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MATHEMATICS IN APPLIED RESEARCH Volume 3 (2022)

Kolej Pengajian Pengkomputeran, Informatik dan Media

Universiti Teknologi MARA Cawangan Negeri Sembilan

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The Lane-Emden equations are singular initial value problems associated with ordinary differential equations of the second order (ODEs), (Zhou, 1986). The equations have been used to model a variety of mathematical physics and astrophysics phenomena. The equation is essential in mathematical physics, celestial mechanics, and computer science. Various approximations have been used to solve multiple variations of the Lane-Emden problem: Legendre Wavelets Approximations, Hybrid Adomian Decomposition, Method-Successive Linearization Method (ADM-SLM), q-homotopy analysis, Laplace transform method (q-HATM), and others. How- ever, not all solutions to the Lane-Emden equation are accurate. In this study, an example of the Lane-Emden equation is solved using Pade’s approximation method, and the result will be compared with the exact and Taylor’s series solutions.

1.1. Lane-Emden Equation

The Lane–Emden equation is Poisson’s equation for the gravitational potential of a self-gravitating, spherically symmetric polytrophic ﬂuid. This equation appears in numerous mathematical physics applications and is a valuable model for many systems in various domains. The Lane–

Emden type equations, initially introduced by Jonathan Homer Lane in 1870 and afterward in- vestigated in depth by Emden, is a second-order ordinary differential equation with an arbitrary index, known as the polytrophic index, in one of its parts. Emden’s equation is a classical nonlinear equation derived from studying the thermal behavior of a spherical cloud of gas according to the classical principles of thermodynamics and functioning under the mutual attraction of its molecules. The expression has variable coefﬁcients and a regular-singular point at x=10.

Following is the general version of the Lane-Emden equations:

y+k

xy+F(x,y) = g(x),where0≤x≤1, k≥0, (1)

y(0) = A, (2)

y(0) = B. (3)

This equation is used in several phenomena, such as stellar structure theory, astrophysics, and the thermal behavior of spherical clouds of gas.

1.2. Example of Lane-Emden Equation

Consider the following Lane-Emden equation in Eq.4, (Qamar et al., 2021).

y+2

xy+y(x) =0,0x1 (4)

with initial conditiony(0) =1,y(0) =0 and the exact solutions issin(x) x .

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1.3. Pade’ series

The Pade’ approximation method can not be applied directly to Eq.4. Therefore, a one-dimensional different transform needs to be used for Eq.4,(Yi˘gider et al., 2011),(Chang and Chang, 2008).

Theu^∗(x)is a solution generated after one-dimensional different transform was applied.

u^∗(x) =1−x² 3!+x⁴

5!+x⁶ 7!+x⁸

9!+x¹0

11!+··· (5)

Then, the power series ofu^∗(x)can be written as follows.

p 5

4

=A0+A1x+A2x²+A3x³+A4x⁴+A5x⁵

B0+B1x+B2x²+B3x³+B4x⁴ =1−x² 3!+x⁴

5!+x⁶ 7!+x⁸

9!+x¹0

11!+··· (6) The Eq.6 was solved by using Pade’ approximation method (Brezinski and Van Iseghem, 1994), (Celik and Bayram, 2003) by lettingB0=1. The solution was shown in Eq.7.

p 5

4

=1− 79

386x²+0.01269x⁴ 1− 13

396x²− 37 33264x⁴

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2. Result and discussing

In this section, the comparison between, exact solution, Pade’s approximation method, and Taylor’s Series method are shown in Table 1, and Figure 1.

Table 1: Exact and Numerical Solution

x Exact Solution Pade’ Approximation Method Taylor’s Series Method

0.0 - 1.000000000 1.0000000000

0.1 0.9983341665 0.9983341665 0.9983341665

0.2 0.9933466540 0.9933466685 0.9933466540

0.3 0.9850673555 0.9850675456 0.9850673555

0.4 0.9735458558 0.9735469741 0.9735458558

0.5 0.9588510772 0.9588554413 0.9588510772

0.6 0.9410707890 0.9410840068 0.9410707891

0.7 0.9203109818 0.9203446712 0.9203109825

0.8 0.8966951136 0.8967708751 0.8966951163

0.9 0.8703632329 0.8705181638 0.8703632416

1.0 0.8414709848 0.8417650587 0.8414710097

Based on Table 1, it can be seen that the results obtained using Pade’s approximation method are almost approaching to exact solution for 0.0x1.0. Meanwhile, Figure 1 indicates that the result for Taylor’s series is close to both exact and Pade’s series

3. Conclusion

In the above discussion, the Padé Approximation method was used to solve an example of the Lane-Emden equation. The relative error was calculated to determine the accuracy and efficacy of both methods. As a result, the Padé Approximation method has smaller relative errors than the Taylor series method. It shows that the Padé Approximation method is more accurate and efficient than the Taylor Series method in solving Lane-Emden equations Eq. 4.

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Figure 1: The value ofU(x)andP(⁵₄)of Pade’ approximation References

Brezinski, C. and Van Iseghem, J. (1994). Pad´e approximations. Handbook of Numerical Analysis, 3:47–222.

Celik, E. and Bayram, M. (2003). Arbitrary order numerical method for solving differential-algebraic equation by pad´e series.Applied Mathematics and Computation, 137(1):57–65.

Chang, S.-H. and Chang, I.-L. (2008). A new algorithm for calculating one-dimensional differential transform of nonlinear functions.Applied Mathematics and Computation, 195(2):799–808.

Qamar, A., Iqbal, S. M., Khoso, F., Uddin, Z., and Arain, A. (2021). Improved numerical computation to solve lane-emden type equations. International Journal of Advanced Trends in Computer Science and Engineering, 10:1980–1984.

Yi˘gider, M., Tabatabaei, K., and C¸ elik, E. (2011). The numerical method for solving differential equations of lane-emden type by pade approximation.Discrete Dynamics in Nature and Society, 2011.

Zhou, J. (1986). Differential transformation and its applications for electrical circuits.

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M A T H E M A T I C S

I N A P P L I E D R E S E A R C H

V O L U M E I I I

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72635.pdf

MATHEMATICS IN APPLIED RESEARCH Volume 3 (2022)

Kolej Pengajian Pengkomputeran, Informatik dan Media

Universiti Teknologi MARA Cawangan Negeri Sembilan

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