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A coupled method of Laplace transform and Legendre wavelets is presented to obtain exact solutions of Lane-Emden-type equations. By employing properties of Laplace transform, a new operator is first introduced and then its Legendre wavelets operational matrix is derived to convert the Lane-Emden equations into a system of algebraic equations. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The results show that the proposed method is very effective and easy to implement.

The Lane-Emden-type equation has attracted much attention from mathematicians and physicists, since it is widely used to investigate the theory of stellar structure, the thermal behavior of spherical cloud gas, and theory of thermionic currents [

The solution of the Lane-Emden equations is numerically challenging due to the singularity behavior at the origin and nonlinearities. Therefore, much attention has been paid to searching for the better and more efficient methods for determining a solution, approximate or exact, analytical or numerical, to the Lane-Emden equations. The existing methods fall into two groups: the analytical methods and the numerical ones. The analytical methods express the exact solution of the equation in the form of elementary functions and convergent function series, such as the Adomian decomposition method (ADM) [

The Laplace transform is a wonderful tool for solving linear differential equations and has enjoyed much success in this realm. However, it is totally incapable of handling nonlinear equations because of the difficulties caused by nonlinear terms. Since Laplace Adomian decomposition method (LADM) was proposed by Khuri [

Wavelets theory, as a relatively new and emerging area in mathematical research, has received considerable attention in dealing with various problems of dynamic systems. The fundamental idea of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifies the problem and reduces the computation cost [

Motivated and inspired by the ongoing research in these areas, we propose a coupled method of Laplace transform and Legendre wavelets to establish exact solutions of (

Legendre wavelets

A function

If the infinite series in (

Block pulse functions (BPFs) form a complete set of orthogonal functions that are defined on the interval

Disjointness: the BPFs are disjoined with each other in the interval

Orthogonality: the BPFs are orthogonal with each other in the interval

Completeness: the BPFs set is complete when

Let

Assuming

By using the property of BPFs in (

The Legendre wavelets can be expanded into

Taking the collocation points as in the following,

The

The operational matrix of product of Legendre wavelets can be obtained by using the properties of BPFs. Let

From (

By employing Lemma

In this section, we will briefly demonstrate the utilization of the LLWM for solving the Lane-Emden equations given in (

By multiplying

Equation (

By integrating both sides of (

Taking the inverse Laplace transform to (

By using the initial conditions from (

For convenience, we define an operator

Now, we will show how to derive the Legendre wavelets operational matrix of operator

Let

Let

From (

Let

From the definition of operator

By further analysis, we obtain

So we finally have

Let

In order to use Legendre wavelets method, we approximate

Substituting (

Finally, we can get

In this section, four different examples are examined to demonstrate the effectiveness and high accuracy of the LLWM.

Consider the lane-Emden equation given in [

Applying the method proposed in Section

When

Numerical solution and absolute error of Example

Consider the lane-Emden equation of index

Applying the method developed in Section

From [

In the case of

Numerical solution and absolute error of Example

Numerical solution and absolute error of Example

Numerical solutions of Example

Consider the Lane-Emden equation given in [

Applying the method developed in Section

When

Numerical solution and absolute error of Example

Consider the following nonlinear Lane-Emden differential equation [

Here, we first expand

Considering only the first five terms we can write

We let

By applying the LLWM, we have

The analytical methods, such as the ADM [

When

Numerical solution of Example

In this paper, we have successfully developed a coupled method of Laplace transform and Legendre wavelets (LLWM) for solution of Lane-Emden equations. The advantage of our method is that only small size operational matrix is required to provide the solution at high accuracy. It can be clearly seen in the paper that the proposed method works well even in the case of high nonlinearity. Compared to the analytical methods, such as the ADM, VIM, and HPM, the LLWM can only get the approximate solution because of the error caused by expanding highly nonlinear terms. However, the developed vector-matrix form makes LLWM a promising tool for Lane-Emden-type equations, because LLWM is computer-oriented and can use many existing fast algorithms to reduce the computational cost.

This work is supported by National Natural Science Foundation of China (Grant no. 41105063). The authors are very grateful to reviewers for carefully reading the paper and for his (her) comments and suggestions which have improved the paper.