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The goal is to solve analytically or numerically the following two integral functions for the two unknowns $f_1(t)$ and $f_2(t)$: \begin{align} \int_0^1 f_1(t) J_{3/2} (\lambda t) \, \mathrm{d} t = F_1(\lambda) \, , \\ \int_0^1 f_2(t) J_{1/2} (\lambda t) \, \mathrm{d} t = F_2(\lambda) \, , \end{align} where the right-hand sides are explicitly given by \begin{align} F_1(\lambda) &= \frac{2\lambda^{3/2}e^{-\lambda/2} \left( (7\lambda+3) e^{-\lambda}-1 \right)}{1+2\lambda e^{-\lambda} - e^{-2\lambda}} \, , \\ F_2(\lambda) &= \frac{2\lambda^{1/2} e^{-\lambda/2} \left( (7\lambda^2-4\lambda-4)e^{-\lambda} + 4 + 8\lambda \right)}{1+2\lambda e^{-\lambda} - e^{-2\lambda}} \, . \end{align} Here, $\lambda \in [0,\infty)$.

i have tried the standard approach of expressing the solutions as $$ f_1(t) = \sum_{k=0}^N a_kt^k \, , \qquad f_2(t) = \sum_{k=0}^N b_kt^k \, , $$ ans solving numerically the resulting linear system of equations for the unknown coefficients $a_k$ and $b_k$, $k = 0, \dots, N$. However, the resulting coefficients are very large and the solution is strongly oscillating.

i was wondering whether an alternative procedure can be employed in order to obtain solutions for the above integral equations.

Any help or hints are highly appreciated and desirable.

Thank you

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1 Answer 1

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Use the Bessel orthogonality integral $$ \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda t)\; J_{\nu}(\lambda u) \;=\; \frac{1}{u} \, \delta(t - u)\, . $$ Multiply both sides of your equation by $\lambda\; J_{\nu}(\lambda u)$, where $u \in (0,1)$, and integrate from $\lambda = 0$ to $\lambda = \infty$ to yield: $$ \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u) \; \int_0^1 dt\; f(t)\; J_{\nu}(\lambda t) \;=\; \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\; F(\lambda) $$ Now assume that one can exchange the order of integration on the left hand side: $$ \int_0^1 dt\; f(t)\; \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\;J_{\nu}(\lambda t) \;=\; \int_0^1 dt\; f(t)\;\frac{1}{u} \, \delta(t - u) \;=\; \frac{f(u)}{u} $$ We are finally left with $f(u)$ as an integral: $$ f(u) \;=\; u \int_0^{\infty} d\lambda\; \lambda\; J_{\nu}(\lambda u)\; F(\lambda) $$ I leave it to you to attempt this integral for your two cases...

Edited to add: For whatever it's worth, I find that Mathematica cannot solve the two resulting integrals, which doesn't necessarily mean they don't have a closed form, just that if they do it won't be straightforward.

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