Polytropic Dynamical Systems with Time Singularity
Abstract
In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.
References
Ala'yed, O., Saadeh, R., & Qazza, A. (2023). Numerical solution for the system of Lane-Emden type equations using cubic B-spline method arising in engineering. AIMS Mathematics, 8(6). https://doi.org/10.3934/math.2023754
Ali, K. K., Mehanna, M. S., Abdelrahman, M. I., & Shaalan, M. A. (2022). Analytical and numerical solutions for the fourth-order Lane-Emden-Fowler equation. Partial Differential Equations in Applied Mathematics, 6. https://doi.org/10.1016/j.padiff.2022.100430
Asadpour, A., Hosseinzadeh, H., & Yazdani, A. (2019). Numerical solution of the Lane-Emden equations with moving least squares method. Applications and Applied Mathematics, 14(2).
Barreira, L., & Valls, C. (2010). Ordinary differential equations. American Mathematical Society.
Bataineh, A. S., Noorani, M. S. M., & Hashim, I. (2009). Homotopy analysis method for singular IVPs of Emden-Fowler type. Communications in Nonlinear Science and Numerical Simulation, 14, 1121-1131. https://doi.org/10.1016/j.cnsns.2008.02.004
Biles, D. C., Robinson, M. P., & Sparker, J. S. (2002). A generalization of the Lane-Emden equation. Journal of Mathematical Analysis and Applications, 654-666. https://doi.org/10.1016/S0022-247X(02)00296-2
Burden, R. L., & Faires, J. D. (2011). Numerical analysis. Brooks/Cole Publishing Company.
Chandrasekhar, S. (1967). An introduction to the study of Stellar structure. Dover Publications.
Collings, A. G., & Tee, G. J. (1979). Stability and accuracy of the generalized Euler method for ordinary differential equations, with reference to structural dynamics problems. Engineering Structures, 2, 99-108. https://doi.org/10.1016/0141-0296(79)90019-1
Datta, B. K. (1996). Analytic solution to the Lane-Emden equation. Nuovo Cimento, 111, 1385-1388. https://doi.org/10.1007/BF02742511
Davis, H. T. (1962). Introduction to nonlinear differential and integral equations. Dover Publications Inc.
Gregory, J., & Hughes, H. R. (2003). New general methods for numerical stochastic differential equations. Utilitas Mathematica, 63, 53-64.
He, J. H. (2003). Variational approach to the Lane-Emden equation. Applied Mathematics and Computation, 143, 539-541. https://doi.org/10.1016/S0096-3003(02)00382-X
Herron, I. H. (2013). Solving singular boundary value problems for ordinary differential equations. Caribb. J. Math. Comput. Sci., 15, 1-30.
Hughes, H. R., & Siriwardena, P. L. (2014). Efficient variable step size approximations for strong solutions of stochastic differential equations with additive noise and time singularity. International Journal of Stochastic Analysis, 2014, Article ID 852962, 6 pp. https://doi.org/10.1155/2014/852962
Karimi, S. V., & Aminataei, A. (2010). On the numerical solution of differential equations of Lane-Emden type. Computers and Mathematics with Applications, 59, 2815-2820. https://doi.org/10.1016/j.camwa.2010.01.052
Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Applications of Mathematics, Stochastic Modelling and Applied Probability (Vol. 23). Springer, New York. https://doi.org/10.1007/978-3-662-12616-5
Koch, O., & Weinmüller, E. B. (2003). The convergence of shooting methods for singular boundary value problems. Mathematics of Computation, 72, 289-305. https://doi.org/10.1090/S0025-5718-01-01407-7
Maruyama, G. (1955). Continuous Markov processes and stochastic equations. Rendiconti del Circolo Matematico di Palermo, 4, 48-90. https://doi.org/10.1007/BF02846028
Motsa, S. S., & Shateye, S. (2012). New analytic solution to the Lane-Emden equation of index 2. Mathematical Problems in Engineering. https://doi.org/10.1155/2012/614796
Parand, K., & Ghaderi, A. (2017). Two efficient computational algorithms to solve the singularly perturbed Lane-Emden problem. arXiv.org.
Peter, J. O. (2014). Introduction to partial differential equations. Springer.
Ramos, J. I. (2005). Linearization techniques for singular initial-value problems of ordinary differential equations. Applied Mathematics and Computation, 161, 525-542. https://doi.org/10.1016/j.amc.2003.12.047
Royden, H. L. (1988). Real analysis. New York.
Russell, R. D., & Shampine, L. F. (1975). Numerical methods for singular boundary value problems. SIAM Journal of Numerical Analysis, 12, 13-36. https://doi.org/10.1137/0712002
Sandile, S. M., & Precious, S. I. (2010). A new algorithm for solving singular IVPs of Lane-Emden type. Latest Trends on Applied Mathematics, Simulation, Modelling, 176-180.
Wazwaz, A. (2005). Adomian decomposition method for a reliable treatment of the Emden-Fowler equation. Applied Mathematics and Computation, 161, 543-560. https://doi.org/10.1016/j.amc.2003.12.048
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