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I am trying to solve for D in the following equation:

$$ \frac{M_t}{M_\infty} = 1 - \frac{32}{\pi} * \sum_{n=1}^{25}\frac{exp(-q^2_nDt/R^2)}{q_n^2} * \sum_{p=0}^{100} \frac{exp(-(2p+1)^2)\pi^2Dt/H^2)}{(2p+1)^2} $$

Where $ q_1 < q_2 < q_3 $ are positive zeroes of the Bessel J function of order 0. It is known that: $(n-\frac{1}{4})\pi < q_n < n\pi$.

My initial instincts were to take the log of both sides of the equation but that leaves me with:

$$ ln((1-\frac{M_t}{M_\infty})\frac{\pi}{32}) = ln(\sum_{n=1}^{25} \frac{exp(-q^2_nDt/R^2)}{q_n^2}) + ln(\sum_{p=0}^{100} \frac{exp(-(2p+1)^2)\pi^2Dt/H^2)}{(2p+1)^2}) $$

This is where I'm stuck. I know that I won't be able to get an exact value of D because of $q_n$ (and will ultimately get an approximation) but I am not sure how to get rid of the exp in the infinite sum as I do not believe there are log rules for addition.

And for reference, I do have an example of the data where D has been found:

$\frac{M_t}{M_\infty}$ = 0.663228201

$t$ = 1123000

$H$ = 0.003235088

$R$ = 0.001268375

$D$ = 1.13816081338369290421042023809E-13

$q_1$ = 2.404825557695772768621631879326454643124244909145967135706

$q_2$ = 5.520078110286310649596604112813027425221865478782909853757

Update:

This is currently unsolved. However, we do now know that we are going to find the approximate value of D by taking the first sum (n) to 25 terms and the second sum (p) to 100 terms. I have updated the equation to reflect this approximation.

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    You use $n$ as a variable on the LHS of the equation & then you use it again as a dummy variable on the RHS ... so I am confused. Not mention a missing subscript in the denominator of the second sum ? – 2017-02-24
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    Sorry, that should be a t! Fixed now. – 2017-02-24
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    cool ... now the question makes sense ... can you give us clue about what we know about the zeros of the zero order Bessel function ? – 2017-02-24
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    Ok, I updated with a more clear explanation of q_n as well as gave a few data examples below. – 2017-02-24
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    Maybe there is some way to pull an exact value out of that, but it seems unlikely to me. The example you offered appears to be a good approximate answer, not an exact value. Why do you think there would be an exact way to solve it? It seems more likely that some approximate method is called for. – 2017-02-24
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    Oh, I know that I won't be able to get an exact value for D, it's getting the approximation that I'm stumped on. Any ideas on that front? – 2017-02-24

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