# Solving integral equations of the form $\int_0^1 f(t)J_\nu(\lambda t)\,\mathrm{d}t = F(\lambda)$.

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

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...