##
**Chebyshev wavelet method for solving radiative transfer equation in a slab medium.**
*(English)*
Zbl 1431.76098

Summary: In this paper, a numerical method for solving the radiative transfer equation in a slab medium with isotropic scattering is presented. By employing the properties of Chebyshev wavelets together with the collocation method, the problem is reduced into a system of algebraic equations and the approximate solutions are computed. Moreover, numerical examples are included to demonstrate the validity and applicability of this method and a comparison is made with the existing results.

### MSC:

76M22 | Spectral methods applied to problems in fluid mechanics |

65T60 | Numerical methods for wavelets |

34B40 | Boundary value problems on infinite intervals for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

### Keywords:

radiative transfer equation; Chebyshev wavelet; integro-differential equation; collocation method
PDF
BibTeX
XML
Cite

\textit{S. Shekarpaz} et al., Tbil. Math. J. 12, No. 1, 17--31 (2019; Zbl 1431.76098)

### References:

[1] | S. P. Ahmad and D. W. Deering, A simple analytical function for bidirectional reectance, Journal of Geophysical Research: Atmospheres, vol. 97, pp. 18867-18886 (1992). |

[2] | J. M. Alam, N. K. R. Kevlahan, and O. V. Vasilyev, Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations, Journal of Computational Physics, vol. 214, no. 2, pp. 829-857 (2006). · Zbl 1089.65103 |

[3] | E. Babolian and F. Fattahzadeh, Numerical computation method in solving integral equations by using chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation, vol. 188, no. 1, pp. 1016-1022 (2007). · Zbl 1114.65366 |

[4] | G. Beylkin and J. M. Keiser, On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases, Journal of Computational Physics, vol. 132, no. 2, pp. 233-259 (1997). · Zbl 0880.65076 |

[5] | J. Canosa and H. R. Penafiel, A direct solution of the radiative transfer equation: Application to rayleigh and mie atmospheres, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 13, no. 1, pp. 21-39 (1973). |

[6] | C. K. Chui, Wavelets: A mathematical tool for signal analysis, SIAM e-books, Society for Industrial and Applied Mathematics(SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), (1997). |

[7] | B. Dai, B. Zheng, Q. Liang, and L. Wang, Numerical solution of transient heat conduction problems using improved meshless local petrov-galerkin method, Applied Mathematics and Computation, vol. 219, no. 19, pp. 10044-10052 (2013). · Zbl 1307.80008 |

[8] | A. Dayan and C. L. Tien, Heat transfer in a gray planar medium with linear anisotropic scattering, ASME Transactions Journal of Heat Transfer, vol. 97, pp. 391-396 (1975). |

[9] | S. A. El Wakil, M. H. Haggag, H. M. Machali and E. A. Saad, Padé approximant in radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 32, no. 2, pp. 173-177 (1984). |

[10] | P. D. I. Fletcher, S. J. Haswell, and V. N. Paunov, Theoretical considerations of chemical reactions in micro-reactors operating under electroosmotic and electrophoretic control, Analyst, vol. 124, pp. 1273-1282 (1999). |

[11] | A. Gachpazan, M. Kerayechian and h. zeidabadi, Finite element method for solving linear volterra integro-differential equations of the second kind, Journal of Information and Computing Science, vol. 9, no. 4, pp. 289-297 (2014). |

[12] | E. Hesameddini, S. Shekarpaz, Wavelet Solutions of the Klein-Gordon Equation, Journal of Mahani Mathematical Research Center, vol. 1, no. 1, pp. 29-45 (2012). · Zbl 1261.65104 |

[13] | M. Tavassoli Kajania, A. Hadi Vencheha and M. Ghasemib, The Chebyshev wavelets operational matrix of integration and product operation matrix, International Journal of Computer Mathematics, vol. 86, no. 7, pp. 1118-1125 (2009). · Zbl 1169.65072 |

[14] | S. Kumar, A. Majumdar and C. L. Tien, The differential-discrete-ordinate method for solutions of the equation of radiative transfer, Journal of Heat Transfer, vol. 112, pp. 424-429 (1990). vol. 15, no. 9, pp. 2284-2292 (2010). |

[15] | E-B. Lin and Y. Al-Jarrah,Wavelet Based Methods for Numerical Solutions of Two Dimensional Integral Equations, Mathematica Aeterna, vol. 4, no. 8, pp. 839-853 (2014). |

[16] | M. P. Mengüç and R. Viskanta, Comparison of radiative transfer approximations for a highly forward scattering planar medium, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 29, no. 5, pp. 381-394 (1983). |

[17] | S. Müller, Adaptive multiscale schemes for conservation laws, Lecture Notes in Computational Science and Engineering, vol. 27, Springer, (2003). · Zbl 1016.76004 |

[18] | K. Parand, Sayyed A. Hossayni, and J. A. Rad, An operation matrix method based on bernstein polynomials for riccati differential equation and volterra population model, Applied Mathematical Modelling, vol. 40, no. 2, pp. 993-1011 (2016). · Zbl 1446.65210 |

[19] | K. Parand, M. Dehghan, and A. Pirkhedri, The sinc-collocation method for solving the thomas-fermi equation, Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 244-252 (2013). · Zbl 1253.65111 |

[20] | K. Parand, M. Dehghan, and A. Taghavi, Modified generalized laguerre function tau method for solving laminar viscous flow: The blasius equation, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 20, no. 7, pp. 728-743 (2010). |

[21] | K. Parand, M. Shahini and M. Dehghan, Solution of a laminar boundary layer flow via a numerical method, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 360-367 (2010). · Zbl 1221.76157 |

[22] | G. C. Pomraning, The milne problem in a statistical medium, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 41, no. 2, pp. 103-115 (1989). |

[23] | J. A. Rad, S. Kazem, M. Shaban, K. Parand and A. Yildirim, Numerical solution of fractional differential equations with a tau method based on legendre and bernstein polynomials, Mathematical Methods in the Applied Sciences, vol. 37, no. 3, pp. 329-342 (2014). · Zbl 1290.26008 |

[24] | J. A. Rad, K. Parand and S. Abbasbandy, Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate european and american options, Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1, pp. 1178-1200 (2015). · Zbl 1329.91142 |

[25] | J. A. Rad, S. Kazem and K. Parand, The meshless method for solving radiative transfer problems in a slab medium based on radial basis functions, arXiv preprint arXiv:1408.2209 (2014). · Zbl 1252.76044 |

[26] | J. A. Rad, K. Parand and S. Abbasbandy, Pricing european and american options using a very fast and accurate scheme: The meshless local petrov-galerkin method, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 85, no. 3, pp. 337-351 (2015). · Zbl 1325.91061 |

[27] | J. A. Rad, K. Parand and L. V. Ballestra, Pricing european and american options by radial basis point interpolation, Applied Mathematics and Computation, vol. 251, pp. 363-377 (2015). · Zbl 1328.91286 |

[28] | K. Rashedi, H. Adibi, J. A. Rad and K. Parand, Application of meshfree methods for solving the inverse one-dimensional stefan problem, Engineering Analysis with Boundary Elements, vol. 40, pp. 1-21 (2014). · Zbl 1297.80005 |

[29] | K. Razi Naqvi, Milne’s problem for a non-capturing medium: Accurate analytic approximations for particle density and emergent angular distribution, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 50, no. 1, 59-64 (1993). |

[30] | M. Razzaghi and S. Yousefi, Legendre wavelets method for the solution of nonlinear problems in the calculus of variations, Mathematical and Computer Modelling, vol. 34, no. 1, pp. 45-54 (2001). · Zbl 0991.65053 |

[31] | M. Razzaghi, S. Oppenheimer and F. Ahmad, A collocation-type method for radiative transfer problems in a slab medium, Microwave and Optical Technology Letters, vol. 28, no. 5, pp. 307-311 (2001). · Zbl 1149.78318 |

[32] | M. Razzaghi, S. Oppenheimer and F. Ahmad, Numerical solution of radiative transfer problems in a slab medium by galerkin-type approximation techniques, Physica Scripta, vol. 64, no. 2, 97 (2001). · Zbl 1149.78318 |

[33] | M. Razzaghi, On the applications of orthogonal functions in pattern recognition, Smart Structures and Materials, International Society for Optics and Photonics, pp. 543-552 (2005). |

[34] | M. Razzaghi, S. Oppenheimer and F. Ahmad, Tau method approximation for radiative transfer problems in a slab medium, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 72, no. 4, pp. 439-447 (2002). |

[35] | C. E. Siewert, J. R. Maiorino and M. N. Özişik, The use of the the \(\textsc{F}_\textsc{N}\) method for radiative transfer problems with reactive boundary conditions, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 23, no. 6, pp. 565-573 (1980). |

[36] | K. Styś and T. Styś, A higher-order finite difference method for solving a system of integro-differential equations, Journal of Computational and Applied Mathematics, vol. 126, no. 12, pp. 33-46 (2000). · Zbl 0979.65124 |

[37] | V. Thomée and N. Y. Zhang, Error estimates for semidiscrete finite element methods for parabolic integro-differential equations, Mathematics of Computation, vol. 53, no. 187, pp. 121-139 (1989). · Zbl 0673.65099 |

[38] | S. T. Thynell and M. N. Özişik, A new efficient method of solution to radiation transfer in absorbing, emitting, isotropically scattering, homogeneous, finite or semi-infinite, plane-parallel media, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 36, no. 1, pp. 39-50 (1986). |

[39] | T. W. Tong and C. L. Tien, Resistance-network representation of radiative heat transfer with particulate scattering, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 24, no. 6, pp. 491-503 (1980). |

[40] | J. R. Tsai, M. N. Özişik and F. Santarelli, Radiation in spherical symmetry with anisotropic scattering and variable properties, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 42, no. 3, pp. 187-199 (1989). |

[41] | Y. Wang, Y. Mu, and P. Ding, A linear spline approximation for radiative transfer problems in slab medium, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 55, no. 1, pp. 1-5 (1996). |

[42] | A. M. Wazwaz, The combined laplace transform adomian decomposition method for handling nonlinear volterra integro-differential equations, Applied Mathematics and Computation, vol. 216, no. 4, pp. 1304-1309 (2010). · Zbl 1190.65199 |

[43] | S. J. Wilson and K. K. Sen, Generalized eddington approximation method for radiative transfer problems in slab medium, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 35, no. 6, pp. 467-472 (1986). |

[44] | X. T. Xiong, C. L. Fu and Z. Qian, Two numerical methods for solving a backward heat conduction problem, Applied Mathematics and Computation, vol. 179, no. 1, pp. 370-377 (2006). · Zbl 1102.65098 |

[45] | C. Yang and J. Hou, Chebyshev wavelets method for solving Bratu’s problem, Boundary value problems, no. 1, pp. 1-9 (2013). Science (2012). · Zbl 1296.65105 |

[46] | F. Yin, T. Tian, J. Song and M. Zhu, Spectral methods using Legendre wavelets for nonlinear Klein Sine-Gordon equations, Journal of Computational and Applied Mathematics, vol. 275, pp. 321-334 (2015). · Zbl 1334.65175 |

[47] | S. Yousefi and M. Razzaghi, Legendre wavelets method for the nonlinear volterra-fredholm integral equations, Mathematics and Computers in Simulation, vol. 70, no. 1, pp. 1-8 (2005). · Zbl 1205.65342 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.