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e^(-2x)(d^2 f/dx^2 +d^2 f/dy^2 ) = m f
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acmuser

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domain and boundary condition?
2Â¥2012-05-22 15:46:32
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acmuser

Òø³æ (СÓÐÃûÆø)

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What is the domain and boundary condition of the problem?
3Â¥2012-05-22 15:47:00
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leedobb

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ÒýÓûØÌû:
2Â¥: Originally posted by acmuser at 2012-05-22 15:46:32:
domain and boundary condition?

ÎÒÒªµÄÊǽâÎöµÄ£¬²»ÐèÒªboundary£¬domainÕû¸öʵÊýÓò¡£

ÆäʵÈô°Ñ×ó±ß¿´³ÉËã×ÓH*f, H= e^(2x) \nabla^2£¬Æäʵ¾ÍÊÇÇóËã×ÓHµÄÌØÕ÷½â¼°ÌØÕ÷Öµm
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4Â¥2012-05-22 15:59:01
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acmuser

Òø³æ (СÓÐÃûÆø)

ÒýÓûØÌû:
4Â¥: Originally posted by leedobb at 2012-05-22 15:59:01:
ÎÒÒªµÄÊǽâÎöµÄ£¬²»ÐèÒªboundary£¬domainÕû¸öʵÊýÓò¡£

ÆäʵÈô°Ñ×ó±ß¿´³ÉËã×ÓH*f, H= e^(2x) \nabla^2£¬Æäʵ¾ÍÊÇÇóËã×ÓHµÄÌØÕ÷½â¼°ÌØÕ÷Öµm

and it is e^(-2x), so the problem is not symmetric in x and y?
when you are talking about eigenvalue problem, so m is not a pre-defined number?
5Â¥2012-05-22 16:35:33
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leedobb

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6Â¥2012-05-22 16:49:00
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leedobb: ½ð±Ò+30, ¡ï¡ï¡ï¡ï¡ï×î¼Ñ´ð°¸, лл 2012-05-23 10:41:33
If you do separation of variables, u=X(x)Y(y), you could obtain Y(y)=exp(iky), and X"(x)/X(x) = mexp(2x)+k^2,
by linearity, we can consider X"(x)/X(x) = m exp(2x) and X"(x)/X(x) separately, for the first equation, the solution is
Bessel(0,sqrt(m)exp(x)) and BesselK(0, sqrt(m)exp(x)), and the second one is exp(k*x),
therefore, a particular solution is
u = exp(iky)*(c1exp(kx)+c2BesselI(0,sqrt(m)exp(x))+c3BesselK(0,sqrt(m)exp(y))
for all m\in R, you need to specify some condition, for example, decay at \infinity to restrict the value of m.
7Â¥2012-05-23 00:16:30
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acmuser

Òø³æ (СÓÐÃûÆø)

ÒýÓûØÌû:
7Â¥: Originally posted by acmuser at 2012-05-23 00:16:30:
If you do separation of variables, u=X(x)Y(y), you could obtain Y(y)=exp(iky), and X"(x)/X(x) = mexp(2x)+k^2,
by linearity, we can consider X"(x)/X(x) = m exp(2x) and X"(x)/X(x) separ

exp(-kx) term is in the general form of u as well.
8Â¥2012-05-23 04:39:04
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leedobb

½ð³æ (ÕýʽдÊÖ)

ÒýÓûØÌû:
7Â¥: Originally posted by acmuser at 2012-05-23 00:16:30:
If you do separation of variables, u=X(x)Y(y), you could obtain Y(y)=exp(iky), and X"(x)/X(x) = mexp(2x)+k^2,
by linearity, we can consider X"(x)/X(x) = m exp(2x) and X"(x)/X(x) separ

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9Â¥2012-05-23 10:43:15
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leedobb

½ð³æ (ÕýʽдÊÖ)

X(x)''/X(x)=mexp(2x)+k^2£¬ºÃÏñ²»ÏßÐÔ¡£ÎÒÔÙÏëÏë°É¡£
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10Â¥2012-05-23 10:57:08
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