JAMC
Accurate and Efficient Computations of the Gordeyev Integral
TOTAL VIEWS: 5574
Abstract
Investigation of the dispersion properties of plasma waves and associated phenomena is important in many fields of science. The dispersion relation of a magnetized hot Maxwellian plasma can be expressed in terms of the transcendental Gordeyev integral, 
with R (ν)>0. In a general physical situation, the variables ν,ω and λ are complex. An accurate and efficient algorithm to calculate this transcendental integral is missing in the literature. In this paper, we present two accurate algorithms to calculate this important function: (a) a reference algorithm in which the Gordeyev integral is reformulated in terms of equivalent integrals of real functions where standard adaptive quadrature can be used to evaluate the integrals; and (b) an accurate and relatively efficient algorithm, in terms of an infinite series sum, employing recent developments in the calculation of the Faddeyeva or plasma dispersion function. The present algorithms can be implemented in any software package or any programming language. Validation and reliability of the present algorithms have been established through comparison between the two independent techniques in several computational platforms including modern Fortran, MapleTM 2015 and MatlabTM in addition to comparison with results of some practical physical problems existing in the literature.
References
[1] Gordeyev, G. V. (1952). Plasma Oscillations in a Magnetic Field. Sov. Phys. JETP 6.
[2] Johnston, G. L. (1973). Representation of Dielectric Function of Magnetized Plasma. The Physics of Fluids, V. 16., No. 9.
[3] Schmitt, J. P. M. (1974). The magnetoplasma dispersion function: Some Mathematical Properties. J. Plasma Phys. 12, 51-59.
[4] Paris, R. B. (1998). The asymptotic expansion of Gordeyev’s integral. Z. angew. Math. Phys. 49, 322-338.
[5] Nijimbere, V. (2019). Analytical and asymptotic evaluations of Dawson’s integral and related functions in mathematical physics. Journal of Applied Analysis, 25(2), 179-188.
[6] Maroli, C., Petrillo, V., and Ganoutas, E. (1988). An asymptotic form of the Gordeyev function for initial value problems. Europhys. Lett., 5(3), 229-233.
[7] Bernstein, I. B. (1958). Waves in a Plasma in a Magnetic Field. Phys. Rev., 109, 10-21.
[8] Matlab. (2017). Version 9.2.0.538062 (R2017a). The Mathworks, Inc. Natick, Massachusetts.
[9] Jones, R. E. (SNLA). (1993). “DGaus8.f”, SLATEC Common Mathematical Library, Version 4.1, July 1993, http://www.netlib.org/slatec/index.html.
[10] Amos, D. E. (1986). ALGORITHM 644 A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order. ACM Transactions on Mathematical Software, Vol. 12, No. 3, Pages 265-273.
[11] Amos, D. E. (1995). A Remark on ALGORITHM 644: “A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”. ACM Transactions on Mathematical Software, Vol. 21, No. 4, Pages 388-393.
[12] Zaghloul, M. R. and Ali, A. N. (2011). Algorithm 916: Computing the Faddeyeva and Voigt Functions. ACM Trans. Math. Software (TOMS). Vol. 38, No. 2, article 15:1-22.
[13] Zaghloul, M. R. (2016). Remark on “Algorithm 916: Computing the Faddeyeva and Voigt Functions”: Efficiency Improvements and Fortran Translation. ACM Trans. Math. Software (TOMS), Vol. 42, No. 3, Article 26:1-9.
[14] Zaghloul, M. R. (2017). Algorithm 985: Simple, Efficient and Relatively Accurate Approximation for the Evaluation of Faddeyeva Function. ACM Transactions on Mathematical Software, Vol. 44, No. 2, Article 22.
[15] Zaghloul, M. R. (2018). A Fortran package for efficient multi-accuracy computations of the Faddeyeva function and related functions of complex arguments arXiv preprint arXiv:1806.01656.
[16] Zaghloul, M. R. (2019). Remark on “Algorithm 680: evaluation of the complex error function”: Cause and Remedy for the Loss of Accuracy near the Real Axis. ACM Transactions on Mathematical Software (TOMS), 45(2), 1-3.
[17] Cody, W. J. (1993). Algorithm 715: SPECFUNA portable Fortran Package of Special Function Routines and Test Drivers. ACM Transactions on Mathematical Software, Vol. 19, No. 1, 22-32.
[18] Maple. (2015). Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.
[19] Pécseli, H. L. (2013). Waves and Oscillations in Plasma. CRC Press by Taylor & Francis.
How to cite this paper
Accurate and Efficient Computations of the Gordeyev Integral
How to cite this paper: Mofreh R. Zaghloul. (2022) Accurate and Efficient Computations of the Gordeyev Integral. Journal of Applied Mathematics and Computation, 6(2), 219-229.
DOI: http://dx.doi.org/10.26855/jamc.2022.06.006
More Articles
- TRIMM: A Propagation Model Based on Authentic Data
- An Inequality with Doubling Measure in (quasi-) Banach Function Spaces
- Research Trends in Adversarial Machine Learning
- A Laplace Adomian Decomposition Method for Fractional Order Infection Model
- The Influence of m-σ-embedded Subgroups on the Structure of Finite Groups
- Research on Signal Timing System Optimization under the Background of Intelligent Transportation