Approximation of Bessel Functions by the Method of Constructing Multilayer Solutions of Differential Equations
Abstract
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.
References
[2] Harrison J. Fast and Accurate Bessel Function Computation. In: 2009 19th IEEE Symposium on Computer Arithmetic. Portland, OR; 2009. p. 104-113. (In Eng.) DOI: https://doi.org/10.1109/ARITH.2009.32
[3] Kadri U. Multiple-location matched approximation for Bessel function J0 and its derivatives. Communications in Nonlinear Science and Numerical Simulation. 2019; 72:59-63. (In Eng.) DOI: https://doi.org/10.1016/j.cnsns.2018.12.005
[4] Maass F., Martin P. Precise analytic approximations for the Bessel function J1(x). Results in Physics. 2018; 8:1234-1238. (In Eng.) DOI: https://doi.org/10.1016/j.rinp.2018.01.071
[5] Martin P., Olivares J., Maass F., Valero E. Analytic approximations for special functions, applied to the modified Bessel functions I2(x) and I2/3(x). Results in Physics. 2018; 11:1028-1033. (In Eng.) DOI: https://doi.org/10.1016/j.rinp.2018.10.029
[6] Maass F., Martin P., Olivares J. Analytic approximation to Bessel function J0(x). Computational and Applied Mathematics. 2020; 39(3):222. (In Eng.) DOI: https://doi.org/10.1007/s40314-020-01238-z
[7] Lazovskaya T., Tarkhov D. Multilayer neural network models based on grid methods. IOP Conf. Series: Materials Science and Engineering. 2016; 158:012061. (In Eng.) DOI: https://doi.org/10.1088/1757-899X/158/1/012061
[8] Vasilyev A., Tarkhov D., Bolgov I., Kaverzneva T., Kolesova S., Lazovskaya T., Lukinskiy E., Petrov A., Filkin V. Multilayer neural network models based on experimental data for processes of sample deformation and destruction. In: CEUR Workshop Proceedings. Selected Papers of the First International Scientific Conference Convergent Cognitive Information Technologies (Convergent 2016). 2016; 1763:6-14. Available at: http://ceur-ws.org/Vol-1763/paper01.pdf (accessed 02.09.2020). (In Russ., abstract in Eng.)
[9] Tarkhov D., Shershneva E. Approximate analytical solutions of Mathieu’s equations based on classical numerical methods. In: Selected Papers of the XI International Scientific-Practical Conference Modern Information Technologies and IT-Education (SITITO 2016.) 2016; 1761: 356-362. Available at: http://ceur-ws.org/Vol-1761/paper46.pdf (accessed 02.09.2020). (In Russ., abstract in Eng.)
[10] Vasilyev A., Tarkhov D., Shemyakina T. Approximate analytical solutions of ordinary differential equations. In: CEUR Workshop Proceedings. Selected Papers of the XI International Scientific-Practical Conference Modern Information Technologies and IT-Education (SITITO 2016). 2016; 1761:393-400. Available at: http://ceur-ws.org/Vol-1761/paper50.pdf (accessed 02.09.2020). (In Russ., abstract in Eng.)
[11] Tarkhov D., Vasilyev A. Semi-empirical Neural Network Modeling and Digital Twins Development. Academic Press, Elsevier; 2020. (In Eng.) DOI: https://doi.org/10.1016/C2017-0-02027-X
[12] Kuznetsov E.B., Leonov S.S., Tarkhov D.A., Vasilyev A.N. Multilayer method for solving a problem of metals rupture under creep conditions. Thermal Science. 2019; 23(S 2):575-582. (In Eng.) DOI: https://doi.org/10.2298/TSCI19S2575K
[13] Budkina E.M., Kuznetsov E.B., Tarkhov D.A., Gomzina A.A., Maltsev S.D. Comparison of solving a stiff equation on a sphere by the multi-layer method and method of continuing at the best parameter. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2018; 14(3):533-541. (In Russ., abstract in Eng.) DOI: https://doi.org/10.25559/SITITO.14.201803.533-541
[14] Kuznetsov E.B., Leonov S.S., Tarkhov D.A., Tsapko E.D., Babintseva A.A. Numerical methods for solving problems with contrast structures. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2018; 14(3):542-551. (In Russ., abstract in Eng.) DOI: https://doi.org/10.25559/SITITO.14.201803.542-551
[15] Kartavchenko A.E., Tarkhov D.A. Comparison of methods of construction of approximate analytical solutions of differential equations considering on the example of elementary functions. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2017; 13(3):16-23. (In Russ., abstract in Eng.) DOI: https://doi.org/10.25559/SITITO.2017.3.440
[16] Tarkhov D.A., Subbota A.D. Surikov I.Yu. The use of multilayer approximate methods of solving differential equations to the problem of stabilization of an inverted pendulum. In: V. A. Sukhomlin (ed.) Proceedings of the II International Scientific Conference and the XII International Scientific and Practical Conference. Moscow, МАКS-Press; 2017. p. 57-67. Available at: https://elibrary.ru/item.asp?id=32661941 (accessed 02.09.2020). (In Russ., abstract in Eng.)
[17] Vasilyev A.N., Tarkhov D.A., Bolgov I.P., Kaverzneva T.T., Kolesova S.A., Lazovskaya T.V., Lukinsky E.V., Petrov A.A., Filkin V.M. Multilayer neural network models based on experimental data for processes of sample deformation and destruction. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2016; 12(1):6-14. Available at: https://www.elibrary.ru/item.asp?id=27539212 (accessed 02.09.2020). (In Russ., abstract in Eng.)
[18] Tarkhov D.A., Shershneva E.A. Approximate analytical solutions of Mathieu’s equations based on classical numerical methods. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2016; 12(3-1):202-208. Available at: https://www.elibrary.ru/item.asp?id=27411993 (accessed 02.09.2020). (In Russ., abstract in Eng.)
[19] Vasilyev A.N., Lozhkin V.N., Lozhkina O.V., Tarkhov D.A., Timofeev V.D. Neural network approach in information process for predicting highway area air pollution by peat fire. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2016; 12(3-2):181-187. Available at: https://www.elibrary.ru/item.asp?id=27705974 (accessed 02.09.2020). (In Russ., abstract in Eng.)
[20] Vasilyev A.N., Tarkhov D.A., Shemyakina T.A. Approximate analytical solutions of ordinary differential equations. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2016; 12(3-2):188-195. Available at: https://www.elibrary.ru/item.asp?id=27705975 (accessed 02.09.2020). (In Russ., abstract in Eng.)
[21] Kaverzneva T.T., Malykhina G.F., Tarkhov D.A. From Differential Equations to Multilayer Neural Network Models. In: H. Lu, H. Tang, Z. Wang (ed.) Advances in Neural Networks – ISNN 2019. ISNN 2019. Lecture Notes in Computer Science. 2019; 11554:19-27. Springer, Cham. (In Eng.) DOI: http://doi.org/10.1007/978-3-030-22796-8_3
[22] Zulkarnay I.U., Kaverzneva T.T., Tarkhov D.A., Tereshin V.A., Vinokhodov T.V., Kapitsin D.R. A Two-layer Semi Empirical Model of Nonlinear Bending of the Cantilevered Beam. IOP Conf. Series: Journal of Physics: Conf. Series. 2017 IMEKO TC1-TC7-TC13 Joint Symposium. 2018; 1044:012005. (In Eng.) DOI: http://doi.org/10.1088/1742-6596/1044/1/012005
[23] Vasilyev A.N., Tarkhov D.A., Tereshin V.A., Berminova M.S., Galyautdinova A.R. Semi-empirical Neural Network Model of Real Thread Sagging. In: B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (ed.) Advances in Neural Computation, Machine Learning, and Cognitive Research. NEUROINFORMATICS 2017. Studies in Computational Intelligence. 2018; 736:138-144. Springer, Cham. (In Eng.) DOI: http://doi.org/10.1007/978-3-319-66604-4_21
[24] Lazovskaya T., Tarkhov D., Vasilyev A. Multi-Layer Solution of Heat Equation. In: B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko (ed.) Advances in Neural Computation, Machine Learning, and Cognitive Research. NEUROINFORMATICS 2017. Studies in Computational Intelligence. 2018; 736:17-22. Springer, Cham. (In Eng.) DOI: https://doi.org/10.1007/978-3-319-66604-4_3
[25] Takhov D.A., Bortkovskaya M.R., Kaverzneva T.T., Kapitsin D.R., Shishkina I.A., Semenova D.A., Udalov P.P., Zulkarnay I.U. Semiempirical Model of the Real Membrane Bending. In: B. Kryzhanovsky, W. Dunin-Barkowski, V. Redko, Y. Tiumentsev (ed.) Advances in Neural Computation, Machine Learning, and Cognitive Research II. NEUROINFORMATICS 2018. Studies in Computational Intelligence. 2019; 799:221-226. Springer, Cham. (In Eng.) DOI: https://doi.org/10.1007/978-3-030-01328-8_26
[26] Verzhbitsky V.M. Chislennye metody. Matematicheskij analiz i obyknovennye differencial'nye uravnenija[Numerical methods. Mathematical analysis and ordinary differential equations]. Moscow, Onyx 21st Century; 2005. (In Russ.)
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