 # The generalized fractional order of the Chebyshev functions on nonlinear boundary value problems in the semi-infinite domain

• and Mehdi Delkhosh
From the journal Nonlinear Engineering

## Abstract

A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

MSC 2010: 34B15; 34B40; 74S25; 34L30

## 1 Introduction

In this section, some necessary preliminary which are useful for our method have been introduced.

### 1.1 The Chebyshev functions

The Chebyshev polynomials have frequently been used in numerical analysis including polynomial approximation, Gauss-quadrature integration, integral and differential equations and spectral methods. Chebyshev polynomials have many properties, for example orthogonal, recursive, simple real roots, complete in the space of polynomials. For these reasons, many researchers have employed these polynomials in their research [1, 2].

Using some transformations, the number of researchers extended Chebyshev polynomials to semiinfinite or infinite domains, for example by using x=tLt+L,L>0 the rational Chebyshev functions are introduced [3, 4].

In the proposed work, by transformation x=12(tη)αα, η > 0 on the Chebyshev polynomials of the first kind, the generalized fractional order of the Chebyshev orthogonal functions (GFCF) in the interval [0, η] have been introduced, that we can use them to solve differential equations.

### 1.2 Basic definitions

In this section, some basic definitions and theorems have been expressed [5-7].

### Definition 1

For any real function f(t), t > 0, if there exists a real number p > μ, such that f(t) = tpf1(t), wherefl(t) ∈ C(0, ∞), is said to be in space Cμ, μ ∈ ℜ, and it is in the space Cμn if and only if f(n)Cμ, nN.

### Definition 2

Suppose that f(t), g(t) ∈ C(0, η) and w(t) is a positive weight function in (0, η), then we define

f(t)w2=0ηf2(t)w(t)dt,f(t),g(t)w=0ηf(t)g(t)w(t)dt.

### Theorem 1

(Taylor’s formula) Suppose that f(k)(t) ∈ C(0, η) where k = 0, 1, ..., m and η > 0. Then we have

f(t)=i=0m1tii!f(i)(0+)+tmm!f(m)(ξ),(1)

with 0 < ξt, ∀t ∈ [0, η]. And thus for M ⩾ |f(m)(ξ)|:

|f(t)i=0m1tii!f(i)(0+)|Mtmm!.(2)

### Proof

See Ref. .  □

We know that, the solution of some equations is generated by fractional powers or the structure of the solution of some equations is not exactly known. For example, one of the famous equations that its solution is generated by fractional powers is Thomas-Fermi equation [14, 15]. Baker  has proved that the solution of Thomas-Fermi equation is generated by the powers of t12. For these reasons, in this paper, we decided to solve equations using the fractional basis, namely the generalized fractional order of the Chebyshev function (GFCF), in order to obtain acceptable results.

The GFCFs are introduced as a new basis for Spectral methods and this basis can be used to develop a framework or theory in Spectral methods. In this research, the fractional basis was used for solving nonlinear ordinary differential equations and it provided insight into an important issue.

Recently, some researchers have introduced the fractional basis of various basic functions and have used them in their research, such as the fractional-order Euler functions , the fractional-order Legendre functions [6, 8], the fractional-order Bessel functions , the fractional-order Jacobi functions [10, 11], and the fractional-order Bernoulli functions .

The organization of the paper is expressed as follows: In section 2, the GFCFs and their properties are obtained. In section 3, the proposed method is applied for solving some nonlinear boundary value problems in the semiinfinite domain. Finally, a brief conclusion is given in the last section.

## 2 Generalized Fractional order of the Chebyshev Functions

In this section, first, the generalized fractional order of the Chebyshev functions (GFCFs) of the first kind have been defined and then some properties and convergence of them for our method have been provided.

### 2.1 The GFCFs definition

The efficient methods have been used by many researchers to solve the differential equations (DE) is based on series expansion of the form i=0nciti, such as Adomian’s decomposition method and Homotopy perturbation method. But the exact solution of some DEs can not be estimated by polynomials basis, for a simple example: the ODE of 4yy=3t,y(0)=y(0)=0, that the exact solution is y(t)=t32, therefore we have decided to define a new basis for Spectral methods to solve them as follows:

Φn(t)=i=0ncitiα

Now by transformation z=12(tη)α;α,η>0 on classical Chebyshev polynomials of the first kind, we is defined the GFCFs in the interval [0, η], which is denoted by ηFTnα(t)=Tn(12(tη)α).

The ηFTnα(t) can be obtained using the recursive relation as follows:

ηFT0α(t)=1,ηFT1α(t)=12(tη)α,ηFTn+1α(t)=(24(tη)α)ηFTnα(t)ηFTn1α(t),n=1,2,.,

The analytical form of ηFTnα(t) of degree is given by

ηFTnα(t)=k=0nβn,k,η,αtαk,t[0,η],(3)

where

βn,k,η,α=(1)kn22k(n+k1)!(nk)!(2k)!ηαkandβ0,k,η,α=1.

Note that ηFTnα(0)=1 and ηFTnα(η)=(1)n.

The GFCFs are orthogonal with respect to the weight function w(t)=tα21ηαtα in the interval (0, η):

0ηηFTnα(t)ηFTmα(t)w(t)dt=π2αcnδmn.(4)

where δmn is Kronecker delta, c0 = 2, and cn = 1 for n ⩾ 1. The Eq. (4) is provable using the properties of orthogonality in the Chebyshev polynomials.

Figs. 1 show graphs of GFCFs for various values of n and α and η = 5. Fig. 1

Graphs of the GFCFs for various values of n and α.

### 2.2 Approximation of functions

Any function y(t) ∈ C[0, η] can be expanded as the follows:

y(t)=n=0anηFTnα(t),

where the coefficients an are obtained by inner product:

y(t),ηFTnα(t)w=n=0anηFTnα(t),ηFTnα(t)w

and using the property of orthogonality in the GFCFs:

an=2απcn0ηηFTnα(t)y(t)w(t)dt,n=0,1,2,.

In practice, we have to use first m-terms GFCFs and approximate y(t):

y(t)ym(t)=n=0m1anηFTnα(t)=ATΦ(t),(5)

where

A=[a0,a1,...,am1]T,(6)
Φ(t)=[ηFT0α(t),ηFT1α(t),...,ηFTm1α(t)]T.(7)

### 2.3 Convergence of the method

The following theorem shows that by increasing m, the approximation solution fm(t) is convergent to f(t) exponentially.

### Theorem 2

Suppose that f(k)(t) ∈ C[0, η] for k = 0,1, ..., m, and ηFmα is the subspace generated by {ηFT0α(t),ηFT1α(t),...,ηFTm1α(t)}. If fm(t) = ATΦ(t) (in Eq. (5)) is the best approximation to f(t) from ηFmα, then the error bound is presented as follows

f(t)fm(t)wηmMm!πΓ(2ma+12)αΓ(2ma+1),(8)

whereM ≥ |f(m)(t)|, t ∈ [0, η].

### Proof

By theorem 1, y(t)=i=0m1tii!f(i)(0+) and

|f(t)y(t)|Mtmm!.

Since ATΦ(t) is the best approximation to f(t) from ηFmα, and y(t)ηFmα, one has

f(t)fm(t)w2f(t)y(t)w2M2m!20ηtα2+2mα1ηαtαdt=M2m!2η2mπΓ(2ma+12)αΓ(2ma+1).

Now, by taking the square roots, the theorem can be proved. Eq. (8) shows that if m → ∞ then ||f(t) - fm(t)||w→0.  □

### Theorem 3

The generalized fractional order of the Chebyshev function, ηFTnα(t), has precisely n real zeros on interval (0, η) in the form

tk=η1cos((2k1)π2n)21α,k=1,2,...,n.

Moreover, ddtηFTnα(t) has precisely n-1 real zeros on interval (0, η) in the following points:

tk=η1cos(kπn)21α,k=1,2,...,n1.

### Proof

The Chebyshev polynomial Tn(x) has n real zeros :

xk=cos((2k1)π2n),k=1,2,...,n.

Therefore Tn(x) can be written as

Tn(x)=(xx1)(xx2)...(xxn).

Using transformation x=12(tη)α yields to

ηFTnα(t)=12tηαx112(tη)αx212tηαxn,

so, the real zeros of ηFTnα(t) are tk=η(1xk2)1α. Also, we know that, the real zeros of ddtTn(t) occurs in the following points:

xk=cos(kπn),k=1,2,...,n1

Same as in the previous, the absolute extremes of ηFTnα(t) are tk=η(1xk2)1α.  □

## 3 Examples

Many problems arising in fluid dynamics, quantum mechanics, astrophysics, and other fields are defined on infinite or semi-infinite domains. There are different approaches for solving this type of equation, such as numerical, analytical, and semi-analytical methods. In this section, we attempt to introduce a numerical method, based on the generalized fractional order of the Chebyshev orthogonal functions for solving this type of equations.

### 3.1 The unsteady isothermal flow of a gas

One of the important nonlinear ordinary differential equations that occurs on semi-infinite domain is the unsteady gas equation:

d2ydt2+2t1βy(t)dydt=0,t[0,),(9)

where 0 ⩽ β ⩽ 1 is a real constant and the boundary conditions are:

y(0)=1,limty(t)=0.(10)

A substantial amount of numerical and analytical work has been invested so far in this model . The main reason of this interest is that the approximation can be used for many engineering purposes. As stated before, the problem of Eq. (9) was handled by Kidder  where a perturbation technique is carried out to include terms of the second order. Wazwaz  has solved this equation nonlinearly by modifying the decomposition method and Pade approximation. Parand et al. [19, 20], and Taghavi et al.  have also applied the rational Jacobi functions, Bessel function collocation method, and modified generalized Laguerre polynomials for solving this equation. Rezaei et al.  have applied two numerical methods based on Sinc and rational Legendre functions to solve gas flow through a micro-nano-porous media. Rad et al.  have solved this equation by two numerical and analytical solutions based on Homotopy analysis method and Hermite functions collocation method. Recently, Parand et al.  have used the combination of the quasilinearization method and the rational Jacobi function collocation method, and have obtained an accurate solution to the equation, the value of initial slope is calculated as -1.1917906497194217341228284 for β = 0.50.

Now, we solve this equation by using GFCF collocation method.

For satisfying the boundary conditions, we satisfy the Eq. (10) as follows:

ym^(t)=λt2+λ+te2tym(t),(11)

where λ is an arbitrary real constant and ym(t) is defined in Eq. (5). So, ym^(0)=1 and ym^(t)=0when t tends to ∞ and the boundary conditions (10) are satisfied for all λ > 0, and is defined in the semi-infinite domain.

To apply the collocation method, we construct the residual function by substituting ym^(t) in Eq. (11) for y(t) in the unsteady gas equation (9):

Res(t)=d2dt2ym^(t)+2t1βym^(t)ddtym^(t).(12)

The equations for obtaining the coefficient {ai}i=0m1 arise from equalizing Res(t) to zero on m collocation points:

Res(ti)=0,i=0,1,...,m1.

In this study, the roots of the GFCFs in the interval [0, η] (Theorem 3) are used as collocation points. By solving the obtained set of equations, we have the approximating function ym^(t).

Table 1 shows the value of y′(0) by the present method and comparison it with the values of Bessel function collocation (BFC) , Shooting method , and RBF  for β = 0.25, 0.50, 0.75 with m = 35 and α = 0.50.

Table 1

Comparison of the obtained values of y′(0) for β = 0.25, 0.50, 0.75 by the present method, Shooting method, BFC , and RBF , with m = 35 and α = 0.50

βλPresentBFCRBFShootingAbs. Err.
0.250.1754-1.156572055-1.156275-1.156557-1.1565725.514e-8
0.500.1755-1.191790893-1.191718-1.191498-1.1917915.078e-7
0.750.0167-1.239760055-1.239760-1.239671-1.2397605.565e-8

Table 2 shows the obtained values of y’(0) and Resw2 by the present method for various values of m, β = 0.50, and α = 0.50.

Table 2

Obtained values of y′(0) by the present method for various values of m, β = 0.50, and α = 0.50

mλy′(0)Resw2
150.3005-1.1917940972.2591e-02
250.1788-1.1917731559.2586e-05
350.1755-1.1917908931.0379e-05

Table 3 shows the values of y(t) for various values of t by the present method and comparison it with the values of perturbation (PB) , Bessel function collocation (BFC) , modified generalized Laguerre (MGL) , Runge-Kutta method (RK) , RBF , and finite difference (FD) , with m = 35 and α = 0.50.

Table 3

Obtained values of y(t) for β = 0.5, α = 0.5, m = 35, and λ = 0.1755, comparison between the present method, BFC , RBF , FD , PB , RK , and MGL 

tPresentBFC RBF FD PB RK MGL 
0.10.8813649790.881371780.881398020.881364650.881658820.881364610.90931873
0.20.7658289980.765828740.765880290.765828820.766307670.765828740.81748763
0.30.6560001410.656021430.656069280.656000690.656537990.656000580.71522344
0.40.5538976860.553925930.553994310.553898940.554402400.553898810.60982075
0.50.4609431640.460975510.461072020.460942760.461365020.460942600.51632348
0.60.3779810560.378019570.378143800.377981580.378310930.377981400.41932385
0.70.3053509820.305395030.305541740.305352320.305597650.305352120.40982377
0.80.2429534300.243132540.243165540.242954370.243132540.242954160.31999068
0.90.1903336780.190384710.190564190.190334210.190462360.190333990.20820285
1.00.1467726180.146826910.147020480.146773280.158768980.146773050.21991074

Figure 2 shows the graphs of y(t) and y′(t) from the solution of unsteady gas equation for β = 0.25, 0.50, and 0.75 with m = 35, α = 0.5 and η = 10. Fig. 2

Obtained graphs of unsteady gas equation for β = 0.25, 0.50, 0.75

Figure 3 shows the graphs of residual errors of Eq. (12) and the logarithm of coefficients |ai| for β = 0.50, 0.75, with m = 35 and α = 0.50, to show the convergence of the method. Fig. 3

Graphs of residual errors and log(|ai| for β = 0.50(Blue) and β = 0.75(Red)

### 3.2 The third grade fluid in a porous half space

The boundary value problem modelling the steady state flow of a third grade fluid in a porous half space on the semi-infinite domain is as follows:

d2ydt21+b1(dydt)2b1b23ydydt2b2y=0,t[0,),(13)

where b1 and b2 are real constants and the boundary conditions for this equation:

y(0)=1,limty(t)=0.(14)

Some researchers approximate the third grade fluid equations in a porous half space; for example, Ahmad  by applying the Homotopy analysis method, Kazem et al.  by applying the radial basis functions collocation method, Parand and Hajizadeh  by applying the modified rational Christov functions collocation method, Baharifard et al.  by applying the rational and exponential Legendre Tau method.

For satisfying the boundary conditions, we satisfy the Eq. (14) as follows:

ym^(t)=λt2+λ+te2tym(t),(15)

where λ is an arbitrary real constant and ym(t) is defined in Eq. (5). So, ym^(0)=1, and ym^(t)=0 when t tends to ∞, and the boundary conditions (14) are satisfied for all λ > 0, and is defined in the semi-infinite domain. We construct the residual function as follows:

Res(t)=d2ym^dt21+b1(dym^dt)2b1b23ym^dym^dt2b2ym^.(16)

As before, by solving the obtained set of equations, we have the approximating function ym^(t).

Table 4 shows the obtained values of y′(0) by the present method and comparison it with the values of Gaussian RBF (G-RBF) , rational Christov functions (RCF) , Shooting method , rational Legendre Tau method (RLT) , and exponential Legendre Tau method (ELT)  for various values of b1 and b2, with m = 20 and α = 0.50.

Table 4

Comparison of the obtained values of y′(0) for various values of b1 and b2 by the present method, shooting method, RLT, ELT, RCF, and G-RBF

b1b2λPresentShootingRLT ELT RCF G-RBF
0.30.50.80-0.69128036-0.691280-0.691493-0.691279
0.60.80-0.67830298-0.678301-0.678511-0.678302-0.6783017-0.6783013
0.90.80-0.66732796-0.667327-0.667528-0.667327
1.20.80-0.65783825-0.657838-0.658029-0.657837
0.60.30.81-0.53331017-0.533303-0.533545-0.533302
0.60.81-0.73800824-0.738008-0.738116-0.738007
0.90.81-0.88746913-0.887467-0.887350-0.887467
1.20.81-1.00865474-1.008653-1.008516-1.008653

Table 5 shows the obtained values of y′(0) and Resw2 by the present method for various values of m, b1 = 0.60, b2 = 0.50, and α = 0.50.

Table 5

Obtained values of y′(0) by the present method for various values of m and α = 0.50

mλy′(0)Resw2
100.4-0.67572664616.098e-03
150.7-0.67829358575.147e-05
200.8-0.67830298291.198e-05

Table 6 shows the obtained values of y′(0) and Resw2 by the present method for various values of α, b1 = 0.60, b2 = 0.50, and m = 20.

Table 6

Obtained values of y′(0) by the present method for various values of α and m = 20

αλy′(0)Resw2
0.250.4-0.678276279.6011e-04
0.500.8-0.678302981.1986e-05
0.751.0-0.678441961.1074e-06
1.001.8-0.678301712.7556e-08

Table 7 shows the obtained values of y(t) for various values of t by the present method and comparison it with the values of Homotopy analysis method (HAM) , Gaussian RBF (G-RBF) , rational Christov functions (RCF) , rational Legendre Tau method (RLT) , and exponential Legendre Tau method (ELT) , with m = 20 and α = 0.50.

Table 7

Comparison of the obtained values of y(t) for various values of t by the present method, RLT, ELT, RCF, G-RBF, and HAM

tPresentRLT ELT RCF G-RBFHAM 
0.20.872610200.872610.872610.8726085960.872652640.87220
0.40.760625620.760630.760630.7606268050.760748430.76010
0.60.662432200.662430.662430.6624311760.662614880.66190
0.80.576503400.576500.576500.5765022980.576714950.57600
1.00.501437230.501430.501440.5014361650.501645420.50100
1.20.435951390.435950.435950.4359503970.436133220.43560
1.60.329203990.329200.329200.3292020520.329306790.32890
2.00.248386910.248370.248380.2483843480.248427020.24820
2.50.174551430.174550.174550.1745476340.174560330.17440
3.00.122617900.122640.122610.1226123860.122626520.12250
3.50.086121240.086170.08611
4.00.060484710.060540.060470.0604734180.060490380.06042
4.50.042481740.042520.04247
5.00.029842410.029840.029820.0298191280.029824460.02979

Figure 4 shows the graphs of residual error of Eq. (16) and the logarithm of coefficients |ai| for b1 = 0.60, b2 = 0.50, with m = 20 and α = 0.50, to show the convergence of the method. Fig. 4

Obtained graphs of residual error and the logarithm of coefficients |ai for b1 = 0.60 and b2 = 0.50, to show the convergence of the method.

### 3.3 The Blasius equation

Another of important third-order nonlinear ordinary differential equations that occurs in the semi-infinite domain is the Blasius equation:

2d3ydt3+yd2ydt2=0,t[0,),(17)

where the boundary conditions for this equation are as follows:

y(0)=0,y(0)=0,limty(t)=1.(18)

In recent years, different methods have been used to solve the Blasius equation. For example, Liao  by applying the Homotopy analysis method (HAM), Yu and Chen  by applying the differential transformation method, Wang  by applying the Adomian decomposition method (ADM), Hashim  by applying the ADM Pade approach, Cortell  by applying the Runge-Kutta algorithm, Wazwaz  by applying the modified Adomian decomposition method, Parand et al. [36, 37] by applying Bessel functions of the first kind and the rational Chebyshev functions. In 2008, Boyd  has solved the Blasius equation and has reported the accurate solution of 0.33205733621519630 for f″(0).

For satisfying the boundary conditions, we satisfy the Eq. (18) as follows:

ym^(t)=t3(t+λ)2+t2e2t2ym(t),(19)

where λ is an arbitrary real constant and ym(t) is defined in Eq. (5). So, ym^(0)=0,ym^(0)=0, and ym^(t)=1 when t tends to ∞, and the boundary conditions (18) are satisfied for all λ > 0, and is defined in the semi-infinite domain. We construct the residual function as follows:

Res(t)=2d3ym^dt3+ym^d2ym^dt2.(20)

Table 8 shows the value of y″(0) by the present method with m = 20, α = 0.50 and λ = 0.9859 and comparison it with the values of Liao , Parand et al. [3, 36], and Boyd .

Table 8

Comparison of the obtained values of y″(0) between the present method, Boyd , Parand et al. [3, 36], and Liao 

PresentBoyd Parand Parand Liao 
0.33205730490.332057336215196300.332057336215195420.332057330.33206

Table 9 shows the obtained values of y″(0) and Resw2 by the present method for various values of m andα = 0.50.

Table 9

Obtained values of y″(0) by the present method for various values of m and α = 0.50

mλy′(0)Resw2
100.770.332389312.0196e-01
151.090.332135941.0365e-03
200.980.332053672.8918e-08

Table 10 shows the obtained values of y″(0) and Resw2 by the present method for various values of α and m = 20.

Table 10

Obtained values of y″(0) by the present method for various values of α and m = 20

αλy″(0)Resw2
0.251.01640.33206001701.1698e-02
0.500.98590.33205730492.8930e-08
0.751.71910.33207183553.4739e-08
1.000.91680.33205735411.5661e-06

Tables 11-13 show the values of y(t), y′(t) and y″(t) for various values of t by the present method and comparison it with the values of Parand et al. [3, 36], Cortell , and Howarth , with m = 20, α = 0.50 and λ = 0.9859.

Table 11

Comparison of the obtained values of y(t) for various values of t by the present method, Parand et al. [3, 36], Howarth , and Cortell 

tPresentParand Howarth Cortell Parand 
1.00.16557173460.16557170.165570.165570.1655724
2.00.65002441890.65002430.650030:650030.6500351
3.01.39680833091.39680821.396821.396821.3968223
4.02.30574657422.30574642.305762.305762.3057618
5.03.28327380233.28327363.283293.283303.2832910

Table 12

Comparison of the obtained values of y′(t) for various values of t by the present method, Parand et al.[3, 36], Howarth , and Cortell 

tPresentParand Howarth Cortell Parand 
1.00.32978008070.329780030.329790.329780.3297963
2.00.62976578680.629765730.629770.629770.6297763
3.00.84604449870.846044440.846050.846050.8460595
4.00.95551828490.955518230.955520.955520.9555236
5.00.99154136060.991541900.991550.991550.9915546

Table 13

Comparison of the obtained values of y″(t) for various values of t by the present method, Parand et al. [3, 36], Howarth , and Cortell 

tPresentParand Howarth Cortell Parand 
0.00.33205730490.33205733620.332060.332060.3320571
1.00.32300740620.32300711680.323010.323010.3230136
2.00.26675151440.26675154560.266750.266750.2667557
3.00.16136031850.16136031940.161360.161360.1613637
4.00.06423411870.06423412090.064240.064230.0642411
5.00.01590261750.01590679850.015910.015910.0159134

Figure 5 shows the graph of residual error of Eq. (20) and the graph of obtained solution for the Blasius equation by the present method with m = 20 and α = 0.50. Fig. 5

Obtained graphs of residual error and the Blasius solution.

### 3.4 The field equation determining the vortex profile

Finally, we consider the field equation determining the vortex profile:

d2ydt2+1tdydt+1n2t2y(t)y3(t)=0,t[0,),(21)

where the boundary conditions for this equation are as follows:

limt0y(t)=kntn+O(tn+2),limty(t)=1.(22)

where y(t) is a real function and nZ.

A major problem of the this boundary conditions amounts to find the value(s) of the free parameter, kn, to ensure the boundary conditions (22) of y(t) at t = ∞.

This equation is examined in the static, rotationally symmetric global vortex in a Ginzburg-Landau effective theory , which has numerous applications ranging from condensed matter to cosmic strings . Some researchers approximate the field equation determining the vortex profile; for example, Boisseau et al.  by applying the analytical method based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients, and Amore & Fernandez  by applying the Pade-Hankel method.

For satisfying the boundary conditions, we satisfy the Eq. (22) as follows:

ym^(t)=tntn+λ+tne2t2ym(t),(23)

where λ is an arbitrary real constant and ym(t) is dehned in Eq. (5). So the boundary conditions (22) are satished for all λ > 0, and is dehned in the semi-infinite domain. We construct the residual function as follows:

Res(t)=d2ym^dt2+1tdym^dt+1n2t2ym^(t)ym^3(t).(24)

Table 14 shows the values of kn by the present method with m = 40 and α = 0.50, and comparison it with the values of Shooting method , Boisseau et al. , and Amore & Fernandez .

Table 14

Comparison of the obtained values of kn between the present method, Shooting method , Boisseau et al. , and Amore & Fernandez .

knλPresentShooting Boisseau Amore 
k10.10.5831897174890.58318949590.58318949360.5831894958601
k220.1530991028290.15309910290.153090.15309910286
k3230.0261834202090.026183420720.0261850.0261834207
k4450.0033271733390.003327173400.00330.0033271734

Table 15 shows the obtained values of k2 and Resw2 by the present method for various values of m and α = 0.50.

Table 15

Obtained values of k2 by the present method for various values of m and α = 0.50

mλk2Resw2
1020.1529258473004.3939e-05
2020.1530990508703.5332e-10
3020.1530991055357.3141e-13
4020.1530991028291.5483e-16

Table 16 shows the obtained values of k2 and Resw2 by the present method for various values of α and m = 40. As can be seen, the results for α = 0.50 and α = 1.00 are almost identical.

Table 16

Obtained values of k2 by the present method for various values of α and m = 40

αλk2Resw2
0.2520.1530921052542.7708e-04
0.5020.1530991028291.5483e-16
0.7520.1530687747736.7629e-08
1.0020.1530991036728.0842e-16
1.2510.1516963048178.8242e-05
1.5010.1370874138817.2232e-04

Table 17 shows the obtained values of y(t) by the present method for various values of n and t with m = 40 and α = 0.50.

Table 17

Obtained values of y(t) by the present method for various values of n and t.

tn = 1n = 2n = 3n = 4
0.50.282821730.037483810.003222100.00020535
1.00.520051630.140783580.024587440.00316426
1.50.692087350.285314190.076629910.01503527
2.00.804956820.440087510.162245060.04343598
2.50.874970190.579723300.273831620.09430307
3.00.917481070.691547540.396619140.16902827
3.50.943377030.774355810.514957660.26313174
4.00.959468290.833004590.617880700.36737260
6.00.984731110.936704600.848453200.71026581
8.00.991845330.966818210.922947130.85614939
10.0.994873780.979303590.952658490.91367234

Figure 6 shows the graphs of residual errors of Eq. (24) and the graphs of the obtained solutions for the field equation by the present method with m = 40 and α = 0.50. Fig. 6

Obtained graphs of the residual errors and the obtained solutions for various values of n.

Figure 7 shows the graphs of 1n!y(n)(t) for n = 1, 2, 3, and 4, and the graphs of log(|ai|) for n = 2, 3, and 4 to show the convergence of the method. Fig. 7

(a) Graphs of 1n!y(n)(t) for n = 1, 2, 3, 4 (b) Graphs of log(|ai|) for n = 2, 3, 4 to show the convergence.

## 4 Conclusion

The main goal of this paper is introducing a new orthogonal fractional basis, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCF), for solving nonlinear boundary value problems in the semi-infinite domain. Solving these problems is difficult because they have a boundary condition in the infinite. But we used these new basis to solve them and obtained the good results. The present results show that new basis for the collocation Spectral method is efficient and applicable. A comparison was made of the numerical solution of other researchers and the present method. It has been shown that the present method has provided an acceptable approach for solving these types of equations.

## Acknowledgement

The authors are very grateful to reviewers and editor for carefully reading the paper and for their comments and suggestions which have improved the paper.

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