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Glossary

Preface

This secton provides a stream of examples demonstating applications of power series method for solving initial value problems for second order differential equations.

 

Elementary Functions

Some variable coefficient linear differential equations have a solution expressed via a familiar or elementary functions. Some examples of such equations are presented.

Example 1: A linear differential equation with a parameter 𝑎∈ℝ

\[ x^2 y'' -x \left( x + a \right) y' , + a\,y = 0 , \qquad y' = {\text d}y/{\text d}x , \tag{1.1} \]
admits a solution
\[ y(x) = x^a e^x . \tag{1.2} \]
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Example 2: A linear differential equation with two real parameters

\[ x\, y'' + \left( a - bx \right) y' , - b\,y = 0 , \qquad y' = {\text d}y/{\text d}x , \tag{2.1} \]
admits a solution
\[ y(x) = x^{1-a} e^{bx} . \tag{2.2} \]
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Lane--Emden Equations

Lane--Emden type equations are nonlinear ordinary differential equations on semi-infinite domain (0, ∞). They are categorized as singular initial value problems. These equations describe the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. The polytropic theory of stars essentially follows out of thermodynamic considerations, that deals with the issue of energy transport, through the transfer of material between different levels of the star. These equations are one of the basic equations in the theory of stellar structure and has been the focus of many studies.

The famous form of the Lane--Emden equations, given in the following

\begin{equation} \label{EqLane.1} \frac{1}{x^2} \,\frac{\text d}{{\text d}x} \left( x^2 \frac{{\text d}y}{{\text d}x} \right) + y^m =0 \qquad\mbox{or} \qquad y'' + \frac{2}{x}\, y' + y^m , \qquad x>0 . \end{equation}
At this point, it is also important to introduce the boundary conditions which are based upon the following boundary conditions for hydrostatic equilibrium and normalization con-sideration of the newly introduced quantities x and y. What follows for r = 0 is
\[ r=0 \qquad \Longrightarrow \qquad y(0) = 1. \]
This boundary condition does not guarantee uniqueness of the solution unless some regularity condition at x = ∞ is imposed. However, imposing the initial conditions at x = 0 will lead to non-existence of the solution. Since the Lane--Emden equations is derived from Poisson's equation (PFE of elliptic type), there should be only one boundary condition at x = 0. Note that the Dirichlet boundary condition y(0) = 1 actually requires the solution derivative at the singular point x = 0 to be zero.

The values of m that are physically interesting, lie in the interval [0,5]. The main difficulty in the analysis of this type of equation is the singularity behavior occurring at x = 0. Exact soloutions for Eq.\eqref{EqLane.1} are known only for m = 0, 1, and 5. For other values of m the standard Lane--Emden equation is to be integrated numerically.

  1. Eltayeb1, H., Adem Kiliçman, A., and Bachar, I., Modified Sumudu decomposition method for Lane--Emden-type differential equations, Advances in Mechanical Engineering, 2015, Vol. 7(12) 1--6. doi: 10.1177/1687814015620086
  2. Hasan, A., A new development to the Adomian decomposition for solving singular IVPs of Lane-Emden Type, United States of America Research Journal (USARJ) Vol. 2, No.3, 2014.
  3. Parand, K., Rezaei, A.R., Taghavi, A., Lagrangian method for solving Lane--Emden type equation arising in astrophysics on semi-infinite domains, Mathematical Physics, 2010, 18pages. doi:10.1016/j.actaastro.2010.05.015

 

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