• 论文 •

### 第六类正交Chebyshev多项式混合Block-Pulse函数求解分数阶Lane-Emden微分方程

1. 东华理工大学理学院, 南昌 330013
• 收稿日期:2022-01-19 出版日期:2023-03-14 发布日期:2023-03-16
• 基金资助:
国家自然科学基金（11601070）；江西省自然科学基金（20202BABL201006）；东华理工大学博士科研启动项目（DHBK2019213）资助.

Huang Yingjie, Zhou Fengying, Xu Xiaoyong. USING THE SIXTH KIND OF ORTHOGONAL CHEBYSHEV POLYNOMIAL MIXED BLOCK PULSE FUNCTION TO SOLVE FRACTIONAL LANE-EMDEN DIFFERENTIAL EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2023, 44(1): 95-112.

### USING THE SIXTH KIND OF ORTHOGONAL CHEBYSHEV POLYNOMIAL MIXED BLOCK PULSE FUNCTION TO SOLVE FRACTIONAL LANE-EMDEN DIFFERENTIAL EQUATIONS

Huang Yingjie, Zhou Fengying, Xu Xiaoyong

1. School of Sciences, East China University of Technology, Nanchang 330013, China
• Received:2022-01-19 Online:2023-03-14 Published:2023-03-16

Based on the sixth kind of orthogonal Chebyshev polynomial mixed block pulse function, a numerical method for solving the numerical solutions of fractional Lane-Emden differential equations is obtained. The mixed functions consist of the sixth kind of orthogonal Chebyshev polynomials and block pulse functions. Under the definition of Rieman-Liouville fractional order integral, the fractional integral formulas of mixed functions are derived by Laplace transform. Using the mixed functions integral formulas and the effective collocation method, the fractional Lane-Emden differential equation with boundary conditions is transformed into an algebraic equation system, and then the iterative method is used for numerical solution. At the same time, the uniform convergence analysis and error estimation of mixed function expansion are also given. Numerical examples and results verify the effectiveness and accuracy of the method.

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 [1] Chandrasekhar S. An Introduction to the Study of Stellar Structure[M]. NewYork:Dover Publi cations, 1967.[2] Richardson O U. The Emission of Electricity From Hot Bodies[M]. London:Longmans Green and Company, 1921.[3] Sahu P K, Saha R S. Two-dimensional Legendre wavelet method for the numerical solutions of fuzzy integro-differential equations[J]. Journal of Intelligent and Fuzzy Systems, 2015, 28(3):1271-1279.[4] Nouh M I, Abdel-Salam E A B. Approximate Solution to the Fractional Lane-Emden Type Equations[J]. Iran J Sci Technol Trans Sci, 2018, 42:2199-2206.[5] Randhir S, Himanshu G, Vandana G. Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions[J]. Journal of Computational Applied Mathematics, 2019, 346:150-161.[6] Roul P, Kumari T, Goura V M K P. An efficient numerical approach for solving a general class of nonlinear singular boundary value problems[J]. Journal of Mathematical Chemistry, 2021, 59:1977-1993.[7] Mahboubeh M A, Zahra S. Application of Compact Finite Difference Method for Solving Some Type of Fractional Derivative Equations[J]. International Journal of Circuits, Systems and Signal Processing, 2021, 15(143):1324-1335.[8] Zhang B, Tang Y, Zhang X. Numerical solution of fractional differential equations using hybrid Bernoulli polynomials and block pulse functions[J]. Math Sci., 2021, 15:293-304.[9] Meng Z J, Yi M X, Huang J. Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials[J]. Applied Mathematics and Computation, 2018, 336:454-464.[10] Haider A M, Tahrier N. New Algorithm of the Optimal Homotopy Asymptotic Method for Solving Lane-Emden Equations[J]. Journal of Applied Mathematics and Computation, 2021, 5(4):237-246.[11] Delkhosh M, Rahmanzadeh A, Shafiei S F. Solving nonlinear differential equations in astrophysics and fluid mechanics using the generalized pseudo spectral method[J]. SeMA, 2021, 78:457-474.[12] Parand K, Mehdi D, Rezaei A R. An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method[J]. Computer Physics Communications, 2010, 181(6):1096-1108.[13] Mirshafiee S, Kajani M T, Goughery H S. Numerical solving of nonlinear differential equations using a hybrid method on a semi-infinite interval[J]. Comp. Appl. Math., 2021, 40:284.[14] Parand K, Hashemi S. RBF-DQ method for solving non-linear differential equations of LaneEmden type[J]. Ain Shams Engineering Journal, 2018, 9(4):615-629.[15] Youssri Y H, Elhameed A W M, Doha E H. Ultraspherical wavelets method for solving LaneEmden type equations[J]. Rom. J. Phys, 2015, 60(9):1298-1314.[16] Abdelhalim E. A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method[J]. Journal of Computational and Applied Mathematics, 2010, 235(8):1914-1924.[17] Zhou F Y, Xu X Y. The third kind Chebyshev wavelets collocation method for solving the timefractional convection diffusion equations with variable coefficients[J].Applied Mathematics and Computation, 2016, 280:11-29[18] 张元林,工程数学积分变换(第四版)[M].北京:高等教育出版社, 2003, 67-139.[19] Marzban H R, Malakoutikhah F. Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials[J]. Journal of the Franklin Institute, 2019, 356(15):8182-8215.[20] Sahu P K, Ray S. Hybrid Legendre Block-Pulse functions for the numerical solutions of system of nonlinear Fredholm-Hammerstein integral equations[J].Applied Mathematics and Computation, 2015, 270:871-878.[21] Abd-Elhameed W M, Youssri Y H. Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations[J]. Comp. Appl. Math., 2018, 37(3):2897-2921.[22] Charles F D,Yuan X. Orthogonal Polynomials in Several Varaibles[M]. Cambridge University Press, 2001, Ch1.5.2.[23] Kilicman A, Zhour Z A A A. Kronecker operational matrices for fractional calculus and some applications[J]. Applied Mathematics and Computation, 2007, 187(1):250-265.[24] Zhou F Y, Xu X Y. Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets[J]. Adv Differ Equ., 2016, 2016:17.[25] Kazemi N A, Pashazadeh A Z. A numerical method for solving singular fractional Lane-Emden type equations[J]. Journal of King Saud University-Science, 2018, 30(1):120-130.[26] Saadatmandi A, Ghasemi N A, Eftekhari A. Numerical study of singular fractional Lane-Emden type equations arising in astrophysics[J]. J Astrophys Astron, 2019, 40. doi:10.1007/s12036-019-9587-0
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