On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method

Ali Akgül; Dumitru Baleanu; Mustafa Inc; Fairouz Tchier

doi:10.1515/phys-2016-0077

Article Open Access

On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method

Ali Akgül , Dumitru Baleanu , Mustafa Inc and Fairouz Tchier

Published/Copyright: December 30, 2016

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From the journal Open Physics Volume 14 Issue 1

Abstract

In this manuscript we investigate electrodynamic flow. For several values of the intimate parameters we proved that the approximate solution depends on a reproducing kernel model. Obtained results prove that the reproducing kernel method (RKM) is very effective. We obtain good results without any transformation or discretization. Numerical experiments on test examples show that our proposed schemes are of high accuracy and strongly support the theoretical results.

Keywords: kernel functions; electrohydrodynamic flow; approximate solutions

PACS: 02.30.Hq; 02.30 Gp; 02.30 Tb

1 Introduction

The electrohydrodynamic flow of a fluid is governed by a non-linear ordinary differential equation. The degree of non-linearityis stated by a nondimensional variable α and the equation can be approached by two different linear equations for very small or very large values of α respectively. The electrohydrodynamic flow of a fluid has been researched by McKee [21]. The governing equations were turned to the following problem [20]:

(1)dγφdrγ+1rdβφdrβ+H21−φ1−αφ=0,0<r<1,

with the boundary conditions

(2)φ′(0)=φ(1)=0,

where φ(r) is the fluid speed, r is the radial range from the center of the cylindrical conduit, H is the Hartmann electric number, the parameter α is the size of the power of the nonlinearity and γ = 2, β = 1. Paullet [23]showed the existence and uniqueness of a solution to (1)–(2), and explored an error in the perturbative and numerical solutions given in [21] for large values of α.

Fractional calculus is a 300 years old and has been enhanced progressively up to now. The concept of differentiation to fractional order was described in 19th century by Rieman and Liouville. In several problems of physics, mechanics and engineering, fractional differential equations have been demonstrated to bea valuable tool in modeling many phenomena. However, most fractional order equations do not have analytic solutions. Therefore, there has been an important interest in developing numerical methods for the solutions of fractional-order differential equations [18]. Fractional differential equations, as an important research branch, have attracted much interest recently [9]. We recall that a general solution technique for fractional differential equations has not yet been constituted. Most of the solution methods in this area have been enhanced for significant sorts of problems. Consequently, a single standard method for problems related fractional calculus has not been found. Thus, determining credible and affirmative solution methods along with fast application techniques is beneficial and worthy of further examination [3]. For more details see [10–14, 24, 30].

The goal of this paper is to give approximate solutions to (1)–(2) for all values of the relevant variables using the RKM. Recently, much interest has been dedicated to the work of the RKM to research several scientific models [4]. The book [6] presents an overview for the RKM. Many problems such as population models and complex dynamics have been solved in the reproducing kernel spaces [7, 17, 26, 27]. For more details of this method see [1, 2, 5, 8, 15, 16, 19, 25, 28, 29].

This study is arranged as follows. Section 2 presents useful reproducing kernel functions. Solutions in W23[0,1] and a related linear operator are given in Section 3. This section demonstrates the fundamental results. The exact and approximate solutions of (1)–(2) are given in this section. Examples are shown in Section 4. Some conclusions are given in the final section.

Definition 1.1

A Hilbert space H which is defined on a nonempty set E is denominated a reproducing kernel Hilbert space if there exists a reproducing kernel function K:E×E→C.

2 Construction of reproducing kernel space

Definition 2.1

G21[0,1] is defined by:

G21[0,1]={φ∈AC[0,1]:φ′∈L2[0,1]}.φ,ψG21=φ(0)ψ(0)+∫01φ′(r)ψ′(r)dr,φ,ψ∈G21[0,1]

and

φG21=φ,φG21,φ∈G21[0,1],

are the inner product and the norm in G21[0,1].

Lemma 2.2

(See [6, page 17]. Reproducing kernel functionQθofG21[0,1]is obtained as:

Qθ(r)=1+r,0≤r≤θ≤1,1+θ,0≤θ<r≤1.

Definition 2.3

We denote the space W23[0,1] by

W23[0,1]=φ∈AC[0,1]:φ′,φ″∈AC[0,1],φ(3)∈L2[0,1],φ′(0)=0=φ(1).φ,ψW23=∑i=02φ(i)(0)ψ(i)(0)+∫01φ(3)(r)ψ(3)(r)dr,φ,ψ∈W23[0,1]

and

φW23=φ,φW23,φ∈W23[0,1],

are the inner product and the norm in W23[0,1].

Theorem 2.4

Reproducing kernel functionW23[0,1]is acquired as

(3)Bθ(r)=5624r2θ4−1624r2θ3−5312r2θ3+21104r2θ2+51872r3θ4−11872r3θ5−5936r3θ3+7104r3θ2−53744r4θ4+13744r4θ5+51872r4θ3+5624r4θ2−524r4θ+13744r5θ4−118720r5θ5−11872r5θ3−1624r5θ2−526r2−578r3+5156r4−526θ2−578θ3+5156θ4−1156θ5+313+1520r5,0≤r≤θ≤1,5624θ2r4−1624θ2r3−5312θ2r3+21104θ2r2+51872θ3r4−11872θ3r5−5936θ3r3+7104θ3r2−53744θ4r4+13744θ4r5+51872θ4r3+5624θ4r2−524θ4r+13744θ5r4−118720θ5r5−11872θ5r3−1624θ5r2−526θ2−578θ3+5156θ4−526r2−578r3+5156r4−1156r5+313+1520θ5,0≤θ<r≤1.

Proof

Let φ∈W23[0,1] and 0≤θ≤1. By using the definition 3 and integrating by parts, we acquire

φ,BθW23=∑i=02φ(i)(0)Bθ(i)(0)+∫01φ(3)(r)Bθ(3)(r)dr=φ(0)Bθ(0)+φ′(0)Bθ′(0)+φ″(0)Bθ″(0)+φ″(1)Bθ(3)(1)−φ″(0)Bθ(3)(0)−φ′(1)Bθ(4)(1)+φ′(0)Bθ(4)(0)+∫01φ′(r)Bθ(5)(r)dr.

After substituting the values of Bθ(0),Bθ′(0),Bθ″(0),Bθ(3)(0),Bθ(4)(0),Bθ(3)(1),Bθ(4)(1) into the above equation we get

φ,BθW23=φ(θ).

This completes the proof.

3 Representation of the solutions

The solution of (1)–(2) is acquired in the W23[0,1]. We describe the linear operator T:W23[0,1]→G21[0,1] by

(4)Tφ=dγφdrγ+1rdβφdrβ,φ∈W23[0,1].

The problem (1)–(2) alters to the problem

(5)Tφ=z(r,φ),φ(1)=0,φ′(0)=0,

where z(r,φ)=−H21−φ1−αφ.

Theorem 3.1

T is a bounded linear operator.

Proof

We will show TφG212≤KφW232. We get

TφG212=Tφ,TφG21=Tφ(0)2+∫01Tφ′(r)2dr,

by definition 2.1. By the reproducing property, we obtain

φ(r)=φ(⋅),Br(⋅)W23,

and

Tφ(r)=φ(⋅),TBr(⋅)W23.

Thus,

Tφ(r)≤φW23TBrW23=K1φW23,

where K₁ > 0. Therefore,

Tφ(0)2dr≤K12φW232.

Considering that

(Tφ)′(r)=φ(⋅),(TBr)′(⋅)W23,

then

(Tφ)′(r)≤φW23(TBr)′W23=K2φW23,

where K₂ > 0. Thus, we acquire

(Tφ)′(r)2≤K22φW232,

and

∫01(Tφ)′(r)2dr≤K22φW232.

Therefore, we get

TφG212≤Tφ(0)2+∫01(Tφ)′(r)2dr≤K12+K22φW232=KφW232,

where K=K12+K22>0. This completes the proof.

We denote ϱi(r)=Qri(r)andηi(r)=T∗ϱi(r). The orthonormal system η^i(r)i=1∞ofW23[0,1] is obtained from Gram-Schmidt orthogonalization process of {ηi(r)}i=1∞ and

(6)η^i(r)=∑k=1iσikηk(r),(σii>0,i=1,2,…).

Theorem 3.2

Letrii=1∞be dense in [0, 1] andηi(r)=TθBr(θ)θ=ri. Then, the sequenceηi(r)i=1∞is a complete system inW23[0,1].

Proof

We obtain

ηi(r)=(T∗ϱi)(r)=(T∗ϱi)(θ),Br(θ)=(ϱi)(θ),TθBr(θ)=TθBr(θ)θ=ri.

Therefore, ηi(r)∈W23[0,1]. For each fixed φ(r)∈W23[0,1],letφ(r),ηi(r)=0,(i=1,2,...), i.e.,

φ(r),(T∗ϱi)(r)=Tφ(⋅),ϱi(⋅)=(Tφ)(ri)=0.

Thus, (Tφ)(x) = 0 and φ ≡ 0. This completes the proof.

Theorem 3.3

If φ(r) is the exact solution of (5), then

(7)φ(r)=T−1z(r,φ)=∑i=1∞∑k=1iσikz(rk,φ(rk))η^i(r),

where{(ri)}i=1∞is dense in [0, 1].

Proof

We acquire

φ(r)=∑i=1∞φ(r),η^i(r)W23η^i(r)=∑i=1∞∑k=1iσikφ(r),ηk(r)W23η^i(r)=∑i=1∞∑k=1iσikφ(r),T∗ϱk(r)W23η^i(r)=∑i=1∞∑k=1iσikTφ(r),ϱk(r)G21η^i(r),

from (6). By uniqueness of the solution of (5), we acquire

φ(r)=∑i=1∞∑k=1iσikz(r,φ),QrkG21η^i(r)=∑i=1∞∑k=1iσikz(rk,φ(rk))η^i(r).

The approximate solution ϕ_n(r) is achieved as

(8)φn(r)=∑i=1n∑k=1iσikz(rk,φ(rk))η^i(r).

Theorem 3.4

Let φ be any solution of (1) inW23[0,1].Then

φn−φW23→0,n→∞.

Moreover the sequence φn−φW23 is monotonically decreasing in n.

Proof

We obtain

φn−φW23=∑i=n+1∞∑k=1iσikz(rk,φ(rk))η^i(r)W23,

by (7) and (8). Therefore

φn−φW23→0,n→∞.

Furthermore

φn−φW232=∑i=n+1∞∑k=1iσikz(rk,φ(rk))η^i(r)W232=∑i=n+1∞∑k=1iβikΨ^i2.

Obviously, φn−φW23 is monotonically decreasing in n.

4 Numerical experiments

We solve (1)–(2) numerically in this section. Tables 1–2 present the approximate solutions of the problem (1)–(2) for different values of γ,β and α. Figures 1–2 show the approximate solutions for several values of the intimate variables. The results depend on both H and α. We use MAPLE to solve the BVP. In figures 1–2 we give numerical solutions of the BVP for values of α = 0.5, 1.0 and H²= 0.5, 1.0, 2.0.

Figure 1

Graph of numerical results for α = 0.5, γ = 2, β = 1 and several values of H.

Figure 2

Graph of numerical results for γ= 2, β = 1, α = 1.0 and several values of H.

Table 1

Approximate solutions of (1)–(2) when α = 0.5.

r	γ = 1.9, β = 0.9	γ = 1.9, β = 1.0
0.0	0.381236310	0.3771173839
0.1	0.374950080	0.3713730420
0.2	0.359968470	0.3575225829
0.3	0.342105510	0.3401839144
0.4	0.315723980	0.3146921558
0.5	0.284128540	0.2835844577
0.6	0.244546771	0.2445250520
0.7	0.197105564	0.1974472101
0.8	0.141053509	0.1415539908
0.9	0.075613972	0.0760178037
1.0	2.9691 × 10^–11	–5.856 × 10^–10

Table 2

Approximate solutions of (1)–(2) when α = 1.0.

r	γ = 1.9, β = 0.9	γ = 1.9, β = 1.0
0.0	0.317659270	0.3348212962
0.1	0.311127694	0.3288793293
0.2	0.297816605	0.3162240251
0.3	0.285820932	0.3028615829
0.4	0.265485319	0.2812364427
0.5	0.245783017	0.2579063018
0.6	0.215406643	0.2247358955
0.7	0.176723589	0.1834843558
0.8	0.128333833	0.1327839543
0.9	0.069612052	0.0718612844
1.0	2.858 × 10^–11	–7.983 × 10^–9

Remark 4.1

A Spectral Method [22] and Homotopy analysis method [20] have been applied to the electrohydrodynamic flow. Our results are in good agreement with the results obtained by these methods. Therefore the RKM is a reliable method for electrohydrodynamic flow.

5 Conclusion

In this work, the reproducing kernel method (RKM) has been performed to acquire solutions for a nonlinear boundary value problems. We came across an important challenge in regard to attaining solutions however, we have shown that the solutions obtained are convergent. We obtained good results for different values of α, β and γ in (1)–(2). Reproducing kernel functions were found to be very useful to get these results and they prove that the RKM is very effective.

Competing interests: The authors declare that they have no competing interests.

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Received: 2016-8-23

Accepted: 2016-11-3

Published Online: 2016-12-30

Published in Print: 2016-1-1

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