The Application of Bessel Function in the Definite Solution Problem of Cylindrical Coordinate System
Applied and Computational Mathematics
Volume 8, Issue 3, June 2019, Pages: 58-64
Received: Jul. 12, 2019;
Accepted: Aug. 10, 2019;
Published: Aug. 23, 2019
Views 648 Downloads 251
Wenjie He, School of Mathematics and Physics, North China Electric Power University, Baoding, China
Meiling Zhao, School of Mathematics and Physics, North China Electric Power University, Baoding, China
The variable separation method is an important method to solve the definite solution problems, especially the definite solution problems of cylinder and sphere regions. This method can solve these problems on cylinder and sphere regions, but the solving procedures are very difficult in the practical application. It is often solved by combining the properties of Bessel functions. In this paper, we propose a method combining Bessel function to solve homogeneous definite solution problem on the cylindrical coordinate system and give the algorithm of solving a definite problem. This algorithm is easy to implement and simplifies the process of calculation. Firstly, the definition and properties of Bessel function are briefly recalled, which are the first and essential step to solve the definite solution problem. Then we give the basic process of solving homogeneous definite solution problem, where consider the problem of the definite solution of the homogeneous wave equation, homogeneous heat conduction equation and Laplace equation. We analyze the solution of the Bessel equation definite solution problem under three kinds of boundary conditions and conclude the algorithm of solving a definite problem. At last, two numerical examples are provided to validate the feasibility of the proposed method.
The Application of Bessel Function in the Definite Solution Problem of Cylindrical Coordinate System, Applied and Computational Mathematics.
Vol. 8, No. 3,
2019, pp. 58-64.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
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A. Mitchell, R. Pearce, Explicit difference methods for solving the cylindrical heat conduction equation. Mathematics of Computation, 1963, vol. 17, no. 84, pp. 426-432.
C. Rossetti, Approximate expressions for the Bessel functions and their zeros. Nuovo Cimento B, 1987, vol. 100, no. 4, pp. 515–536.
E. Karatsuba. Fast evaluation of Bessel functions. Integral Transforms and Special Functions, 1993, vol. 1, no. 4, pp. 269-276.
H. Bateman, The solution of the wave equation by means of definite integrals. Bulletin of the American Mathematical Society, 1918, vol. 24, no. 6, pp. 296-301.
J. Harrison, Fast and accurate Bessel function computation. Proceedings of the 19th IEEE international symposium on computer arithmetic, 2009, pp. 104-113.
M. Higgins, A theory of the origin of microseisms. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1950, vol. 243, no. 857, pp. 1-35.
R. Higdon, Numerical absorbing boundary conditions for the wave equation. Mathematics of Computation, 1987, vol. 49, no. 179, pp. 65-90.
S. Liu, S. Liu, Special function. Meteorological press, 1988.
S. Zhou, S. Zhang, S. Sun, Special function applied in mechanical analysis. Journal of Shandong University of Technology, 1994, vol. 4, pp. 306-311.
T. Zhan, Inquiring into fixed answers to the thermal transmission equation. Journal of Dalian university, 1998, vol. 2, pp. 34-37.
W. Cheng, Application of Bessel functions in solving parabolic partial differential equations. Mathematics Learning and Research: Teaching Research Edition, 2017, vol. 14, pp. 9-10.
Y. Taitel, On the parabolic, hyperbolic and discrete formulation of the heat conduction equation. International Journal of Heat and Mass Transfer, 1972, vol. 15, no. 2, pp. 369-371.
Z. Wang, D. Guo, Introduction to special functions. Science press, 1965.
H. Wang, A general solution to the common eigenvalue problem, Journal of Qingdao University of Science and Technology, 2018, vol. 39, no. 1, pp. 134-138.
K. Parand, M. Nikarya, Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation, Nonlinear Engineering, 2019, vol. 8, pp. 688-694.
R. Gauthier, A. Mohammed, Cylindrical space fourier-Bessel mode solver for Maxwell’s wave equation. Advances in Materials, 2013, vol. 2, no. 3, pp. 32-35.