An efficient collocation scheme for new type of variable-order fractional Lane–Emden equation

Document Type : Research Article

Authors

• H. Azin ¹
• A. Habibirad ¹
• E. Hesameddini ²
• M.H. Heydari ²

¹ Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

² Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.

10.22067/ijnao.2024.87501.1419

Abstract

The fractional Lane–Emden model illustrates different phenomena in astrophysics and mathematical physics. This paper involves the Vieta–Lucas (Vt-L) bases to solve types of variable-order (V-O) fractional Lane–Emden equation (linear and nonlinear). The operational matrix of the V-O fractional derivative is obtained for the Vt-L polynomials. In the established approach, these polynomials are applied to transform the main problem into an algebraic equations system. To indicate the performance and capability of the scheme, a number of examples are presented for various types of V-O fractional Lane–Emden equations. Also, for one example, a comparison is done between the calculated results by our technique and those obtained via the Bernoulli polynomials. Overall, this paper introduces a new methodology for solving V-O fractional Lane–Emden equations using Vt-L bases. The derived operational matrix and the transformation to an algebraic equation system offer practical advantages in solving these equations efficiently. The presented examples and comparative analysis highlight the effectiveness and validity of the proposed technique, contributing to the understanding and advancement of fractional Lane–Emden models in astrophysics and mathematical physics.

Keywords

• Lane–Emden equation
• Vieta–Lucas polynomials
• Variable-order fractional differential equation

Main Subjects

• Computational Mathematics

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References

[1] Abu Arqub, O., El-Ajou, A., Bataineh, A.S., and Hashim, I. A representation of the exact solution of generalized Lane–Emden equations using a new analytical method, In Abstr. Appl. Anal. (Vol. 2013, No. 1, p. 378593), Hindawi, 2013.

[2] Agarwal, P. and El-Sayed, A.A. Vieta–Lucas polynomials for solving a fractional-order mathematical physics model, Adv. Diff. Equ. 2020(1) (2020) 1–18.
[3] Bu, W., Tang, Y., Wu, Y. and Yang, J. Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations, J. Comput. Phys. 293 (2015) 264–279.
[4] Caglar, N. and Caglar, H. B-spline solution of singular boundary value problems, Appl. Math. Comput. 182(2) (2006) 1509–1513.
[5] Chambré, P.L. On the solution of the Poisson-Boltzmann equation with application to the theory of thermal explosions, J. Chem. Phys. 20(11) (1952) 1795–1797.
[6] Chandrasekhar, S. Eddington: The most distinguished astrophysicist of his time, Cambridge University Press, 1983.
[7] Choudhury, M.D., Chandra, S., Nag, S., Das, S. and Tarafdar, S. Forced spreading and rheology of starch gel: Viscoelastic modeling with fractional calculus, Colloids Surf. A: Physicochem. Eng. Asp. 407 (2012) 64–70.
[8] Coimbra, C.F.M. Mechanics with variable-order differential operators, Annal. Phys. 12(11-12) (2003) 692–703.
[9] Cooper, G.R.J. and Cowan, D.R. Filtering using variable order vertical derivatives, Comput. Geosci. 30(5) (2004) 455–459.
[10] Davis, H.T. Introduction to nonlinear differential and integral equations, US Government Printing Office, 1961.
[11] Dehghan, M. and Shakeri, F. Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron. 13(1) (2008) 53–59.
[12] Flandoli, F. and Tudor, C.A. Brownian and fractional Brownian stochastic currents via Malliavin calculus, J. Funct. Anal. 258(1) (2010) 279–306.

[13] Habibirad, A., Hesameddini, E., Heydari, M.H. and Roohi, R. An efficient meshless method based on the moving Kriging interpolation for twodimensional variable-order time fractional mobile/immobile advectiondiffusion model, Math. Method. Appl. Sci. 44(4) (2021) 3182–3194.
[14] He, J. H. and Ji, F. Y. Taylor series solution for Lane–Emden equation, J. Math. Chem. 57 (2019) 1932–1934.
[15] Heydari, M.H., Avazzadeh, Z. and Razzaghi, M. Vieta–Lucas polynomials for the coupled nonlinear variable-order fractional Ginzburg–Landau equations, Appl. Numer. Math. 165 (2021) 442–458.
[16] Heydari, M.H., Avazzadeh, Z. and Yang, Y. A computational method for solving variable-order fractional nonlinear diffusion-wave equation, Appl. Math. Comput. 352 (2019) 235–248.
[17] Heydari, M.H., Hooshmandasl, M.R. and Ghaini, F.M.M. An efficient computational method for solving fractional biharmonic equation, Comput. Math. Appl. 68(3) (2014) 269–287.
[18] Izadi, M. A discontinuous finite element approximation to singular Lane–Emden type equations, Appl. Math. Comput. 401 (2021) 126115.
[19] Izadi, M., Yüzbaşı, Ş. and Ansari, K.J. Application of Vieta–Lucas series to solve a class of multi-pantograph delay differential equations with singularity, Symmetry, 13(12) (2021) 2370.
[20] Karimi Dizicheh, A., Salahshour, S., Ahmadian, A. and Baleanu, D. A novel algorithm based on the Legendre wavelets spectral technique for solving the Lane–Emden equations, Appl. Numer. Math. 153 (2020)443–456.
[21] Khalique, C. M. and Muatjetjeja, B. Lie group classification of the generalized Lane–Emden equation, Appl. Math. Comput. 210(2) (2009) 405–410.
[22] Kilicman, A., Shokhanda, R. and Goswami, P. On the solution of (n+1)-dimensional fractional m-burgers equation, Alex. Eng. J. 60(1) (2021) 1165–1172. 

[23] Kumar, M. and Singh, N. Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems, Comput. Chem. Eng. 34(11) (2010)1750–1760.
[24] Lin, R., Liu, F., Anh, V. and Turner, I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput. 212(2) (2009) 435–445.
[25] Magin, R.L., Ingo, C., Colon-Perez, L., Triplett, W. and Mareci, T.H. Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy, Micropor. Mesopor. Mat. 178 (2013) 39–43.
[26] Moghaddam, B.P., Yaghoobi, S. and Machado, J.A.T. An extended predictor-corrector algorithm for variable-order fractional delay differential equations, J. Comput. Nonlinear Dyn. 11(6) (2016).
[27] Noor, Z.A., Talib, I., Abdeljawad, T. and Alqudah, M.A. Numerical study of Caputo fractional-order differential equations by developing new operational matrices of Vieta–Lucas polynomials, Fract. Fract. 6(2) (2022) 79.
[28] Parand, K., Yousefi, H. and Delkhosh, M. A numerical approach to solve Lane–Emden type equations by the fractional order of rational Bernoulli functions, Rom. J. Phys. 62(104) (2017) 1–24.
[29] Patnaik, S., Hollkamp, J.P. and Semperlotti, F. Applications of variableorder fractional operators: A review, Proc. R. Soc. A, 476(2234) (2020) 20190498.
[30] Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[31] Robbins, N. Vieta’s triangular array and a related family of polynomials, Int. J. Math. Math. Sci. 14(2) (1991) 239–244.

[32] Ross, B. and Samko, S. Fractional integration operator of variable order in the holder spaces hλ (x), Int. J. Math. Math. Sci. 18(4) (1995) 777–788.
[33] Sahu, P.K. and Mallick, B. Approximate solution of fractional order Lane–Emden type differential equation by orthonormal Bernoulli’s polynomials, Int. J. Appl. Comput. Math. 5(3) (2019) 1–9.
[34] Samko, S.G. and Ross, B. Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct. 1(4) (1993) 277–300 1993.
[35] Shokhanda, R. and Goswami, P. Solution of generalized fractional burgers equation with a nonlinear term, Int. J. Appl. Comput. Math. 8(5) (2022) 1–14.
[36] Shokhanda, R., Goswami, P., He, J.H. and Althobaiti, A. An approximate solution of the time-fractional two-mode coupled burgers equation, Fract. Fract. 5(4) (2021) 196.
[37] Singh, R., Das, N. and Kumar, J. The optimal modified variational iteration method for the Lane–Emden equations with Neumann and Robin boundary conditions, Eur. Phys. J. Plus, 132 (2017) 1–11.
[38] Sun, H., Chen, W. and Chen, Y.Q. Variable-order fractional differential operators in anomalous diffusion modeling, Physica A Stat. Mech. Appl. 388(21) (2009) 4586–4592.
[39] Sweilam, N.H., Nagy, A.M., Assiri, T.A. and Ali, N.Y. Numerical simulations for variable-order fractional nonlinear delay differential equations, J. Frac. Calc. Appl. 6(1) (2015) 71–82.
[40] Tseng, C.C. Design of variable and adaptive fractional order fir differentiators, Signal Process. 86(10) (2006) 2554–2566.
[41] VanGorder, R.A. An elegant perturbation solution for the Lane–Emden equation of the second kind, New Astron. 16(2) (2011) 65–67.

[42] Wazwaz, A.M. and Rach, R. Comparison of the Adomian decomposition method and the variational iteration method for solving the Lane–Emden equations of the first and second kinds, Kybernetes, 40(9-10) (2011)1305–1318.
[43] Xie, L.J., Zhou, C.L. and Song Xu, S. Solving the systems of equations of Lane–Emden type by differential transform method coupled with Adomian polynomials, Math. 7(4) (2019) 377.
[44] Zhuang, P., Liu, F., Anh, V. and Turner, I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal. 47(3) (2009) 1760–1781.