Numerical Solutions for the Time-Dependent Emden-Fowler-Type Equations by B-Spline Method

A numerical method based on B-spline is developed to solve the time-dependent Emden-Fowler-type equations. We also present a reliable new algorithm based on B-spline to overcome the difficulty of the singular point at 0 x = . The error analysis of the method is described. Numerical results are given to illustrate the efficiency of the proposed method.


Introduction
In recent years, a lot of attentions have been devoted to the study of B-spline method to investigate various scientific models.The efficiency of the method has been formally proved by many researchers [1]- [7].
Spline functions have some attractive properties.Due to the being piecewise polynomial, they can be integrated and differentiated easily.Since they have compact support, numerical methods in which spline functions are used as a basis function lead to matrix systems including band matrices.Such systems have solution algorithms with low computational cost.
In this paper, we employ the B-spline method to solve the time dependent partial differential equations.For clarity, the method is presented for the heat equation ( ) ( ) (2) and the initial condition Some forms of the above equations model several phenomena in mathematical physics and astrophysics such as the diffusion of heat perpendicular to the surface of parallel planes, theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas sphere and theory of thermionic currents [8]- [10].
The solution of the time-dependent Emden-Fowler equation as well as a variety of linear and nonlinear singular IVPs in quantum mechanics and astrophysics is numerically challenging because of singularity behavior at the origin.The singularity behavior that occurs at the point is the main difficulty in the analysis of Equations ( 1)- (3).
The paper is organized as follows.In Section 2, we review some basic facts about the B-spline that are necessary for the formulation of the discrete linear and nonlinear system.In Section 3, the error analysis of the method is described.In Section 4, we formulate our B-spline collocation method to the solution of (1)-(3).In Section 5, numerical experiments are tested to demonstrate the viability of the proposed method.

Some Properties of B-Spline
B-splines are mathematically more sophisticated than other types of splines, so we start with a gentle introduction.We first use basic assumptions to derive the expressions for the cubic uniform B-splines directly and without mentioning knots.We then show how to extend the derivations to uniform B-splines of any order.Following this, we discuss a different, recursive formulation of the weight functions of the uniform B-splines.
The third-degree B-spline is used to construct numerical solutions to a given problem (1).A detailed description of third-degree B-spline functions can be found in [13] [17].The third-degree B-splines are defined as ( ) ( ) ( ) ( ) We can simply check that each of the function x is twice continuously differentiable on the entire real line.Also, ( ) Similarly, we can show that:

Governing Equation
where ( ) ( ) , we consider the linear non-homogeneous Emden-Fowler differential Equation ( 9), difference schemes for this problem considered as following: or Suppose ( ) S x is the cubic B-spline interpolating ( ) at the ( ) Theorem 3.1.Let the collocation approximation ( ) S x to the solution ( ) u x of the boundary value problem ( 1) is (12) then the following relation is hold

Truncation Error of Time Dependent Emden-Fowler Equation
Theorem 3.2.By using the combination of the finite difference approximation and cubic B-spline method, the truncation error is ( ) we can calculate the truncation error which is defined as ( ) as: ( ) ( )

B-Spline Solutions for Time-Dependent Emden-Fowler
In this section, we introduce a reliable algorithm based on B-spline method to handle singular initial value problems (IVPs) in a realistic and efficient way considering Emden Fowler equation as a model problem.

Linear Time-Dependent Emden-Fowler
Let ( ) ( ) be an approximate solution of Equation ( 1 Proof: If we replace each term of (10) with its corresponding approximation given by ( 14) and ( 15) and substituting Then use them with the system of Equation ( 16), which can be written in the matrix form = AC F (18) where ( ) Also the matrix A can be written as It is easily seen that the matrix A is strictly diagonally dominant and hence nonsingular.Now we have a linear system of .We can obtain the coefficient of the approximate solution by solving this linear system by Q-R method.

Non-Linear Time-Dependent Emden-Fowler
, we consider the nonlinear non-homogeneous Emden-Fowler differential equation We can use finite difference method Theorem 4.2.If the assumed approximate solution of the problem ( 20) is ( 14), then the discrete collocation system for the determination of the unknown coefficients is given by , .
Then the boundary condition (17) with the system of Equation ( 21).Now we have a nonlinear system of 3 n + equations in the 3 n + unknown coefficients.We can obtain the coefficients of the approximate solution by solving this nonlinear system by Newton's method.

Examples and Comparisons
In this section, we will present three of our numerical results of time dependent problems using the method outlined in the previous section.The performance of the B-spline method is measured by the maximum absolute error k ε which is defined as  is overcome here.Finally, we conclude that the B-spline method is a promising tool for both linear and nonlinear singular time-dependent Emden-Fowler-type equations.
of Equation (1), we first modify Equation (1) at 0. x = By L'Hospital rule, the boundary value problem (1) is transform into

1 nTheorem 4 . 1 .
+ grid points in the interval [ ] If the assumed approximate solution of the problem (10) is(14), then the discrete collocation system for the determination of the unknown coefficients { }

Table 1 .
Maximum absolute errors for Example 1.
+ =The maximum absolute errors at different n and different time t with

Table 2 .
Maximum absolute errors for Example 2.

Table 3 .
Maximum absolute errors for Example 3.