x s
w(x,t)=∫ ∫ u_t(ξ,t)dξds ,
0 0
a=sqrt(2)/6 , b=1/sqrt(12) , c=sqrt(2/3)
sinh(b*(ξ-ct) )
u_t(ξ,t)=-a(-------------------) = -a*(sech^2(b (ξ-ct) ))(tanh(b (ξ-ct) ))
cosh^3(b*(ξ-ct) )
x s
=>w(x,t)=-a∫∫(sech^2(b*(ξ-ct) ))(tanh(b*(ξ-ct) ))
0 0
令u=tanh(b*(ξ-ct)) , du=b*sech^2((b*(ξ-ct) ))dξ
s
=>(1/b)∫udu = (1/2b)*u^2 =(1/2b)( (tanh^2(b*(s-ct))-tanh^2(b*(0-ct ))) )
0
=(1/2b)*( (tanh^2(b*(s-ct))-tanh^2(bct)) )
x
-a*(1/2b)∫ ( (tanh^2(b*(s-ct))-tanh^2(bct)) )ds
0
x
(-a/2b)(∫ ( 1-sech^2(b*(s-ct)) )ds-s*tanh^2(bct))
0
=(-a/2b)(x-(1/b)( tanh(b-(x-ct))-tanh(b-(0-ct)) )-xtanh^2(bct)
w(x,t)=(-a/2b)*( x-(1/b)( tanh(b-(x-ct))-tanh(b-(0-ct)) )-xtanh^2(bct) )
w(x,0)=(-sqrt(6)/6)*( x-(2*sqrt(3))*tanh( x/(2*sqrt(3)) ) )
想請問各位大大我這樣的計算過程是否有誤!? <(__)>
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