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Numerical Study of Singular Fractional Lane–Emden Type Equations Arising in Astrophysics

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Numerical Study of Singular Fractional Lane–Emden Type Equations Arising in Astrophysics J. Astrophys. Astr. (2019) 40:27 © Indian Academy of Sciences https://doi.org/10.1007/s12036-019-9587-0 Numerical study of singular fractional Lane–Emden type equations arising in astrophysics ABBAS SAADATMANDI∗ , AZAM GHASEMI-NASRABADY and ALI EFTEKHARI Department of Applied Mathematics, University Kashan, Kashan 87317-53153, Iran. ∗Corresponding author. E-mail: [email protected] MS received 20 December 2018; accepted 13 April 2019; published online 17 June 2019 Abstract. The well-known Lane–Emden equation plays an important role in describing some phenomena in mathematical physics and astrophysics. Recently, a new type of this equation with fractional order derivative in the Caputo sense has been introduced. In this paper, two computational schemes based on collocation method with operational matrices of orthonormal Bernstein polynomials are presented to obtain numerical approximate solutions of singular Lane–Emden equations of fractional order. Four illustrative examples are implemented in order to verify the efficiency and demonstrate solution accuracy. Keywords. Fractional Lane–Emden type equations—orthonormal Bernstein polynomials—operational matrices—collocation method—Caputo derivative. 1. Introduction hydraulic pressure at a certain radius r. Due to the contribution of photons emitted from a cloud in radi- The Lane–Emden equation has been used to model ation, the total pressure due to the usual gas pressure many phenomena in mathematical physics, astrophysics and photon radiation is formulated as follows: and celestial mechanics such as the thermal behavior of 1 RT a spherical cloud of gas under mutual attraction of its P = ξ T 4 + , 3 V molecules, the theory of stellar structure, and the theory ξ, , of thermionic currents (Chandrasekhar 1967). This is a where the parameters T R and V stand for the radia- singular nonlinear ordinary differential equation (ODE) tion constant, the absolute temperature, the gas constant ( ) which was first introduced by U.S. astrophysicist, and the specific volume respectively. Let M r be the , ,φ Jonathan Homer Lane (Lane 1870) who was interested mass of a sphere of radius r and G g respectively in computing both the temperature and the density of denote the constant of gravitation, acceleration of grav- mass on the surface, and 37 years later was studied ity and gravitational potential of gas. By having the in more detail by Emden (1907). In astrophysics, this relation GM(r) dφ ODE is a dimensionless version of Poisson’s equation g = =− , for the gravitational potential of a simple stellar model r 2 dr (Momoniat & Harley 2006). We now briefly explain one can determine P and φ by meeting the following mathematical modeling of the thermal behavior of a three conditions: spherical cloud which leads to the classic Lane–Emden dP =−gρdr = ρdφ, (1) equation. See Chandrasekhar (1967) and Parand et al. 2 (2010) for a comprehensive coverage of the subject. ∇ φ =−4πGρ, (2) γ P = Kρ . (3) 1.1 Modeling of the thermal behavior of a spherical Here, ρ is the density of gas at a distance r from the cen- cloud of gas tre of the spherical cloud and the constants γ = 1+1/m and K are experimentally determined. The constant m For the sake of simplicity, we consider a spherical gas is called the polytropic index and is related to the ratio cloud as shown in Figure 1. Let P(r) denote the total of specific heats of gas comprising the star. Using the 27 Page 2 of 12 J. Astrophys. Astr. (2019) 40:27 1 ( + ) 2 K m 1 1 −1 a = λ m , 4πG the differential equation (4) reduces to 2 y + y + ym = 0, x > 0. (5) x In physics, this singular differential equation with the initial conditions y(0) = 1, y (0) = 0(6) Figure 1. A spherical gas cloud. is known as the classic Lane–Emden problem. In quan- tum mechanics and astrophysics, the values of m are conditions (1)–(3), with a straightforward calculation, physically meaningful and interesting and lie in the the following relations are revealed: interval [0, 5]. In the literature, the exact solutions of dP GM(r) problems (5)–(6) are known only for m = 0, 1, 5 =−gρ =−ρ , dr r 2 (Parand et al. 2010) and for other values of m, with the dφ M(r) approach of handling singularity in the presence of the =−g =−G , 2 origin, several methods have been efficiently employed dr r d2φ 1 dM(r) M(r) to approximate the solution of this problem. A fairly =−G − 2 . dr 2 r 2 dr r 3 complete discussion of these methods are presented in Parand et al. (2010, 2017); Dehghan & Shakeri By substituting the above expressions into Equation (2), (2008); Abd-Elhameed et al. (2014) and references we obtain therein. 1 dM(r) M(r) 2 M(r) In recent years, considerable attention has been −G − 2 + −G r 2 dr r 3 r r 2 devoted to develop the applications of fractional calcu- =−4πGρ, lus in the fields of science and engineering. By the use of differential equations of non-integer order, dynamic and consequently, behavior of different phenomena in engineering sci- dM(r) ences, physics and other branches of science can be = 4πρr 2. dr studied more precisely (Saadatmandi & Dehghan 2010, Therefore, the hydrostatic equilibrium conditions 2011; Saadatmandi 2014; Doha et al. 2012). Hence, this (HEC) are as follows: motivates us to study the efficient low-cost numerical ⎧ ( ) algorithms for solving the singular Lane–Emden equa- ⎨⎪ dP =−ρ GM r , dr r2 tions of fractional order as follows (Nasab et al. 2018): ⎪ α L β ⎩ dM(r) = πρ 2, D y(x) + D y(x) + g(x, y(x)) = h(x), (7) dr 4 r xα−β Eliminating M from the recent equations yields with the initial conditions: 1 d r 2 dP y(0) = A, y (0) = B, (8) =−4πGρ, 2 ρ r dr dr where 0 < x ≤ 1, 1 <α≤ 2and0<β≤ 1. Also, α where the pressure P and the density ρ = V −1 vary A and B are constants and D denotes the fractional with r. If we insert the above expression into the first derivative of order α in the Caputo sense and defined equation for HEC, we obtain the following differential later in the text. Moreover, g(x, y(x)) is a continuous equation: real-valued function and h(x) ∈ C[0, 1].Theexis- ( + ) tence and uniqueness of the solution of problems (7)–(8) K m 1 1 −1 1 d 2 dy m λ m r =−y , (4) are discussed in Ibrahim (2013) and Taleb & Dahmani π 2 4 G r dr dr (2016). To the best of our knowledge, there are only a where λ represents the central density of the gas cloud few numerical algorithms available in the literature for and y is a dimensionless quantity that are both related to the numerical solution of fractional Lane–Emden equa- ρ via ρ = λym. By imposing additional dimensionless tions compared to the classical one. For instance, we variable r = ax via the relation can refer to the reproducing kernel method (Akgül et al. J. Astrophys. Astr. (2019) 40:27 Page 3 of 12 27 2015), collocation method (Mechee & Senu 2012a), the The layout of this paper is as follows. In Section 2, least square method (Mechee & Senu 2012b), and mod- some basic definitions and introductory concepts of ified differential transform method (Marasi et al. 2015). fractional calculus and some properties of OBPs are More recently, Nasab et al. (2018) applied a hybrid presented. Section 3 is devoted to obtaining opera- numerical method combining Chebyshev wavelets and tional matrices of fractional derivatives and integrals a finite difference approach to obtain solutions of sin- for OBPs. In Section 4, with the aid of operational gular fractional Lane–Emden equations of the form (7). matrices of fractional integral and differential and col- Unfortunately, most of the differential equations of location approach, two numerical techniques are driven fractional order do not have an analytical expression to solve the singular fractional Lane–Emden equation of solution and the analytical solution, if any, is not (7) subject to the initial conditions (8). In Section 5, the suitable for numerical purposes. Therefore, in recent proposed methods are applied to several examples and years, researchers have focused on the development of the numerical results are compared with those existing suitable and efficient numerical methods for solving in the literature. Finally, concluding remarks are given linear/nonlinear fractional ordinary/partial differential in Section 6. Note that we have implemented our algo- equations. One of the most powerful and adaptive tools rithms in Maple software. for obtaining numerical solutions of these equations is the use of operational matrices that have been widely extended in the field of fractional calculus over the last 2. Basic definitions and concepts few years (Bhrawy et al. 2015). In fact, by the use of an operational matrix technique, the problem of solving 2.1 A brief review of fractional calculus a fractional differential equation can readily lead to the determination of solution of a system of algebraic equa- In this section, we present some essential basic defi- tions. This approach reduces the computational error nitions and properties of fractional calculus for subse- and the complexity of operations, speeds up comput- quent discussion. The fractional calculus was born more ing, makes programming more easy and suitable, and than 300 years ago and its history goes back to the begin- does not have difficulties arising from problem-solving ning of the differential and integral calculus. Various directly. popular definitions for integral and derivative of frac- In the past few years, the abilities of the operational tional order are given.
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