| Summary: | The construction of multilayer approximate solutions of differential equations based on classical numerical methods is used to approximate special functions as solutions of the corresponding differential equations. In this paper, we investigate the Bessel equation. Multilayer methods were introduced by the authors earlier as a way to construct approximate solutions in an analytical form similar to deep learning neural networks without the need, but with the possibility of such training. The problem of approximating Bessel functions is considered classical, but it remains relevant due to the requirements of modern physics and related calculations. In this paper, we construct unified parametric approximate solutions for Bessel functions of different orders and give examples of specific approximations for negative and positive orders of Bessel functions of the first kind, including for half-integer values. Both explicit and implicit methods are considered as basic methods for constructing multi-layer solutions. The use of explicit methods has been studied in two different settings. The advantages of the obtained results in comparison with the truncated power series are shown. The possibility of constructing approximations with any given accuracy is illustrated. A method for increasing the approximation accuracy by using the optimal starting point for explicit schemes is proposed.
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