An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering

Yalçın Öztürk

doi:10.1515/nleng-2018-0062

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An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering

Yalçın Öztürk

Published/Copyright: November 2, 2018

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From the journal Nonlinear Engineering Volume 8 Issue 1

Abstract

The purpose of this paper is to propose an efficient numerical method for solving system of Lane–Emden type equations using Chebyshev operational matrix method. This method transforms the system of Lane-Emden type equation into the system of algebraic equations with unknown Chebyshev coefficients. Some illustrative examples are given to demonstrate the efficiency and validity of the proposed algorithm.

Keywords: Lane-Emden system; engineering problem; Chebyshev polynomials; operational matrix method

1 Introduction

In this paper, we consider the system of the Lane-Emden type equations as:

P(t)d2y1(t)dt2+αtdy1(t)dt+y2p(t)=g1(t)H(t)d2y2(t)dt2+βtdy2(t)dt+y1q(t)=g2(t)(1)

with initial conditions

y1(0)=λ0, y1′(0)=λ1y2(0)=y0, y2′(0)=y1(2)

where p, q are positive integers. Eq. (1) is the Lane-Emden equation of the first kind, or of index q, which is a basic equation in the theory of stellar structure, in the sequence of papers [1–28]. The Lane–Emden equation of the first kind describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of thermodynamics. Notice that the Lane–Emden equation is linear for q = 0, 1 and nonlinear otherwise. In astrophysics, the Lane–Emden equation is Poisson’s equation for the gravitational potential of a self-gravitating spherically symmetric and polytropic fluid at hydrostatic equilibrium for [22]. Moreover, one of the important fields of application of this equation is the analysis of the diffusive transport and chemical reaction of species inside a porous catalyst particle [22].

The modelling of several physical phenomea such as pattern formation, population evolution, chemical reactions, and so on [7] gives rise to the systems of Lane-Emden equations, and have attracted much attention in recent years. Several authors have proved existence and uniqueness results for the Lane-Emden systems [8, 9].

Several methods for the solution of Lane-Emden equations and the system of Lane-Emden equations have been presented, which are sinc-collocation method [1], the variational iteration method [2], collocation method [10], the modified Homotopy analysis method [11], the variational iteration method [12], the Adomian decomposition method [13], Legendre operational matrix method [14], second kind Chebyshev operational algorithm [15], ultra-spherical wavelets method [16], shifted Legendre operational matrix method [17], spectral second kind Chebyshev wavelets [18] other methods, in the sequence of papers [19–30].

Chebyshev polynomials are the most famous bases of polynomials space. It is well known that there are four kinds of Chebyshev polynomials, and all of them are special cases of the more widest class of Jacobi polynomials. These polynomials have reliable advantage such as easy to compute, rapid convergence and completeness, since Chebyshev polynomials (first kind) is the Fourier cosine series [31, 32]. From this reasons, we use the first kind shifted Chebyshev polynomials in approximation.

Spectral methods play prominent roles in numerical mathematics to solve many more important problems. In this article, we use the operational method which is the kind of spectral method. Operational method have become increasingly popular to solve various problems in applied mathematics. A large number of authors utilize this method to solve various equations such as linear and nonlinear hyperbolic telegraph type equations [33], fractional order differential equation [34], higher-order ordinary differential equations [35, 36], Abel equation [37], two point boundary value problems [38], singularly perturbed boundary value problems of fractional multi-order [39], integro-differential equation [40].

The aim of this study is to get solution as truncated Chebyshev series defined by

y1N(t)=∑n=0N′anTn*(t),y2N(t)=∑n=0N′bnTn*(t)(3)

where Tn*(t)=cos(nθ), 2t−1=cosθ, 0≤t≤1, denotes the shifted Chebyshev polynomials of the first kind; ∑′ denotes a sum whose first term is halved; a_n, b_n (0 ≤ n ≤ N) are unknown Chebyshev coefficients, and N is chosen any positive integer such that N ≥ m.

2 The first kind shifted Chebyshev polynomials

The Chebyshev polynomials of the first kind T_n(x) is a polynomials in x of degree n, defined by relation [31, 32]

Tn(x)=cosnθ,when x=cosθ

If the range of the variable x is the interval [−1, 1], the range the corresponding variables θ can be taken [0, π]. We map the independent variable t in [0, 1] to the variable s in [−1, 1] by transformation

s=2x−1 or x=12(s+1)

and this lead to the shifted Chebyshev polynomial of the first kind Tn*(x) of degree n in x on [0, 1] given by [31]

Tn*(x)=Tn(s)=Tn(2x−1).

These polynomials may be generated by using recurrence relation

Tn*(x)=2(2x−1)Tn−1*(x)−Tn−2*(x)

with initial conditions

T0*(x)=1, T1*(x)=2x−1,

Any given function, y(x) ∊ L²[0, 1], can be approximated as a sum of shifted Chebyshev polynomials as:

y(x)=∑n=0∞anTn*(x)

where

cn=〈y(x),Tn*(x)〉=∫01y(x)Tn*(x)dx,n=0,1,…,.

3 Fundamental relations

Let us consider Eq. (1) and find the matrix forms of the equation. First we can convert the solution y1,2N(t) defined by a truncated shifted Chebyshev series (3) and its derivative (y1,2N(t))(k) to matrix forms

y1N(t)=T*(t)A, 1N(k)(t)=T*(k)(t)A, k=0,1,2(4)

y2N(t)=T*(t)B, y2N(k)(t)=T*(k)(t)B, k=0,1,2(5)

where

T*(t)=[T0*(t) T1*(t) ... TN*(t)]T*(k)(t)=[T0*(k)(t) T1*(k)(t) ... TN*(k)(t)]A=[12a0 a1... aN]TB=[12b0 b1... bN]T

It is well known that the relation between the powers xⁿ and the shifted Chebyshev polynomials Tn*(x) is

xn=2−2n+1∑k=0n′(2nk)Tn−k*(x), 0≤x≤1(6)

where Σ′ denotes a sum whose first term is halved.

By using the expression (6) and taking n=0,1,...,N we find the corresponding matrix relation as follows:

(X(t))T=D(T*(t))T and X(t)=T*(t)DT(7)

where

X(t)=[1 t … tN]D=[20(00)000...02−2(21)2−1(20)00⋯02−4(42)2−3(41)2−3(40)0⋯02−6(63)2−5(62)2−5(61)2−5(60)⋯0⋮⋮⋮⋮⋱⋮2−2N(2NN)2−2N+1(2NN−1)2−2N+1(2NN−2)2−2n+1(2NN−3)⋯2−2N+1(2N0)]

Then, by taking into account (7) we obtain

T*(t)=X(t)(D−1)T(8)

and

(T*(t))(k)=X(k)(t)(D−1)T,k=0,1,2,

To obtain the matrix X^(k)(t) in terms of the matrix X(t), we can use the following relation:

X(1)(t)=X(t)C

X(2)(t)=X(1)(t)C=X(t)(C)2(9)

where

C=[000…0100…0020…0……………000N0]

Consequently, by substituting the matrix forms (8) and (9) into (4)-(5) we have the matrix relation

y1N(t)(k)=X(t)Ck(DT)−1A, k=0,1,2(10)

y2N(t)(k)=X(t)Ck(DT)−1B, k=0,1,2(11)

4 Method of Solution

Firstly, we deal with singularity in Eq. (1). If they set in Eq. (1) by [16]

g1(t)=tf1(t) g2(t)=tf2(t),

Then Eq. (1) is turned into

tP(t)d2y1(t)dt2+αdy1(t)dt+ty2p(t)=f1(t)tH(t)d2y2(t)dt2+βdy2(t)dt+ty1q(t)=f2(t)(12)

y1(0)=λ0, y1′(0)=λ1y2(0)=y0, y2′(0)=y1(13)

Now, we present the numerical solution method for Eq. (12) with initial conditions Eq. (13). We can give the expansion of f_i(x) by the shifted Chebyshev polynomials as:

fi(t)≈FiTX(t)(DT)−1, i=1,2(14)

Using matrix representation of approximate solution and its derivatives, Eq. (12) can be written as:

tP(t)X(t)C2(DT)−1A+αX(t)C(DT)−1A+t(X(t)(DT)−1B)p=f1(t)(15)

tH(t)X(t)C2(DT)−1B+βX(t)C(DT)−1B+t(X(t)(DT)−1A)q=f2(t)(16)

The residual R_i(t) for Eq. (15) can be written as

R1(t)≈tP(t)X(t)C2(DT)−1A+αX(t)C(DT)−1A+t(X(t)(DT)−1B)p−f1(t)(17)

R2(t)≈tH(t)X(t)C2(DT)−1B+βX(t)C(DT)−1B+t(X(t)(DT)−1A)q−f2(t)(18)

and if it is applied typical tau method which is used in the sense of a particular form of the Petrov-Galerkin method [33–40], Eqs. (16) – (17) can be converted in 2 × (N − 1) nonlinear equations by calculating

〈Ri(t),Tn*(t)〉=∫01Ri(t)Tn*(t)dt=0, n=0,1,…,N−1(19)

The initial conditions (13) lead to the four equations

y1N(0)=X(0)(DT)−1A=λ0, (y1N(x))′(0)=X(0)C(DT)−1A=λ1(20)

y2N(0)=X(0)(DT)−1B=y0, (y2N)′(0)=X(0)C(DT)−1B=y1(21)

Therefore, we get the 2 × (N + 1) sets of equations with 2 × (N + 1) unknowns by Eq. (18) and Eqs. (19) – (20). We write the program in the Maple 13 and solve the 2 × (N + 1) sets of equations with 2 × (N + 1) unknowns, and so approximate solution yiN(x), i=1,2 can be calculated.

4.1 Error estimation and convergence analysis

Assume that H = L²[0, 1], where PN={T0*(t),T1*(t),...,TN*(t)}⊂H be the set of polynomials of n − th degree and W = Span(P_N). Clearly, W is a finite dimensional vector space. Let f є H, then f has the unique best approximation out of W such that g₀ є W, that is [41]

‖f−g0‖2≤‖f−g‖2, ∀g∈W(22)

where ‖f‖22=<f,f>. There exist unique coefficients A = [a₀a₁ ... a_N] for g₀ such that

f≈g0=∑k=0NakTk*(t)=T(t)A(23)

where T(x)=[T0*(t),T1*(t),...,TN*(t)] and coefficient matrix A can be given by the following equation

A<T(t),T(t)>=<f,T(t)>

where

<f,T(t)>=∫01f(t)T(t)Tdx=[<f,T0*(t)> <f,T1*(t)> ... <f,TN*(t)>]

and < T(t), T(t) > is an (N + 1) × (N + 1) matrix and

ϕ= <T(t),T(t)> =∫01T(t)T(t)Tdt

and so

A=(∫01f(t)T(t)Tdt)ϕ−1(24)

Theorem 1. [41] Let assume that H is an Hilbert space, W is a closed subspace of H such that dim W is finite and {y₁, y₂, y₃, ..., y_N} is an y basis for W. Let f be an arbitrary element in H and g₀ be the unique best approximation to f out of W. Then we have

‖f−g0‖22=D(f,y1,y2,...,yN)D(y1,y2,...,yN)(25)

where

D(f,y1,y2,...,yN)=[<f,f><f,y1>⋯<f,yN><y1,f><y1,y1>⋯<y1,yN>⋮⋮⋱⋮<yN,f><yN,y1>⋯<yN,yN>]

It is defined the inner product in <f,g>=∫abf(t)g(t) and the subspace W = Span(P_N), so the presented absolute error in Theorem 1 can be written

‖f−g0‖=det(∫01Ψ(t)Ψ(t)Tdt)det(∫01Φ(t)Φ(t)Tdt)(26)

for which Φ(t)T=[T0*(t) T1*(t) ... TN*(t)] and Ψ(x)T=[f T0*(t) T1*(t) ... TN*(t)].

Theorem 2. Assume that the function g : [0, 1] → R is (N + 1) times continuously differentiable, g є C^N⁺¹[0, 1] and W=Span{T0*(t),T1*(t),⋯,TN*(t)}. If AΦ is the best approximation to g out of W, then a bound for absolute error is presented by

‖g−AΦ‖≤M222N((N+1)!)2

where M = max_x∊[0,1](g_i(t)^(N+1).

Proof. We consider the interpolation polynomial g^*(t) is the interpolating polynomial to g at t_i, where t_i, i = 0, 1, ..., N are the Chebyshev-Gauss grid points, then we have

g(t)−g*(t)=g(N+1)(λ)(N+1)!∏i=0N(t−ti), λ∈[0,1](27)

Since, ‖TN*(t)‖∞=1, we conclude that if we choose the grid nodes (t_i)_0≤i≤N to be zero the (N+1) zeroes of the Chebyshev polynomials TN*(t), we have [25, 26]

‖∏i=0N(t−ti)‖=12N(28)

this is the smallest possible value. From (25), we obtain

‖g(t)−g*(t)‖∞≤12N(N+1)!‖g(t)(N+1)‖∞.(29)

Since AΦ is the best approximation to g out of W, considering g^* є W and using (28), we have

‖g−AΦ‖≤‖g−g*‖=∫01|g(t)−g*(t)|2dt≤∫01(M2N(N+1)!)2dx≤M222N((N+1)!)2

Another comparison can be given for quality of the approximation by well known algorithm [32]. Since the approximate solution y1,2N(t) is the approximate solution of Eq. (1), Eq. (1) must be approximately satisfied by the function y_N(x), that is;

|P(t)d2y1N(t)dt2+αtdy1N(t)dt+(y2N(t))p−g1(t)|≈0(30)

|H(t)d2y2N(t)dt2+βtdy2N(t)dt+(y1N(t))q−g2(t)|≈0(31)

This comparison should be advised strongly by [32]. Then the error can be estimated by the error function [32]

E1N=|P(t)d2y1N(t)dt2+αtdy1N(t)dt+(y2N(t))p−g1(t)|(32)

E2N=|H(t)d2y2N(t)dt2+βtdy2N(t)dt+(y1N(t))q−g2(t)|(33) □

5 Examples

In this section, several numerical examples are given to illustrate the accuracy and effectiveness properties of the method and all of them were performed on the computer using a program written in Maple 13. Let us define

EiN,L=(∫01(y(t)−yN(t))2dt)1/2

where y_i(x) and yiN(x), i=1,2 denote the approximate solution obtained by the present method and the exact solution, respectively. In Tables, Ne=|yi(t)−yiN(t)| are absolute error for selected points.

Example 1. Let us consider the following linear systems of Lane-Emden equations [26]

y″1(t)+3ty′1(t)−4(y1(t)+y2(t))=0y″2(t)+2ty′2(t)+3(y1(t)+y2(t))=0(34)

subject to initial conditions

y1(0)=1, y2(0)=1,y′1(0)=0, y′2(0)=0.

For N = 2, We have the residual RiN(t) for this problem

R1≈tX(t)C2(DT)−1A+3X(t)C(DT)−1A−4tX(t)(DT)−1A−4tX(t)(DT)−1B−F1TX(t)(DT)−1(35)

R2≈tX(t)C2(DT)−1B+2X(t)C(DT)−1B+3tX(t)(DT)−1A+3tX(t)(DT)−1B−F2TX(t)(DT)−1(36)

where

D=[1001/21/203/81/21/8] C=[010002000]X(x)=[1xx2] F1[000] F2=[000] A=[a0a1a2] B=[b0b1b2]

Then, using Eqs.(34)-(35) we obtain the linear algebraic system

−2a0+163a1+263a2−2b0−23b1+23b2=0(37)

32a0+12a1−12a2+32b0+92b1+152b2=0(38)

with conditions

y12(0)=X(0)(DT)−1A=a0−a1+a2=1(39)

y22(0)=X(0)(DT)−1B=2a1−8a2=0(40)

(y12(x))′(0)=X(0)C(DT)−1A=b0−b1+b2=1(41)

(y22(x))′(0)=X(0)C(DT)−1A2=2b1−8b2=0(42)

Solving Eqs. (36) – (41), we obtain

a0=1.375, a1=0.5, a2=0.125b0=0.625, b1=−0.5, b2=−0.125

Subsituting these coefficients in Eqs. (4) – (5), then we get

y12(x)=1+t2

y22(x)=1−t2

These solutions are exact solution of this problem.

Example 2. Let us consider the linear, nonhomogeneous systems of Lane-Emden equations [26]

y″1(t)+2ty′1(t)−(4t2+6)y1(t)+y2(t)=t4−t3y″2(t)+8ty′2(t)+ty2(t)+y1(t)=et2+t5−t4+44t2−30t

subject to conditions

y1(0)=1, y′1(0)=0, y2(0)=0, y′2(0)=0

The comparison among present method and exact solution are shown in Table 1 and Table 2. In Table 3, the computational results of the L²-norm error between the approximate solutions and exact solution for different N and truncated errors are summarized. Figs. 1 – 2 represent the error between exact and approximate solutions for N = 6, 8. These figures show that errors are descreasing when N increasing.

Fig. 1

Comparison of absolute errors function for y₁(t)

Fig. 2

Comparison of absolute errors function for y₂(t)

Table 1

Numerical result for approximate solution of y₁(t) in Example 2.

t	Exact Solution	N=5	N_e=5	N=6	N_e=6	N=8	N_e=8
0.0	1.000000	0.999999	0.800E-8	0.999999	0.500E-9	1.000000	0.000E-0
0.2	1.040810	1.040834	0.238E-4	1.040810	0.135E-6	1.040810	0.102E-6
0.4	1.173510	1.173384	0.126E-3	1.173517	0.690E-5	1.173510	0.261E-6
0.6	1.433329	1.433539	0.209E-3	1.433298	0.305E-4	1.433329	0.471E-6
0.8	1.896480	1.895792	0.688E-2	1.896583	0.102E-3	1.896481	0.909E-6
1.0	2.718281	2.686791	0.314E-1	2.712165	0.611E-3	2.718083	0.197E-3

Table 2

Numerical result for approximate solution of y₂(t) in Example 2.

t	Exact Solution	N=5	N_e=5	N=6	N_e=6	N=8	N_e=8
0.0	0.0000	0.000E-0	0.000E-0	0.000E-0	0.000E-0	0.00000	0.000E-0
0.2	-0.0064	-0.006400	0.436E-7	-0.006400	0.189E-9	-0.00640	0.122E-9
0.4	-0.0384	-0.038399	0.343E-6	-0.038400	0.358E-7	-0.03840	0.399E-9
0.6	-0.0864	-0.086399	0.770E-5	-0.086399	0.102E-6	-0.08640	0.138E-8
0.8	-0.1024	-0.102400	0.620E-5	-0.102400	0.259E-6	-0.10240	0.455E-8
1.0	0.0000	-0.419E-4	0.419E-4	-0.722E-5	0.722E-5	0.168E-6	0.168E-6

Table 3

Numerical result for Example 2.

Present method	ENL		ENT
	y₁(t)	y₂(t)	y₁(t)	y₂(t)
N = 5	0.615773×10⁻²	101474×10⁻⁴	10⁻²	10⁻⁶
N = 6	0.103500×10⁻²	0.148370×10⁻⁵	10⁻³	10⁻⁸
N = 8	0.263404×10⁻⁴	0.256832×10⁻⁷	10⁻⁵	10⁻¹⁰

Example 3. Let us consider the nonlinear equation [26]

y″1(t)+5ty′1(t)+8(ey1(x)+2e−y2/2)=0y″2(t)+3ty′2(t)−8(e−y2+ey1/2)=0

with initial conditions

y1(0)=1, y′1(0)=0, y2(0)=0, y′2(0)=0

Above algorithm is applied to this problem, we have y₁(x) = x² − 2 ln 1, y₂(x) = x² + 2 ln 1 which is the exact solution of this problem.

Example 4. Let us consider the following nonlinear problem [26]

y″1(t)+1ty′1(t)−y23(t)(y12(t)+1)=0y″2(t)+3ty′2(t)+y25(t)(y12(t)+3)=0

subject to conditions

y1(0)=1, y′1(0)=1, y2(0)=0, y′2(0)=0

with exact solutions

y1=1+t2, y2=11+t2,

Applying our method for N = 4, 5, 6, obtained numerical results are displayed in Table 4 and Table 5. Also, absolute errors are displayed in Figures 3 and 4. The tables and the figures show that proposed method is in good agreement with the analytical solution.

Fig. 3

Comparison of absolute errors function for y₁(t) in Ex. 4

Fig. 4

Comparison of absolute errors for y₂(t) in Ex. 4

Table 4

Numerical result for approximate solution of y₁(t) in Example 5.

t	Solution Exact	N=4	N_e=4	N=5 N_e=5		N=6	N_e=6
0.0	1.00000	1.000000	0.000E-0	1.00000	0.000E-0	1.00000	0.000E-0
0.2	1.019803	1.020313	0.509E-3	1.019803	0.565E-4	1.019803	0.642E-5
0.4	1.077032	1.077661	0.628E-3	1.077032	0.216E-4	1.077032	0.220E-5
0.6	1.166190	1.166468	0.277E-3	1.166190	0.557E-5	1.166190	0.611E-5
0.8	1.280624	1.280897	0.272E-3	1.280624	0.738E-4	1.280624	0.471E-5
1.0	1.414213	1.414858	0.644E-3	1.414213	0.746E-4	1.414213	0.556E-5

Table 5

Numerical result for approximate solution of y₂(t) in Example 5.

t	Exact Solution	N=4	N_e=4	N=5 N_e=5		N=6	N_e=6
0.0	1.000000	1.000000	0.000E-0	1.000000	0.000E-0	1.000000	0.000E-0
0.2	0.980580	0.979703	0.887E-3	0.980580	0.165E-4	0.980580	0.467E-5
0.4	0.928476	0.928249	0.227E-3	0.928476	0.151E-4	0.928476	0.405E-5
0.6	0.857492	0.858494	0.100E-3	0.857492	0.166E-4	0.857492	0.185E-5
0.8	0.780868	0.781561	0.692E-3	0.780868	0.167E-4	0.780868	0.327E-5
1.0	0.707106	0.706844	0.262E-3	0.707106	0.608E-4	0.707106	0.134E-6

6 Conclusion

An operational matrix method for the solution of systems of Lane-Emden equations has been proposed and investigated. Moreover, the convergence and error analysis have been determined. The presented numerical examples have exhibited the high accuracy, applicability, and efficiency of the proposed algorithms. The proposed method has been given to find the analytical solutions if the problem has exact solutions that are polynomial functions. This method has some considerable advantage that only small size operational matrix is required to provide the solution of high accuracy because most of matrix involves more numbers of zeroes and thus, reduces the run time such as 1.2 sn for Example 3 and lower operation count results in reduction of cumulative truncation errors and improvement of overall accuracy. It follows from the numerical results that the accuracy of the solutions obtained using the operational method is quite good in comparison with the exact solution. Moreover, in Tables 3, we give the L²-norm errors for Example 2.

Afterwards, the method can also be extended to the system of integro-differential equations, but some modifications are required.

Acknowledgement

Author is very grateful to the reviewers for their valuable comments and suggestions which has helped immensely in improving the quality of this manuscript.

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Received: 2018-03-26

Revised: 2018-05-11

Accepted: 2018-08-03

Published Online: 2018-11-02

Published in Print: 2019-01-28

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2018-0062

Keywords for this article

Lane-Emden system; engineering problem; Chebyshev polynomials; operational matrix method

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