I've been trying to solve a similar integral equation to the following :
$$\int_{-\infty }^{+\infty} \int_{0}^{r*}\frac{1}{t}\int_{0}^{t}\left ( \sum_{n=0}^{\infty} J_{0}(\frac{\beta_{n}}{d}r)e^{\frac{\beta_{n}^{2}Dt}{d^{2}}} \right )dtdr\left ( \frac{1}{\sqrt{4\pi Dt}}e^{\frac{u^{2}}{4Dt}} \right )du$$
Here, $J_{0}$ is a Bessel function of order 1 and $\beta_{n}$ is $n^{th}$ root of Bessel function of order 1; D, and d are constants.
I tried solving the first integral manually, got the following solution to the first integral as:
$$\int_{-\infty }^{+\infty} \int_{0}^{r*}\frac{1}{t}\sum_{n=0}^{\infty} J_{0}(\frac{\beta_{n}}{d}r)\frac{(1-e^{\frac{\beta_{n}^{2}Dt}{d}})}{\frac{\beta_{n}^{2}D}{d}}dr\left ( \frac{1}{\sqrt{4\pi Dt}}e^{\frac{u^{2}}{4Dt}} \right )du$$
For second integral since only the Bessel function is changing with respect to $dr$, we could consider the rest as constant for second integration.
Now, for second integral I used two methods using Wolfram and by hand using table of integral.
1st Method:
I directly tried to solve in Wolfram, using the Integrate function. But, I faced some error in solving the third integral. By applying Integrate[L, {u, -Infinity, Infinity}, Assumptions -> v > 0 && x >= 0 && d > 0 && k >= 0 && t >= 0], where L represent the function of interest.
Please refer the attached notebook for the same.
2nd Method:
I tried solving the second integral using the table of integral. Where, definite integration of Bessel function is directly expressed as:
$$\int_{0}^{a}J_{v} (x)dx = 2\sum_{k=0}^{\infty}J_{v+2k+1}(a)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[Re\: v > -1] $$
Which would be valid in my case since, v = 0. Using the above expression for solving the second integral.
Now, for third integral, I again used Wolfram, and faced same issue for solving third integral. Please refer to the attached notebook for the same.
Please let me know the reason for the problem and how I could rectify it to get the desired solution. I would really appreciate your help. Thank you.
Attachments:
