r******f
发帖数: 413 | 1 t^(3/2)=integal from 0 to infinity ( log(1-exp(t,x))) + g(t)
at x=0, there is a singulariy. g(t) has form t^1/2*log(t)+
t
It seems analytical method is impossible. So I am trying to
solve it by using numerical method. It's very hard to me.
Experienced DAXIA, please give me some suggestion...
Which method is best for solving such kind of equations?
Thanks a lot... That's the last part of my MS degree. If I
could not finish quickly, I would have to postpone my
graduation.
That's my third year as | s***e
发帖数: 911 | 2
由于你对函数exp(t,x)没有定义清楚, 以下凭猜测给出一点思路:
设你的函数exp(t,x)是e^(t*x), t<=-1. 不影响下面的思路,
这里设t=-1.
则x>0时e^(-x)<1. 于是对函数log(1-e^(-x))作无穷级数展开:
Log[1+z]=z-(z^2/2)+(z^3/3)+......, z=e^(-x).
自此以上积分可以积成求和形式. 我有印象这积分结果多半和Gamma函数有关,以前
在统计物理里碰见过. 无论如何, 你可以直接用数值方法大致检验一下积分部分
的和形式是否收敛, 然后给定一个合理的求和项, 再用牛顿法求方程的解t.
BTW: Integrate[Log[1-Exp[-t*x]],{x,0,Infinity}]对t>1是:
-Pi^2/(6*t)
刚刚用Matehmatica作的.
【在 r******f 的大作中提到】
: t^(3/2)=integal from 0 to infinity ( log(1-exp(t,x))) + g(t)
: at x=0, there is a singulariy. g(t) has form t^1/2*log(t)+
: t
: It seems analytical method is impossible. So I am trying to
: solve it by using numerical method. It's very hard to me.
: Experienced DAXIA, please give me some suggestion...
: Which method is best for solving such kind of equations?
: Thanks a lot... That's the last part of my MS degree. If I
: could not finish quickly, I would have to postpone my
: graduation.
| s***e
发帖数: 911 | 3
下面是一点math的片断:
a=2;b=1;t0=1;h=0.2;
w=(-x^2-a+Sqrt[a^2+b*x^2])/t;
y=Exp[w];
s=Series[Log[1-z],{z,0,10}];(这里取十项,你可以取11想看看最后积分是否收敛).
tmp=Normal[s]/.z->y;
tab=Table[tmp/.{t->t0+n*h},{n,1,20}];
intg=Table[{t0+k*h,Integrate[tab[[k]]{x,0,Infinity}]},{k,1,20}];
fint=Interpolation[intg,InterpolationOrder->3];
这里fint就是一个t的可导函数了(也就是那个积分). 你可以用fint[t]来访问它.
然后用FindRoot来找你方程的根.
以上的命令我check过了, 可以work.你自己调所需参数.
【在 r******f 的大作中提到】
: t^(3/2)=integal from 0 to infinity ( log(1-exp(t,x))) + g(t)
: at x=0, there is a singulariy. g(t) has form t^1/2*log(t)+
: t
: It seems analytical method is impossible. So I am trying to
: solve it by using numerical method. It's very hard to me.
: Experienced DAXIA, please give me some suggestion...
: Which method is best for solving such kind of equations?
: Thanks a lot... That's the last part of my MS degree. If I
: could not finish quickly, I would have to postpone my
: graduation.
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