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

Kourosh Parand; Mehdi Delkhosh

doi:10.1515/nleng-2017-0030

Open Access Published by De Gruyter May 20, 2017

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

Kourosh Parand and Mehdi Delkhosh

From the journal Nonlinear Engineering

https://doi.org/10.1515/nleng-2017-0030

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.

Keywords: Generalized fractional order of the Chebyshev functions; Unsteady isothermal flow of a gas equation; Third grade fluid equation; Blasius equation; Field equation; Semi-Infinite domain

PACS: 02.60.Lj; 02.70.Jn

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=t−Lt+L,L>0 the rational Chebyshev functions are introduced [3, 4].

In the proposed work, by transformation x=1−2(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) = t^pf₁(t), wheref_l(t) ∈ C(0, ∞), is said to be in space C_μ, μ ∈ ℜ, and it is in the space Cμn if and only if f⁽ⁿ⁾ ∈ C_μ, n ∈ N.

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=0m−1tii!f(i)(0+)+tmm!f(m)(ξ),(1)

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

|f(t)−∑i=0m−1tii!f(i)(0+)|⩽Mtmm!.(2)

Proof

See Ref. [13]. □

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 [15] 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 [4], the fractional-order Legendre functions [6, 8], the fractional-order Bessel functions [9], the fractional-order Jacobi functions [10, 11], and the fractional-order Bernoulli functions [12].

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=1−2(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(1−2(tη)α).

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

ηFT0α(t)=1 , ηFT1α(t)=1−2(tη)α,ηFTn+1α(t)=(2−4(tη)α) ηFTnα(t)− ηFTn−1α(t), n=1,2,⋯.,

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

ηFTnα(t)=∑k=0nβn,k,η,αtαk,t∈[0,η],(3)

where

βn,k,η,α=(−1)kn22k(n+k−1)!(n−k)!(2k)!ηαk and β0,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)=tα2−1ηα−tα in the interval (0, η):

∫0η ηFTnα(t) ηFTmα(t)w(t)dt=π2αcnδmn.(4)

where δ_mn is Kronecker delta, c₀ = 2, and c_n = 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=0∞an ηFTnα(t),

where the coefficients a_n are obtained by inner product:

〈y(t), ηFTnα(t)〉w=〈∑n=0∞an ηFTnα(t), ηFTnα(t)〉w

and using the property of orthogonality in the GFCFs:

an=2απcn∫0η ηFTnα(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=0m−1an ηFTnα(t)=ATΦ(t),(5)

where

A=[a0,a1,...,am−1]T,(6)

Φ(t)=[ηFT0α(t), ηFT1α(t),..., ηFTm−1α(t)]T.(7)

2.3 Convergence of the method

The following theorem shows that by increasing m, the approximation solution f_m(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),...,ηFTm−1α(t)}. If f_m(t) = A^TΦ(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)∥w≤ηmMm!πΓ(2ma+12)αΓ(2ma+1),(8)

whereM ≥ |f^(m)(t)|, t ∈ [0, η].

Proof

By theorem 1, y(t)=∑i=0m−1tii!f(i)(0+) and

|f(t)−y(t)|≤Mtmm!.

Since A^TΦ(t) is the best approximation to f(t) from ηFmα, and y(t)∈ ηFmα, one has

∥f(t)−fm(t)∥w2≤∥f(t)−y(t)∥w2≤M2m!2∫0ηtα2+2mα−1ηα−tαdt=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) - f_m(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=η1−cos⁡((2k−1)π2n)21α, k=1,2,...,n.

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

tk′=η1−cos⁡(kπn)21α, k=1,2,...,n−1.

Proof

The Chebyshev polynomial T_n(x) has n real zeros [16]:

xk=cos⁡((2k−1)π2n), k=1,2,...,n.

Therefore T_n(x) can be written as

Tn(x)=(x−x1)(x−x2)...(x−xn).

Using transformation x=1−2(tη)α yields to

ηFTnα(t)=1−2tηα−x11−2(tη)α−x2…1−2tηα−xn,

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

xk′=cos⁡(kπn), k=1,2,...,n−1

Same as in the previous, the absolute extremes of ηFTnα(t) are tk′=η(1−xk′2)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+2t1−βy(t)dydt=0, t∈[0,∞),(9)

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

y(0)=1, limt→∞y(t)=0.(10)

A substantial amount of numerical and analytical work has been invested so far in this model [17]. 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 [17] where a perturbation technique is carried out to include terms of the second order. Wazwaz [18] has solved this equation nonlinearly by modifying the decomposition method and Pade approximation. Parand et al. [19, 20], and Taghavi et al. [21] have also applied the rational Jacobi functions, Bessel function collocation method, and modified generalized Laguerre polynomials for solving this equation. Rezaei et al. [22] 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. [23] have solved this equation by two numerical and analytical solutions based on Homotopy analysis method and Hermite functions collocation method. Recently, Parand et al. [19] 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+t e−2t ym(t),(11)

where λ is an arbitrary real constant and y_m(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)+2t1−βym^(t)ddtym^(t).(12)

The equations for obtaining the coefficient {ai}i=0m−1 arise from equalizing Res(t) to zero on m collocation points:

Res(ti)=0, i=0,1,...,m−1.

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) [20], Shooting method [24], and RBF [24] 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[24], BFC [20], and RBF [24], with m = 35 and α = 0.50

β	λ	Present	BFC	RBF	Shooting	Abs. Err.
0.25	0.1754	-1.156572055	-1.156275	-1.156557	-1.156572	5.514e-8
0.50	0.1755	-1.191790893	-1.191718	-1.191498	-1.191791	5.078e-7
0.75	0.0167	-1.239760055	-1.239760	-1.239671	-1.239760	5.565e-8

Table 2 shows the obtained values of y’(0) and ∥Res∥w2 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

m	λ	y′(0)	∥Res∥w2
15	0.3005	-1.191794097	2.2591e-02
25	0.1788	-1.191773155	9.2586e-05
35	0.1755	-1.191790893	1.0379e-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) [17], Bessel function collocation (BFC) [20], modified generalized Laguerre (MGL) [21], Runge-Kutta method (RK) [24], RBF [24], and finite difference (FD) [25], 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 [20], RBF [24], FD [25], PB [17], RK [24], and MGL [21]

t	Present	BFC [20]	RBF [24]	FD [25]	PB [17]	RK [24]	MGL [21]
0.1	0.881364979	0.88137178	0.88139802	0.88136465	0.88165882	0.88136461	0.90931873
0.2	0.765828998	0.76582874	0.76588029	0.76582882	0.76630767	0.76582874	0.81748763
0.3	0.656000141	0.65602143	0.65606928	0.65600069	0.65653799	0.65600058	0.71522344
0.4	0.553897686	0.55392593	0.55399431	0.55389894	0.55440240	0.55389881	0.60982075
0.5	0.460943164	0.46097551	0.46107202	0.46094276	0.46136502	0.46094260	0.51632348
0.6	0.377981056	0.37801957	0.37814380	0.37798158	0.37831093	0.37798140	0.41932385
0.7	0.305350982	0.30539503	0.30554174	0.30535232	0.30559765	0.30535212	0.40982377
0.8	0.242953430	0.24313254	0.24316554	0.24295437	0.24313254	0.24295416	0.31999068
0.9	0.190333678	0.19038471	0.19056419	0.19033421	0.19046236	0.19033399	0.20820285
1.0	0.146772618	0.14682691	0.14702048	0.14677328	0.15876898	0.14677305	0.21991074

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 |a_i| 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(|a_i| 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)2−b1b23ydydt2−b2y=0,t∈[0,∞),(13)

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

y(0)=1, limt→∞y(t)=0.(14)

Some researchers approximate the third grade fluid equations in a porous half space; for example, Ahmad [26] by applying the Homotopy analysis method, Kazem et al. [27] by applying the radial basis functions collocation method, Parand and Hajizadeh [28] by applying the modified rational Christov functions collocation method, Baharifard et al. [29] 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+t e−2t ym(t),(15)

where λ is an arbitrary real constant and y_m(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)2−b1b23ym^dym^dt2−b2ym^.(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) [27], rational Christov functions (RCF) [28], Shooting method [29], rational Legendre Tau method (RLT) [29], and exponential Legendre Tau method (ELT) [29] 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

b₁	b₂	λ	Present	Shooting	RLT [29]	ELT [29]	RCF [28]	G-RBF[27]
0.3	0.5	0.80	-0.69128036	-0.691280	-0.691493	-0.691279
0.6		0.80	-0.67830298	-0.678301	-0.678511	-0.678302	-0.6783017	-0.6783013
0.9		0.80	-0.66732796	-0.667327	-0.667528	-0.667327
1.2		0.80	-0.65783825	-0.657838	-0.658029	-0.657837
0.6	0.3	0.81	-0.53331017	-0.533303	-0.533545	-0.533302
	0.6	0.81	-0.73800824	-0.738008	-0.738116	-0.738007
	0.9	0.81	-0.88746913	-0.887467	-0.887350	-0.887467
	1.2	0.81	-1.00865474	-1.008653	-1.008516	-1.008653

Table 5 shows the obtained values of y′(0) and ∥Res∥w2 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

m	λ	y′(0)	∥Res∥w2
10	0.4	-0.6757266461	6.098e-03
15	0.7	-0.6782935857	5.147e-05
20	0.8	-0.6783029829	1.198e-05

Table 6 shows the obtained values of y′(0) and ∥Res∥w2 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

α	λ	y′(0)	∥Res∥w2
0.25	0.4	-0.67827627	9.6011e-04
0.50	0.8	-0.67830298	1.1986e-05
0.75	1.0	-0.67844196	1.1074e-06
1.00	1.8	-0.67830171	2.7556e-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) [26], Gaussian RBF (G-RBF) [27], rational Christov functions (RCF) [28], rational Legendre Tau method (RLT) [29], and exponential Legendre Tau method (ELT) [29], 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

t	Present	RLT [29]	ELT [29]	RCF [28]	G-RBF[27]	HAM [26]
0.2	0.87261020	0.87261	0.87261	0.872608596	0.87265264	0.87220
0.4	0.76062562	0.76063	0.76063	0.760626805	0.76074843	0.76010
0.6	0.66243220	0.66243	0.66243	0.662431176	0.66261488	0.66190
0.8	0.57650340	0.57650	0.57650	0.576502298	0.57671495	0.57600
1.0	0.50143723	0.50143	0.50144	0.501436165	0.50164542	0.50100
1.2	0.43595139	0.43595	0.43595	0.435950397	0.43613322	0.43560
1.6	0.32920399	0.32920	0.32920	0.329202052	0.32930679	0.32890
2.0	0.24838691	0.24837	0.24838	0.248384348	0.24842702	0.24820
2.5	0.17455143	0.17455	0.17455	0.174547634	0.17456033	0.17440
3.0	0.12261790	0.12264	0.12261	0.122612386	0.12262652	0.12250
3.5	0.08612124	0.08617	0.08611	—	—	—
4.0	0.06048471	0.06054	0.06047	0.060473418	0.06049038	0.06042
4.5	0.04248174	0.04252	0.04247	—	—	—
5.0	0.02984241	0.02984	0.02982	0.029819128	0.02982446	0.02979

Figure 4 shows the graphs of residual error of Eq. (16) and the logarithm of coefficients |a_i| 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 |a_i 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, limt→∞y′(t)=1.(18)

In recent years, different methods have been used to solve the Blasius equation. For example, Liao [30] by applying the Homotopy analysis method (HAM), Yu and Chen [31] by applying the differential transformation method, Wang [32] by applying the Adomian decomposition method (ADM), Hashim [33] by applying the ADM Pade approach, Cortell [34] by applying the Runge-Kutta algorithm, Wazwaz [35] 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 [38] 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+t2 e−2t−2 ym(t),(19)

where λ is an arbitrary real constant and y_m(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 [30], Parand et al. [3, 36], and Boyd [38].

Table 8

Comparison of the obtained values of y″(0) between the present method, Boyd [38], Parand et al. [3, 36], and Liao [30]

Present	Boyd [38]	Parand [36]	Parand [3]	Liao [30]
0.3320573049	0.33205733621519630	0.33205733621519542	0.33205733	0.33206

Table 9 shows the obtained values of y″(0) and ∥Res∥w2 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

m	λ	y′(0)	∥Res∥w2
10	0.77	0.33238931	2.0196e-01
15	1.09	0.33213594	1.0365e-03
20	0.98	0.33205367	2.8918e-08

Table 10 shows the obtained values of y″(0) and ∥Res∥w2 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

α	λ	y″(0)	∥Res∥w2
0.25	1.0164	0.3320600170	1.1698e-02
0.50	0.9859	0.3320573049	2.8930e-08
0.75	1.7191	0.3320718355	3.4739e-08
1.00	0.9168	0.3320573541	1.5661e-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 [34], and Howarth [39], 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 [39], and Cortell [34]

t	Present	Parand [36]	Howarth [39]	Cortell [34]	Parand [3]
1.0	0.1655717346	0.1655717	0.16557	0.16557	0.1655724
2.0	0.6500244189	0.6500243	0.65003	0:65003	0.6500351
3.0	1.3968083309	1.3968082	1.39682	1.39682	1.3968223
4.0	2.3057465742	2.3057464	2.30576	2.30576	2.3057618
5.0	3.2832738023	3.2832736	3.28329	3.28330	3.2832910

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 [39], and Cortell [34]

t	Present	Parand [36]	Howarth [39]	Cortell [34]	Parand [3]
1.0	0.3297800807	0.32978003	0.32979	0.32978	0.3297963
2.0	0.6297657868	0.62976573	0.62977	0.62977	0.6297763
3.0	0.8460444987	0.84604444	0.84605	0.84605	0.8460595
4.0	0.9555182849	0.95551823	0.95552	0.95552	0.9555236
5.0	0.9915413606	0.99154190	0.99155	0.99155	0.9915546

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 [39], and Cortell [34]

t	Present	Parand [36]	Howarth [39]	Cortell [34]	Parand [3]
0.0	0.3320573049	0.3320573362	0.33206	0.33206	0.3320571
1.0	0.3230074062	0.3230071168	0.32301	0.32301	0.3230136
2.0	0.2667515144	0.2667515456	0.26675	0.26675	0.2667557
3.0	0.1613603185	0.1613603194	0.16136	0.16136	0.1613637
4.0	0.0642341187	0.0642341209	0.06424	0.06423	0.0642411
5.0	0.0159026175	0.0159067985	0.01591	0.01591	0.0159134

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+1−n2t2y(t)−y3(t)=0, t∈[0,∞),(21)

where the boundary conditions for this equation are as follows:

limt→0y(t)=kntn+O(tn+2), limt→∞y(t)=1.(22)

where y(t) is a real function and n ∈ Z.

A major problem of the this boundary conditions amounts to find the value(s) of the free parameter, k_n, 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 [40], which has numerous applications ranging from condensed matter to cosmic strings [41]. Some researchers approximate the field equation determining the vortex profile; for example, Boisseau et al. [42] 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 [43] by applying the Pade-Hankel method.

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

ym^(t)=tntn+λ+tn e−2t−2 ym(t),(23)

where λ is an arbitrary real constant and y_m(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+1−n2t2ym^(t)−ym^3(t).(24)

Table 14 shows the values of k_n by the present method with m = 40 and α = 0.50, and comparison it with the values of Shooting method [42], Boisseau et al. [42], and Amore & Fernandez [43].

Table 14

Comparison of the obtained values of k_n between the present method, Shooting method [42], Boisseau et al. [42], and Amore & Fernandez [43].

k_n	λ	Present	Shooting [42]	Boisseau [42]	Amore [43]
k₁	0.1	0.583189717489	0.5831894959	0.5831894936	0.5831894958601
k₂	2	0.153099102829	0.1530991029	0.15309	0.15309910286
k₃	23	0.026183420209	0.02618342072	0.026185	0.0261834207
k₄	45	0.003327173339	0.00332717340	0.0033	0.0033271734

Table 15 shows the obtained values of k₂ and ∥Res∥w2 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

m	λ	k₂	∥Res∥w2
10	2	0.152925847300	4.3939e-05
20	2	0.153099050870	3.5332e-10
30	2	0.153099105535	7.3141e-13
40	2	0.153099102829	1.5483e-16

Table 16 shows the obtained values of k₂ and ∥Res∥w2 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

α	λ	k₂	∥Res∥w2
0.25	2	0.153092105254	2.7708e-04
0.50	2	0.153099102829	1.5483e-16
0.75	2	0.153068774773	6.7629e-08
1.00	2	0.153099103672	8.0842e-16
1.25	1	0.151696304817	8.8242e-05
1.50	1	0.137087413881	7.2232e-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.

t	n = 1	n = 2	n = 3	n = 4
0.5	0.28282173	0.03748381	0.00322210	0.00020535
1.0	0.52005163	0.14078358	0.02458744	0.00316426
1.5	0.69208735	0.28531419	0.07662991	0.01503527
2.0	0.80495682	0.44008751	0.16224506	0.04343598
2.5	0.87497019	0.57972330	0.27383162	0.09430307
3.0	0.91748107	0.69154754	0.39661914	0.16902827
3.5	0.94337703	0.77435581	0.51495766	0.26313174
4.0	0.95946829	0.83300459	0.61788070	0.36737260
6.0	0.98473111	0.93670460	0.84845320	0.71026581
8.0	0.99184533	0.96681821	0.92294713	0.85614939
10.	0.99487378	0.97930359	0.95265849	0.91367234

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(|a_i|) for n = 2, 3, and 4 to show the convergence of the method.

$Fig. 7 (a) Graphs of 1n!y(n)(t) $\frac{1}{n!}y^{(n)}(t)$ for n = 1, 2, 3, 4 (b) Graphs of log(|ai|) for n = 2, 3, 4 to show the convergence.$

Fig. 7

(a) Graphs of 1n!y(n)(t) for n = 1, 2, 3, 4 (b) Graphs of log(|a_i|) 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.

References

[1] M.R. Eslahchi, M. Dehghan, S. Amani, Chebyshev polynomials and best approximation of some classes of functions, J.Numer. Math., 23 (1) (2015) 41-50.10.1515/jnma-2015-0004Search in Google Scholar

[2] J.A. Rad, S. Kazem, M. Shaban, K. Parand, A. Yildirim, Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials, Math. Method. Appl. Sci., 37 (3) (2014) 329-342.10.1002/mma.2794Search in Google Scholar

[3] K. Parand, A. Taghavi, M. Shahini, Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane-Emden and unsteady gas equations, Acta Phys. Pol. B, 40(12) (2009) 1749-1763.Search in Google Scholar

[4] K. Parand, H. Yousefi, M. Delkhosh, A. Ghaderi, A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation, Eur. Phys. J. Plus, 131 (2016) 228.10.1140/epjp/i2016-16228-xSearch in Google Scholar

[5] M. Delkhosh, Introduction of Derivatives and Integrals of Fractional order and Its Applications, Appl. Math. Phys., 1 (4) (2013) 103-119.Search in Google Scholar

[6] S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37 (2013) 5498-5510.10.1016/j.apm.2012.10.026Search in Google Scholar

[7] K Parand, M Nikarya, JA Rad, Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocation method, Celest. Mech. Dyn. Astr., 116 (1) (2013) 97-107.10.1007/s10569-013-9477-8Search in Google Scholar

[8] J. Eshaghi, H. Adibi, S. Kazem, Solution of nonlinear weakly singular Volterra integral equations using the fractional-order Legendre functions and pseudospectral method, Math. Method. Appl. Sci., 39(12) (2016) 3411-3425.10.1002/mma.3788Search in Google Scholar

[9] K. Parand, A. Ghaderi, H. Yousefi, M. Delkhosh, A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions, Electronic J.Differential Equations 2016 (2016) 331:1-18.Search in Google Scholar

[10] A.H. Bhrawy, M.A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Appl. Math. Modell. 40 (2016) 832-845.10.1016/j.apm.2015.06.012Search in Google Scholar

[11] K. Parand, P. Mazaheri, H. Yousefi, M. Delkhosh, Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation, Eur. Phys. J. Plus, 132(2)(2017) 77.10.1140/epjp/i2017-11351-xSearch in Google Scholar

[12] K Parand, H Yousefi, M Delkhosh, A Numerical Approach to Solve Lane-Emden Type Equations by the Fractional Order of Rational Bernoulli Functions, Romanian J. Phys. 62 (2017) 104:1-24.Search in Google Scholar

[13] Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math.Inform. 26 (2008) 15-27.Search in Google Scholar

[14] K. Parand, M. Delkhosh, Accurate solution of the Thomas-Fermi equation using the fractional order of rational Chebyshev functions, J. Comput. Appl. Math., 317 (2017) 624-642.10.1016/j.cam.2016.11.035Search in Google Scholar

[15] E.B. Baker, The application of the Fermi-Thomas statistical model to the calculation of potential distribution in positiveions, Quart. Appl. Math., 36 (1930) 630-647.10.1103/PhysRev.36.630Search in Google Scholar

[16] M.A. Darani, M. Nasiri, A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations, Comp. Meth. Diff. Equ., 1 (2013) 96-107.Search in Google Scholar

[17] R.E. Kidder, Unsteady flow of gas through a semi-infinite porous medium, J. Appl. Mech. 24 (1957) 329-332.10.1115/1.4011542Search in Google Scholar

[18] A.M.Wazwaz, The modified decomposition method applied to unsteady flow of gas through a porous medium, Appl. Math.Comput. 118 (2001) 123-132.10.1016/S0096-3003(99)00209-XSearch in Google Scholar

[19] K. Parand, P. Mazaheri, M. Delkhosh, A. Ghaderi, New numerical solutions for solving Kidder equation by using the rational Jacobi functions, SeMA J., (2017) 10.1007/s40324-016-0103-z.Search in Google Scholar

[20] K. Parand, M. Nikarya, Solving the Unsteady Isothermal Gas Through a Micro-Nano Porous Medium via Bessel Function Collocation Method, J. Comput. Theor. NanoSci., 11 (2014) 1-6.10.1166/jctn.2014.3327Search in Google Scholar

[21] A. Taghavi, K. Parand, H. Fani, Lagrangian method for solving unsteady gas equation, World Acad. Sci. Eng. Tech., 3 (2009) 11-23.Search in Google Scholar

[22] A.R. Rezaei, K. Parand, and A. Pirkhedri, Numerical study on gas flow through a micro-nano porous media based on special functions, J. Comput. Theor. NanoSci., 8 (2010) 282-288.10.1166/jctn.2011.1690Search in Google Scholar

[23] J.A. Rad, S.M. Ghaderi, and K. Parand, Numerical and analytical solution of gas flow through a micro-nano porous media: A comparison, J. Comput. Theor. NanoSci., 8 (2011) 2033-2041.10.1166/jctn.2011.1922Search in Google Scholar

[24] S. Kazem, J. A. Rad, K. Parand, M. Shaban, and H. Saberi, The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method, Int. J. Comput. Math., 89(16) (2012) 2240-2258.10.1080/00207160.2012.704995Search in Google Scholar

[25] S. Abbasbandy, Numerical Study on Gas Flow through a Micro-Nano Porous Media, Acta Phy. Pol. A, 121 (2012) 581-585.10.12693/APhysPolA.121.581Search in Google Scholar

[26] F. Ahmad, A simple analytical solution for the steady flow of a third grade fluid in a porous half space, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 2848-2852.10.1016/j.cnsns.2008.09.029Search in Google Scholar

[27] S. Kazem, J.A. Rad, K. Parand, S. Abbasbandy, A New Method for Solving Steady Flow of a Third-Grade Fluid in a Porous Half Space Based on Radial Basis Functions, Z. Naturforsch. A, 66 (2011) 591-598.10.5560/zna.2011-0014Search in Google Scholar

[28] K. Parand, E. Hajizadeh, Solving Steady Flow of a Third-Grade Fluid in a Porous Half Space via Normal and Modified Rational Christov Functions Collocation Method, Z. Naturforsch. A, 69 (2014) 188-194.10.5560/zna.2014-0006Search in Google Scholar

[29] F. Baharifard, S.Kazem, K. Parand, Rational and Exponential Legendre Tau Method on Steady Flow of a Third Grade Fluid in a Porous Half Space, Int. J. Appl. Comput. Math, 10.1007/s40819-015-0096-9, (2015) 1-20.Search in Google Scholar

[30] S. Liao, An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int. J. Nonlin. Mech. 34 (1999) 759-778.10.1016/S0020-7462(98)00056-0Search in Google Scholar

[31] L.T. Yu and C.K. Chen, The solution of the blasius equation by the differential transformation method, Math. Comput. Model. 28 (1998) 101-111.10.1016/S0895-7177(98)00085-5Search in Google Scholar

[32] L. Wang, A new algorithm for solving classical Blasius equation, Appl. Math. Comput. 157 (2004) 1-9.10.1016/j.amc.2003.06.011Search in Google Scholar

[33] I. Hashim, Comments on “A new algorithm for solving classical Blasius equation” by L. Wang, Appl. Math. Comput. 176 (2006) 700-703.10.1016/j.amc.2005.10.016Search in Google Scholar

[34] R. Cortell, Numerical solutions of the classical Blasius flat-plate problem, Appl. Math. Comput. 170 (2005) 706-710.10.1016/j.amc.2004.12.037Search in Google Scholar

[35] A.M. Wazwaz, The modified decomposition method and Pade approximants for a boundary layer equation in unbounded domain, Appl. Math. Comput. 177 (2006) 737-744.10.1016/j.amc.2005.09.102Search in Google Scholar

[36] K. Parand, M. Nikarya, J.A. Rad, F. Baharifard, A new reliable numerical algorithm based on the first kind of Bessel functions to solve Prandtl-Blasius laminar viscous flow over a semi-infinite flat plate, Z. Naturforsch. A, 67(12) (2012) 665-673.10.5560/zna.2012-0065Search in Google Scholar

[37] K. Parand, M. Shahini, M. Dehghan, Solution of a laminar boundary layer flow via a numerical method, Commun. Nonlinear Sci. Numer. Simul., 15(2) (2010) 360-367.10.1016/j.cnsns.2009.04.007Search in Google Scholar

[38] J.P. Boyd, The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems, SIAM. Rev. 50(4) (2008) 791-804.10.1137/070681594Search in Google Scholar

[39] L. Howarth, On the calculation of steady flow in the boundary layer near the surface of a cylinder in a stream, Aero Res. Counc. Lond. RM, 164 (1935) 16-32.Search in Google Scholar

[40] V.L. Ginzburg, L.P. Pitaevskii, On the theory of superfluidity, Sov. Phys. JETP, 7 (5) (1958) 858-861.Search in Google Scholar

[41] A. Vilenkin, E.P.S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge: Cambridge University Press, (1994).Search in Google Scholar

[42] B. Boisseau, P. Forgacs, H. Giacomini, An analytical approximation scheme to two-point boundary value problems of ordinary differential equations, J. Phys. A: Math. Theor., 40(2007) F215-F221.10.1088/1751-8113/40/9/F01Search in Google Scholar

[43] P. Amore, F.M. Fernandez, Rational approximation to the solutions of two-point boundary value problems, Acta Poly., 51 (4)(2011) 9-13.10.14311/1392Search in Google Scholar

Received: 2016-10-20

Accepted: 2017-4-15

Published Online: 2017-5-20

Published in Print: 2017-9-26

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Abstract

1 Introduction

1.1 The Chebyshev functions

1.2 Basic definitions

Definition 1

Definition 2

Theorem 1

Proof

2 Generalized Fractional order of the Chebyshev Functions

2.1 The GFCFs definition

2.2 Approximation of functions

2.3 Convergence of the method

Theorem 2

Proof

Theorem 3

Proof

3 Examples

3.1 The unsteady isothermal flow of a gas

3.2 The third grade fluid in a porous half space

3.3 The Blasius equation

3.4 The field equation determining the vortex profile

4 Conclusion

Acknowledgement

References

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