An effective numerical algorithm for coupled systems of Emden-Fowler equations via shifted airfoil functions of the first kind
Abstract
The present paper deals with the designation of a new efficient numerical scheme for numerical solution of a class of coupled systems of Emden-Fowler equations describing several processes in applied sciences and technology. This method combines the shifted version of airfoil functions of the first kind (AFFK) and quasilinearization technique. More specifically, the quasilinearization technique is first applied to the original problem and the shifted AFFK (SAFFK) collocation matrix technique is then constructed for obtaining the solution of resulting family of submodels. We derive the error bound and analyze the convergence properties of the SAFFK. Computational and experimental simulations are carried out to describe the applicability and accuracy of the new technique. To show the benefit of the new approach, the computed numerical outcomes are compared with those results obtained via the Bernoulli and Haar wavelets collocation methods. It is evident from the numerical illustrations that the new developed scheme is superior to the existing available ones. The elapsed CPU time of the proposed method is provided.
Keyword : collocation points, convergent analysis, coupled ODEs, Emden–Fowler equations, shifted airfoil functions
How to Cite
Izadi, M., & Roul, P. (2024). An effective numerical algorithm for coupled systems of Emden-Fowler equations via shifted airfoil functions of the first kind. Mathematical Modelling and Analysis, 29(4), 781–800. https://doi.org/10.3846/mma.2024.19540
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P. Roul and K. Thula. A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Comput. Math., 96(1):85–104, 2019. https://doi.org/10.1080/00207160.2017.1417592
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Z. Sabir, M.A.Z. Raja, D. Baleanu and J.L. Guirao. Design of Gudermannian Neuroswarming to solve the singular Emden–Fowler nonlinear model numerically. Nonlinear Dyn., 106:3199–3214, 2021. https://doi.org/10.1007/s11071-021-06901-6
S.C. Shiralashetti and S. Kumbinarasaiah. Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations. Appl. Math. Comput., 315:591–602, 2017. https://doi.org/10.1016/j.amc.2017.07.071
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J. Stewart. Single Variable Essential Calculus: Early Transcendentals. Cengage Learning, 2012.
A.M. Wazwaz. The variational iteration method for solving systems of equations of Emden–Fowler type. Commun. Nonlinear Sci. Numer. Simul., 88(16):3406– 3415, 2011. https://doi.org/10.1080/00207160.2011.587513
A.M. Wazwaz, R. Rach and J.S. Duan. A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method. Math. Methods Appl. Sci., 37(1):10–19, 2014. https://doi.org/10.1002/mma.2776
J.S. Wong. On the generalized Emden–Fowler equation. SIAM Rev., 17(2):339– 360, 1975.
S. Yuzbasi and M. Izadi. Bessel-quasilinearization technique to solve the fractional-order HIV-1 infection of CD4+ T-cells considering the impact of antiviral drug treatment. Appl. Math. Comput., 431:127319, 2022. https://doi.org/10.1016/j.amc.2022.127319
H. Zou. A priori estimates for a semilinear elliptic system without variational structure and their applications. Math. Ann., 323:713–735, 2002. https://doi.org/10.1007/s002080200324
W. M. Abd-Elhameed and H.M. Ahmed. Tau and Galerkin operational matrices of derivatives for treating singular and Emden–Fowler thirdorder-type equations. Int. J. Modern Phys. C, 33(05):2250061, 2022. https://doi.org/10.1142/S0129183122500619
K. Aghigh, M. Masjed-Jamei and M. Dehghan. A survey on third and fourth kind of Chebyshev polynomials and their applications. Math. Methods Appl. Sci., 199(1):2–12, 2008. https://doi.org/10.1016/j.amc.2007.09.018
S. Ahmed, S. Jahan and K.S. Nisar. Hybrid Fibonacci wavelet method to solve fractional-order logistic growth model. Math. Methods Appl. Sci., 46(15):16218–16231, 2023. https://doi.org/10.1002/mma.9446
A.A. Alsulami, M. AL-Mazmumy, H.O. Bakodah and N. Alzaid. On coupled Lane–Emden equations arising in dusty fluid models. J. Phys. Conf. Ser., 14(5):843, 2022. https://doi.org/10.3390/sym14050843
J. Biazar and K. Hosseini. An effective modification of Adomian decomposition method for solving Emden–Fowler type systems. Natl. Acad. Sci. Lett., 40:285– 290, 2017. https://doi.org/10.1007/s40009-017-0571-4
R.N. Desmarais and S.R. Bland. Tables of properties of airfoil polynomials. Nasa Reference, Publication 1343, 1995. Available on Internet: https://ntrs.nasa.gov/api/citations/19960001864/downloads/19960001864.pdf
E.H. Doha, W.M. Abd-Elhameed and M.A. Bassuony. On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kinds. Acta Math. Sci., 35(2):326–338, 2015. https://doi.org/10.1016/S0252-9602(15)60004-2
D. Flockerzi and K. Sundmacher. On coupled Lane–Emden equations arising in dusty fluid models. J. Phys. Conf. Ser., 268:012006, 2011. https://doi.org/10.1088/1742-6596/268/1/012006
M. Izadi. An approximation technique for first Painlev´e equation. TWMS J. App. Eng. Math., 11(3):739–750, 2021. Available on Internet: https://jaem.isikun.edu.tr/web/images/articles/vol.11.no.3/12.pdf
M. Izadi and P. Roul. A highly accurate and computationally efficient technique for solving the electrohydrodynamic flow in a circular cylindrical conduit. Appl. Numer. Math., 181:110–124, 2022. https://doi.org/10.1016/j.apnum.2022.05.016
M. Izadi and P. Roul. Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications. Appl. Math. Comput., 429:127226, 2022. https://doi.org/10.1016/j.amc.2022.127226
M. Izadi, S. Yuzbasi and W. Adel. Accurate and efficient matrix techniques for solving the fractional Lotka–Volterra population model. Physica A, 600:127558, 2022. https://doi.org/10.1016/j.physa.2022.127558
S. Kumbinarasaiah, G. Manohara and G. Hariharan. Bernoulli wavelets functional matrix technique for a system of nonlinear singular Lane–Emden equations. Math. Comput. Simul., 204:133–165, 2023. https://doi.org/10.1016/j.matcom.2022.07.024
H. Madduri and P. Roul. A fast-converging iterative scheme for solving a system of Lane–Emden equations arising in catalytic diffusion reactions. J. Math. Chem., 57(2):570–582, 2019. https://doi.org/10.1007/s10910-018-0964-8
J. Mason and D. Handscomb. Chebyshev Polynomials. Chapman and Hall, New York, NY, CRC, Boca Raton, 2002. https://doi.org/10.1201/9781420036114
E. Momoniat and C. Harley. Approximate implicit solution of a Lane–Emden equation. New Astron., 11:520–526, 2006.
B. Muatjetjeja and C.M. Khalique. Noether, partial Noether operators and first integrals for the coupled Lane–Emden system. Math. Comput. Appl., 15:325– 333, 2010. https://doi.org/10.3390/mca15030325
R. Rach, J.S. Duan and A.M. Wazwaz. Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem., 52:255–267, 2014. https://doi.org/10.1007/s10910-013-0260-6
O.W. Richardson. Emission of Electricity from Hot Bodies. Longmans, New York, 1921.
P. Roul and K. Thula. A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Comput. Math., 96(1):85–104, 2019. https://doi.org/10.1080/00207160.2017.1417592
P. Roul, K. Thula and R. Agarwal. Non-optimal fourth-order and optimal sixth-order B-spline collocation methods for Lane–Emden boundary value problems. Appl. Numer. Math., 145(1):342–360, 2019. https://doi.org/10.1016/j.apnum.2019.05.004
S. Sabermahani, Y. Ordokhani and H. Hassani. General Lagrange scaling functions: application in general model of variable order fractional partial differential equations. Comput. Appl. Math., 40(8):269, 2021. https://doi.org/10.1007/s40314-021-01667-4
Z. Sabir, M.A.Z. Raja, D. Baleanu and J.L. Guirao. Design of Gudermannian Neuroswarming to solve the singular Emden–Fowler nonlinear model numerically. Nonlinear Dyn., 106:3199–3214, 2021. https://doi.org/10.1007/s11071-021-06901-6
S.C. Shiralashetti and S. Kumbinarasaiah. Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations. Appl. Math. Comput., 315:591–602, 2017. https://doi.org/10.1016/j.amc.2017.07.071
R. Singh. Analytical approach for computation of exact and analytic approximate solutions to the system of Lane–Emden–Fowler type equations arising in astrophysics. Eur. Phy. J. Plus, 133:320, 2018. https://doi.org/10.1140/epjp/i2018-12140-9
J. Stewart. Single Variable Essential Calculus: Early Transcendentals. Cengage Learning, 2012.
A.M. Wazwaz. The variational iteration method for solving systems of equations of Emden–Fowler type. Commun. Nonlinear Sci. Numer. Simul., 88(16):3406– 3415, 2011. https://doi.org/10.1080/00207160.2011.587513
A.M. Wazwaz, R. Rach and J.S. Duan. A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method. Math. Methods Appl. Sci., 37(1):10–19, 2014. https://doi.org/10.1002/mma.2776
J.S. Wong. On the generalized Emden–Fowler equation. SIAM Rev., 17(2):339– 360, 1975.
S. Yuzbasi and M. Izadi. Bessel-quasilinearization technique to solve the fractional-order HIV-1 infection of CD4+ T-cells considering the impact of antiviral drug treatment. Appl. Math. Comput., 431:127319, 2022. https://doi.org/10.1016/j.amc.2022.127319
H. Zou. A priori estimates for a semilinear elliptic system without variational structure and their applications. Math. Ann., 323:713–735, 2002. https://doi.org/10.1007/s002080200324