنون أنثى*
11-24-2009, 05:53 AM
لو سمحتم أبيكم تساعدون في إثبات معادلات لابلاس في الإحداثيات الكروية والأسطوانية مع شرح خطوات الاثبات بالتفصيل، تعبت وأنا أحاول في حلها وماقدرت أوصل للحل .

Storm Strike
03-27-2012, 05:41 PM
هاي المعادلة



In spherical coordinates (http://mathworld.wolfram.com/SphericalCoordinates.html), the scale factors (http://mathworld.wolfram.com/ScaleFactor.html) are http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline1.gif, http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline2.gif, http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline3.gif, and the separation functions are http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline4.gif, http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline5.gif, http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline6.gif, giving a Stäckel determinant (http://mathworld.wolfram.com/StaeckelDeterminant.html) of http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline7.gif.
The Laplacian (http://mathworld.wolfram.com/Laplacian.html) is

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation1.gif
(1)





To solve Laplace’s equation (http://mathworld.wolfram.com/LaplacesEquation.html) in spherical coordinates (http://mathworld.wolfram.com/SphericalCoordinates.html), attempt separation of variables (http://mathworld.wolfram.com/SeparationofVariables.html) by writing

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation2.gif
(2)





Then the Helmholtz differential equation (http://mathworld.wolfram.com/HelmholtzDifferentialEquation.html) becomes

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation3.gif
(3)





Now divide by http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline8.gif,

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation4.gif
(4)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation5.gif
(5)





The solution to the second part of (5 (http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinate s.html#eqn5)) must be sinusoidal, so the differential equation is

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation6.gif
(6)





which has solutions which may be defined either as a complex (http://mathworld.wolfram.com/ComplexNumber.html) function with http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline9.gif, ..., http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline10.gif

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation7.gif
(7)





or as a sum of real (http://mathworld.wolfram.com/RealNumber.html) sine and cosine functions with http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline11.gif, ..., http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline12.gif

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation8.gif
(8)





Plugging (6 (http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinate s.html#eqn6)) back into (7 (http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinate s.html#eqn7)),

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation9.gif
(9)





The radial part must be equal to a constant

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation10.gif
(10)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation11.gif
(11)





But this is the Euler differential equation (http://mathworld.wolfram.com/EulerDifferentialEquation.html), so we try a series solution of the form (http://mathworld.wolfram.com/OftheForm.html)

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation12.gif
(12)





Then

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation13.gif
(13)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation14.gif
(14)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation15.gif
(15)





This must hold true for all powers (http://mathworld.wolfram.com/Power.html) of http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline13.gif. For the http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline14.gif term (with http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline15.gif),

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation16.gif
(16)





which is true only if http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline16.gif and all other terms vanish. So http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline17.gif for http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline18.gif, http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline19.gif. Therefore, the solution of the http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline20.gif component is given by

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation17.gif
(17)





Plugging (17 (http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinate s.html#eqn17)) back into (◇),

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation18.gif
(18)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation19.gif
(19)





which is the associated Legendre differential equation (http://mathworld.wolfram.com/LegendreDifferentialEquation.html) for http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline21.gif and http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline22.gif, ..., http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline23.gif. The general complex (http://mathworld.wolfram.com/ComplexNumber.html) solution is therefore

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation20.gif
(20)





where

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation21.gif
(21)





are the (complex (http://mathworld.wolfram.com/ComplexNumber.html)) spherical harmonics (http://mathworld.wolfram.com/SphericalHarmonic.html). The general real (http://mathworld.wolfram.com/RealNumber.html) solution is

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation22.gif
(22)





Some of the normalization constants of http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline24.gif can be absorbed by http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline25.gif and http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline26.gif, so this equation may appear in the form

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation23.gif
(23)





where

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation24.gif
(24)






http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation25.gif
(25)





are the even (http://mathworld.wolfram.com/EvenNumber.html) and odd (http://mathworld.wolfram.com/OddNumber.html) (real) spherical harmonics (http://mathworld.wolfram.com/SphericalHarmonic.html). If azimuthal symmetry is present, then http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline27.gif is constant and the solution of the http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline28.gifcomponent is a Legendre polynomial (http://mathworld.wolfram.com/LegendrePolynomial.html) http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/Inline29.gif. The general solution is then

http://mathworld.wolfram.com/images/equations/LaplacesEquationSphericalCoordinate s/NumberedEquation26.gif




انا صراحة ماعرف شو لابلاس لاكن شفتلك من الانترنيت