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$Q=-{\frac {q \left( {{\rm e}^{{\it nb}\,{\it nv}\,\theta\,{\it Td}}}-1 \right) }{1-{{\rm e}^{\theta\,{\it Td}}}} \left( 1-{{\rm e}^{{\frac { Q\theta\,{\it Td}}{p}}}} \right) \left( 1-{{\rm e}^{{\frac {{\it nv} \,\theta\,Q{\it Td}}{p}}}} \right) ^{-1}}$

where nb, nv and td are variables.
nb and nv are integer. td is real.
above equation has not "nice" solution with respect to Q. can I approximate Q? how?

I would be thankful for any help.

Willie Wong
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sara
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    It might be helpful if you know possible range of $nb$, $n\nu$ and other variables, so that you can approximate expotentials. – Kevin Ye Apr 13 '16 at 15:28

1 Answers1

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Write the equation as $$Q = a \dfrac{1-e^{bQ}}{1-e^{cQ}}$$ For example, if $a$ is small we can get a solution as a series in powers of $a$:

$${\frac {b}{c}}a+{\frac {{b}^{2} \left( b-c \right) }{2\,{c}^{2}}}{a}^ {2}+{\frac {{b}^{3} \left( 5\,b-4\,c \right) \left( b-c \right) }{12 \,{c}^{3}}}{a}^{3}+{\frac {{b}^{4} \left( 5\,b-3\,c \right) \left( b- c \right) ^{2}}{12\,{c}^{4}}}{a}^{4}+{\frac {{b}^{5} \left( b-c \right) \left( 331\,{b}^{3}-794\,{b}^{2}c+606\,b{c}^{2}-144\,{c}^{3} \right) }{720\,{c}^{5}}}{a}^{5}+O \left( {a}^{6} \right) $$ Various other expansions are possible. We would need to know which parameters to consider as "large" or "small".

Robert Israel
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  • the range of parameters and variables is as follows 0<theta<1, 1<nb<1000,1<nv<1000, 0<P, 0<Td – sara Apr 13 '16 at 18:14
  • And $q$? That's not the same as $Q$, is it? – Robert Israel Apr 13 '16 at 23:28
  • they are different. q is positive variable and q<500. moreover 1000<P<10000. – sara Apr 14 '16 at 04:13
  • It's rather unlikely that you could get a single simple expression to approximate the solution over the whole range of all parameters and variables. To get a particular value you can use standard numerical methods. – Robert Israel Apr 15 '16 at 17:01