New Technique for Solving Lane-Emden Equation with Vieta-Fibonacci Polynomials

In the present work, new efficient iterative algorithm is presented for solving Lane-Emden equation with initial conditions. The solution is considered as a linear combination of Vieta-Fibonacci polynomials with unknown coefficients. First, the Vieta-Fibonacci polynomials are presented with some new important properties. A new exact formula for finding operation matrix of derivative for the Vieta-Fibonacci polynomials is constructed. Using such new properties the original Lane-Emden equation is transformed to the solution of algebraic equation with small-unknown coefficients. The obtained solution is in the form of a power series with easily computable coefficients. Comparison with some known exact and approximate solutions shows that the proposed solution is highly accurate . MSC. 41A25; 41A35; 41A36


Introduction
The Lane-Emden equation represents the behavior of a spherical cloud of gas acting under the mutual attraction of its molecules depending on thermodynamic rules.They are considered to be as below where g is given function of y with the initial conditions Our aim in this work is to apply new iteration technique for finding the approximate solution to Lane-Emden equation (1)(2) with the values α = 0, 1, 5 for g(y) = y α (t) and k = 2, which is a basic equation in the theory of stellar structure ý́(t) + (2/)ý(t) + y α (t) = 0 (3) Subject to the conditions y(0) = 1, ý(0) = 0 (4) The numerical and approximate solutions of the Lane-Emden equation together with initial conditions and boundary conditions are considered in many articles.Since Taylor series is useful mathematical tool to approximate solution for nonlinear equation, it was used in [1] to find an approximate solution for Lane-Emden type singular value problem with extremely simple technique.In [2,7], the Adomian decomposition method was utilized to solve Lane-Emden equation approximately.The singular Lane-Emden equation was treated in [3] using certain analytic method and obtained an exact solution if the solution of such equation is polynomial type.The Homotopy analysis method was applied to obtain the analytic solution for the classic Lane-Emden equation which describing the thermal behavior of a spherical cloud of gas [4].The authors in [5] used Bernstein polynomials operational matrix of derivative for solving problems arising in astrophysics.While He's homotopy perturbation method was used for treating numerically the singular initial value for treating numerically the singular initial value Lane-Emden problem.
Both the heuristic technique and Galerkin finite element method were applied in [8] and [9] respectively for solving a class of singular Lane-Emden equation.In addition, Bessel matrix representation in [10], was presented to compute a numerical solution for Lane-Emden singular delayed problem the results is a nonlinear fundamental matrix equation.
Our approach based upon the novel Vieta-Fibonacci operational matrix representation for derivative based upon the iterative method which reduces the original model problem into algebraic equations.To testify the validity and applicability of the proposed method, test examples are given.The computational results are accurate as compared with the exact solutions.
The present article is arranged as follows: the definitions of Vieta-Fibonacci polynomials is described in section 2. Some new properties are also included in section 2. Section 3 studies the convergence analysis while section 4 devotes the derivation of operation matrix of derivative for the Vieta-Fibonacci polynomials.A new iterative method is applied to singular initial value Lane-Emden-type equation is discussed in detail in section 5 based upon the obtained Vieta-Fibonacci operational matrix.Numerical results are listed in section 6.Finally, some concluding remarks are appeared in section 7.

Vieta-Fibonacci Polynomials
The explicit representation of the Vieta-Fibonacci polynomials VF(t) of the m th degree are defined on the interval [−2, 2] by the following power formula Furthermore, there polynomials satisfy the orthogonally relation corresponding to the inner product where Note that the weight function Now, assume that the required approximate solution in terms of Vieta-Fibonacci polynomials as follows Here the function y(t)ϵL w 2 [−2, 2] and f i+1 are the unknown coefficients.Truncate the series in Eq. 3 to be or where The vector σ(t) can be expressed as the following expression: where T(t) = [ 1 t t 2 … t m ] T , and V is (m + 1) × (m + 1) matrix which has the following expression The matrix component elements of V are given by In addition, Eq.6 can be written as

Convergence Analysis
The approximate function of y(t) using Vieta-Fibonacci polynomials is given below where F m =< y(t), vF m (t) > Since every continuous function on the closed interval is a bounded function, that is there exists a constant η such that In addition, one can obtain Using Eq. 1, yields The result is given by

Differentiation Operational Matrix 𝐕
A special explicit formula for VF(t) of m th degree operation matrix of derivative is defined in this section.
In other words, where The elements of the Vieta-Fibonacci operational matrix of derivative V can be defined as follows

Iterative Solution Method for the Model (3-4)
In this section, Vieta-Fibonacci iterative method will be proposed to obtain approximate solutions of the following model Assume that Eq. 3 has an approximate solution in the truncated Vieta-Fibonacci series form Eq. 3. In other word, Eq. 5 can be rewritten into a matrix form as The Vieta-Fibonacci polynomials σ(t) can be represented in matrix form as follows σ T (t) = MT T (t) or σ T (t) = T(t) M T , therefore; y(t) = T(t) M T F Now, the matrix representation of ý(t) will be taken the following form or T ́(t) = T(t)B T where In general, y n (t) = T (n) (t)M T F A summary of the suggested Vieta-Fibonacci iterative algorithm can be given by the following steps; Step 1: Approximate the unknown function y(t) in Eq. 3 where and V = ( Step 2: Construct the matrix representation of ý(t) and ý́1(t)as below where Step 3: Use the initial conditions, Eq. 4, yields Step 4: Substitute relation (13) into Eq.11 to obtain Then Put relation 14 into Eq. 3 and get f 3 and the first approximate solution can be defined.
Step 5: Continue the procedure, the following approximate solution in the (n+1) th step is introduced where

Numerical Results
It was shown in physics that important values of α are in the interval 0 ≤ α ≤ S and the exact solution has been known only when α = 0, 1, 5. In this section, the problem in Eqns.3-4 is solved for α = 1 using the proposed iterative method based on Vieta-Fibonacci polynomials.To solve Eqns.3-4 by means of the new iterative method based on Vieta-Fibonacci polynomials, the following approximations are constructed The following first initial approximation is considered The next approximation is obtained from (15) is given by The rest of iterates can be obtained from Eq. 15.
If the index m = 0, the exact solution is y(t) = 1 − t 2   6   To solve Eqns.3-4 with index m=0 by means of Vieta-Fibonacci polynomials, the following first approximation is constructed In this case; there is only one unknown coefficient and its value f 3 = −1
If the index m = 1, the exact solution is y(t) = sin t t Again, the next two iterates approximation is easily obtained from Eq. 15 and is given by Comparisons between the numerical results of the proposed method with existing exact results are carried out in order to show that the new approximation algorithm provides accurate solutions.The proposed iterative algorithm is shown to be highly accurate as well as only a few terms of Vieta-Fibonacci polynomials are required to obtain accurate computable solutions.

Conclusion
In this paper, our aim was to solve the Lane-Emden type equation numerically using Vieta-Fabonacci iterative method, which is a very fast convergent.It was solved for the values of the polytrophic index m = 0, 1, 5.
A new iterative method was applied to singular initial-value Lane-Emden-type problems, and the effectiveness and performance of the method is studied.The proposed method depends on Vieta-Fibonacci polynomials, and when the solution is polynomial, our suggested technique produces the exact solution.It can observe that the method is easy to implement, yields excellent results at a minimum computational cost, and requires less time.Computational results of several test problems are presented to demonstrate the viability and practical usefulness of the method.The results reveal that the method is very effective, straightforward and simple.