mohamedmbd@yahoo.com 1 In this paper, we introduce a method based on replacement of the unknown function Department of Mathematics and Statistics, College of Science, by truncated series of the well-known shifted Chebyshev (of third-kind) expansion of Al Imam Mohammad Ibn Saud functions. We give an approximate formula for the integer derivative of this Islamic University (IMSIU), Riyadh, expansion. We state and prove some theorems on the convergence analysis. By Saudi Arabia Department of Mathematics, means of collocation points the introduced method converts the proposed problem Faculty of Science, Benha University, to solving a system of algebraic equations with shifted Chebyshev coeﬃcients. As an Benha, Egypt application for this eﬃcient numerical method, we employ it in solving the system of ordinary diﬀerential equation that describes the thin ﬁlm ﬂow and heat transfer with the eﬀect of thermal radiation, magnetic ﬁeld, and slip velocity. MSC: 41A04; 65N12; 76S02 Keywords: Liquid ﬁlm; Thermal radiation; Unsteady stretching sheet; Chebyshev collocation method; Convergence analysis 1 Introduction The thin ﬁlm ﬂuid ﬂow has become dependent on many theoretical and experimental studies in recent years due to its widespread applications in industry and engineering such as continuous casting, crystal growing, tinning of copper wires, chemical process- ing equipment, and wire and ﬁber coating. Many authors studied the thin ﬁlm ﬂuid ﬂow and heat transfer under diﬀerent cases; see, for example, [1–10]. In [11], the authors stud- ied the ﬂow of an incompressible liquid ﬁlm down a wavy incline and applied the Galerkin method with only one ansatz function to the Navier–Stokes equations. They derived a second-order weighted residual integral boundary layer equation to describe eddies in the troughs of the wavy bottom. Marin [12] considered a cylinder made of a microstretch ther- moelastic material for which one plane end is subjected to plane boundary data varying harmonically in time. On the lateral surface and other bases, we have zero body force and heat supply. Finally, Melvin and Herman [13] presented algorithmic matters of a computer code to solve linear two-point boundary-value problems. The proposed method used a superposition coupled with an orthonormalization procedure and a variable-step Runge– Kutta–Fehlberg integration scheme. Just and Stempien [14] studied the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. In [15], the authors proposed a modiﬁed and simple algorithm for frac- © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Khader Boundary Value Problems (2018) 2018:77 Page 2 of 11 tional modeling arising in unidirectional propagation of long wave in dispersive media by using the fractional homotopy analysis transform method. The proposed technique can be used to solve nonlinear problems without using the Adomian and He’s polynomials, which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. This modiﬁed method yields an analytical and approximate solution in terms of a rapidly convergent series with easily computable terms. Also, exploiting variational methods and the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator are investigated in [16]. Finally, the radiative eﬀects for some bidimensional thermoelectric problems are investigated in [17]. After these previous publications, a number of researchers have successfully applied several numerical methods in this ﬁeld. Among these numerical methods, the Chebyshev collocation method is a general approximate analytical method used to get the solutions for some of nonlinear diﬀerential equations. The Chebyshev collocation method has some advantages for handling this class of problems, in which the Chebyshev coeﬃcients for the solution can be calculated very easily by numerical programs. For this reason, this method is much faster than the other methods. Chebyshev polynomials are a well-known family of orthogonal polynomials on the interval [–1, 1] with many applications. They are widely used because of their good properties in the approximation of functions [18–20]. Some of these properties take a very concise form in the case of the Chebyshev polyno- mials, making them of leading importance among orthogonal polynomials. The Cheby- shev polynomials belong to an exclusive class of orthogonal polynomials, known as Jacobi α β polynomials, which correspond to weight functions of the form (1 – x) (1 + x) and which are solutions of Sturm–Liouville equations. The Chebyshev collocation method is used to solve many problems in many papers, for example, [18–20]. In this work, we use the properties of Chebyshev polynomials to derive an approximate formula of the integer derivative of the approximate solution and estimate an error upper bound of this formula. Due to a high accuracy of this method, it is inevitable to use it to solve numerically the resulting nonlinear system of ordinary diﬀerential equations, which describe a ﬂow and heat transfer of thin liquid ﬁlm aﬀected by the presence of thermal radiation and magnetic ﬁeld. The basic ordinary diﬀerential equations that describe the thin ﬁlm liquid layer ﬂow and heat transfer can be summarized as follows [21]: f + γ ﬀ – S f – f –(S + M)f =0, (1) 1 S (1 + R)θ + γ f θ –2f θ – 3θ + ηθ =0, (2) Pr 2 f (0) = 0, f (1) = S, θ(0) = 1, (3) f (1) = 0, f (0) = 1 + δf (0), θ (1) = 0, (4) where the prime denotes diﬀerentiation with respect to η, S is the unsteadiness parameter, M is the magnetic parameter, R is the radiation parameter, δ is the slip velocity parameter, γ is the dimensionless ﬁlm thickness, and Pr is the Prandtl number. Here, we must refer Khader Boundary Value Problems (2018) 2018:77 Page 3 of 11 that the previous system of equations is a generalization of the pioneering research of Wang [21]. Our problem can be reduced to the Wang problem by taking M = R = δ =0. 2 Procedure of solution 2.1 Approximate the solution and its convergence analysis The Chebyshev polynomials of the third kind of degree n are deﬁned as cos((n +0.5)ϕ) Ch3 (x)= , x = cos(ϕ), 0 ≤ ϕ ≤ π. cos(0.5ϕ) It well known that this family of polynomials is orthogonal on the interval [–1, 1] with respect to the weight function w(x)= (1 + x)/(1 – x): 1 ⎨ 0if r = s, w(x)Ch3 (x)Ch3 (x) dx = r s –1 π if r = s. These polynomials are widely used because of their good properties in the approximation of functions and can be determined by the following recurrence formula [22]: Ch3 (x)=2xCh3 (x)–Ch3 (x), Ch3 (x)=1, n+1 n n–1 0 Ch3 (x)=2x –1, n = 1,2,.... In this paper, we use these polynomials on the interval [0, 1], that is, the so-called shifted Chebyshev polynomials by introducing the change of variable x =2t –1, T (t)=Ch3 (2t – n n ¯ ¯ ¯ 1), where T (t)=1, T (t)=4t – 3. The most used and useful formula of T (t)is given 0 1 n analytically: (2n +1) (2n – k +1) k 2n–2k n–k T (t)= (–1) 2 t , n = 2,3,.... (k +1) (2n –2k +2) k=0 Every function (t) ∈ L [0, 1] can be expressed as a linear combination of {T (t)} as 2 i i=0 (t)= a T (t)(5) i i i=0 with the coeﬃcients a , i = 0,1,... , given by a = w ¯ (t)(t)T (t) dt, i = 0,1,..., (6) i i where w ¯ (t)= is the shifted weight function. 1–t By taking the ﬁrst m +1 terms of (5) we obtain the following approximation: (t)= a T (t). (7) m i i i=0 Theorem 1 Suppose that the function (t) satisﬁes the following conditions: Khader Boundary Value Problems (2018) 2018:77 Page 4 of 11 1. The second derivative (t) is a square-integrable function on [0, 1], i.e., (t) ∈ L [0, 1]; 2. The second derivative (t) is bounded on [0, 1], i.e., | (t)|≤ for some constant . Then the inﬁnite series (5) of the shifted Chebyshev expansion is uniformly convergent, and |a | < /i , i = 1,2,.... (8) Proof We will use the change of variables in the integration of formula (6)for a by using the transformation 2t –1 = cos(ϕ)to obtain a = y 0.5 1+ cos(ϕ) cos (i +0.5)ϕ cos(0.5ϕ) dϕ. Now, using twice integration by parts and simple calculations, we ﬁnd a = 0.5 1+ cos(ϕ) ρ (ϕ) dϕ, i i 4π where 1 sin(i –1)ϕ sin(i +1)ϕ 1 sin(i)ϕ sin(i +2)ϕ ρ (ϕ)= sin(ϕ) – + – , i i –1 i +1 i +1 i i +2 |c | = 0.5 1+ cos(ϕ) ρ (ϕ) dϕ i i 4π ≤ ρ (ϕ) dϕ 4π 1 1 1 1 1 1 ≤ + + + 4 i i –1 i +1 i +1 i i +2 (2i +2i –1) = . 2(i –1)i(i +1)(i +2) Then, for large i, we can get the required formula (8), and hence it uniformly converges to (t), which completes the proof. Theorem 2 Suppose that (t) ∈ C [0, 1]. Then the error E = (t)– (t) of approxi- m m mation of the function (t) by (t) using formula (7) can be estimated as follows: m+1 (t)– (t) ≤ ,(9) (m +1)! 2 (m+1) m where = max (t) and = max[t , t –t ], and C [0, 1] is the space of all m times t∈[0,1] 0 0 continuously diﬀerentiable functions in the interval [0, 1]. Proof We will consider the Taylor polynomial T (t)of degree m deﬁned by (t – t ) (k) T (t)= (t ) m 0 k! k=0 as an approximation of the function (t), where t ∈ [0, 1]. 0 Khader Boundary Value Problems (2018) 2018:77 Page 5 of 11 We know that the error of this approximation is deﬁned by m+1 (t – t ) (m+1) (t)– T (t) = (τ) , τ ∈ [t , t]. (10) m 0 (m +1)! Now, from these approximations and some facts we can estimate 2 2 (t)– (t) ≤ (t)– T (t) m m = w ¯ (t) (t)– T (t) dt 1 m+1 (t – t ) (m+1) = w ¯ (t) (τ) dt (m +1)! 2 1 2m+2 ≤ w ¯ (t)(t – t ) dt [(m + 1)!] 2 1 2m+2 = (t – t ) dt. [(m + 1)!] 1– t Then, denoting = max[t , t – t ], we have 0 0 2 2m+2 (t)– (t) ≤ dt [(m + 1)!] 1– t 2 2m+2 = . [(m + 1)!] 2 Taking the square roots of both sides, we can get the required formula (9), which completes the proof. In the following theorem,we give the main approximate formula for the integer derivative (n) D (t). Theorem 3 ([23, 24]) Suppose that we approximate the function (t) in the form (7). Then (n) D ( (t)) can be deﬁned as m i–n k 2i–2k (–1) 2 (2n + 1)(2i – k)!(i – k)! (n) (n) i–k–n (n) D (t) = a π t , π = . (11) m i i,k i,k (k!)(2i –2k + 1)!(i – k – n)! i=n k=0 Proof The proof of this theorem can be done directly with the help of formula (7)and some properties of the third-kind shifted Chebyshev polynomials. 2.2 Procedure solution In this section, we propose a method to solve numerically the system of ordinary diﬀeren- tial equations of the form (1)–(2). The unknown functions f (η)and θ(η)may be expanded by ﬁnite series of shifted Chebyshev polynomials as m m ¯ ¯ f (η)= f L (η), θ (η)= θ L (η). (12) m k k m k k k=0 k=0 Khader Boundary Value Problems (2018) 2018:77 Page 6 of 11 From Eqs. (1)–(2)and (12)and formula(11)wehave m i–3 (3) i–k–3 f π η i,k i=3 k=3 m m i–2 m i–2 (2) (2) i–k–2 i–k–2 + γ f L (η) f π η – η f π η i i i i i,k i,k i=0 i=2 k=2 i=2 k=2 m i–1 m i–1 (1) i–k–1 (1) i–k–1 – f π η –(S + M) f π η = 0, (13) i i i,k i,k i=1 k=1 i=1 k=1 m i–2 m m i–1 1+ R (2) i–k–2 (1) i–k–1 θ π η + γ f L (η) θ π η i i i i i,k i,k Pr i=2 k=2 i=0 i=1 k=1 m m i–1 (1) i–k–1 –2 θ L (η) f π η i i i i,k i=0 i=1 k=1 m m i–1 (1) i–k–1 – 3 θ L (η) + η θ π η = 0. (14) i i i i,k i=0 i=1 k=1 We now collocate Eqs. (13)–(14)at (m – n +1) points η , s = 0,1,..., m – n,as m i–3 (3) i–k–3 f π η i,k s i=3 k=3 m m i–2 m i–2 (2) i–k–2 (2) i–k–2 + γ f L (η ) f π η – η f π η i i s i s i i,k s i,k s i=0 i=2 k=2 i=2 k=2 m i–1 m i–1 (1) (1) i–k–1 i–k–1 – f π η –(S + M) f π η = 0, (15) i i i,k s i,k s i=1 k=1 i=1 k=1 m i–2 m m i–1 1+ R (2) (1) i–k–2 i–k–1 θ π η + γ f L (η ) θ π η i i i s i i,k s i,k s Pr i=2 k=2 i=0 i=1 k=1 m m i–1 (1) i–k–1 –2 θ L (η ) f π η i i s i i,k s i=0 i=1 k=1 m m i–1 (1) i–k–1 – 3 θ L (η ) + η θ π η = 0. (16) i i s s i i,k s i=0 i=1 k=1 For suitable collocation points, we use the roots of shifted Chebyshev polynomial L (η). Also, by substituting formula (12) into the boundary conditions (3)–(4)we m–n+1 can obtain the following six equations: m m m i i (–1) (2i +1)f =0, f = , (–1) (2i +1)θ =1, i i i i=0 i=0 i=0 (17) m m m m ¯ ¯ r f – δ r ¯ f =1, r ¯ f =0, r ¯ θ =0, i i i i i i i i i=0 i=0 i=0 i=0 Khader Boundary Value Problems (2018) 2018:77 Page 7 of 11 Table 1 Comparison of γ and –f (0) with δ = M = 0 using the previous work and the Chebyshev collocation method S Data of [25] Present results γ –f (0) γ –f (0) 1.4 0.674089 1.012781 0.6739267 1.0126853 1.6 0.331976 0.642412 0.309138 0.6423921 1.8 0.127013 0.3320138 0.1270089 0.3091378 Figure 1 The velocity distribution with diﬀerent values of S ¯ ¯ ¯ ¯ ¯ ¯ where r = L (0), r ¯ = L (1), r ¯ = L (0), r ¯ = L (1). i i i i i i i i Equations (15)–(16), together with ﬁve equations of the boundary conditions (17), give asystemof (2m + 2) algebraic equations, which can be solved, for the unknowns f , θ , i = i i 0,1,..., m, using the Newton iteration method. In our numerical study, we take m =5, that is, ﬁve terms of the truncated series solution (12)at η =1. 3 Results and discussion Firstly, to verify the accuracy of our numerical scheme, a comparison of the computed val- ues of –f (0) ismadebyNoorand Hashim [25]. This comparison is illustrated in Table 1. Note that there is a close agreement between our data and the previous published work of [25], which lends conﬁdence to the proposed numerical method. Also, in this section, we discuss and analyze the solution of the described problem via ﬁgures that result from using the Chebyshev collocation method. Since the problems that arise in industrial and technological situations are highly nonlinear, it may be diﬃcult to get analytical solutions to this type problems. One such problem, which occurs in the tinning of copper wires, polymer industry, and crystal growing, involves the thin ﬂow of a Newtonian ﬂuid past a stretching sheet. A relatively new method, called the shifted Cheby- shev collocation, is very beneﬁcial for developing solutions that are valid even for moderate to large values of parameters. This method depends on the series solution. One of the chief factors that inﬂuences the convergence of the solution series is the type of base functions used to express the solution. After applying the shifted Chebyshev collocation method, the solutions of the nonlinear system of ODEs (1)–(4) are presented through Figs. 1–7. Figure 1 highlights the eﬀect of S on the dimensionless velocity f (η). These plots re- veal the fact that increasing values of the parameter S results in enhancing the velocity distribution along the thin ﬁlm region, but the reverse is observed for the ﬁlm thickness. The dimensionless temperature for diﬀerent values of unsteadiness parameter S is dis- played in Fig. 2. Note that both the free surface temperature θ(γ ) and the dimensionless temperature increases with the increase in S. Khader Boundary Value Problems (2018) 2018:77 Page 8 of 11 Figure 2 The temperature distribution with diﬀerent values of S Figure 3 The velocity distribution with diﬀerent values of M Figure 4 The temperature distribution with diﬀerent values of M The inﬂuence of magnetic parameter M on the dimensionless velocity is presented in Fig. 3. Observed that, along the sheet, the dimensionless velocity increases with an in- crease in the M parameter, but the reverse trend is away from the sheet. Also, we further ﬁnd that the increasing value of the parameter M leads to decreasing the ﬁlm thickness. In Fig. 4, we depict the eﬀect of the same parameter M on the dimensionless tempera- ture. Note that, with the increasing values of magnetic parameter M, both the free surface temperature θ(γ) and thedimensionless temperatureincreaseinside theﬁlm region. Figure 5 represents the variation of velocity proﬁles for diﬀerent values of the slip veloc- ity parameter δ. From this plot it is evident that, along the sheet, the eﬀect of increasing values of the parameter δ is responsible for thinning the ﬁlm thickness and decreasing the dimensionless velocity. Figure 6 displays the temperature θ(η)proﬁles versus η for various values of the same parameter δ. It is elucidating that both the dimensionless temperature distribution and the free surface temperature θ(γ ) increase with an increase in parameter δ.Figure 7 depicts the eﬀect of the radiation parameter R on the temperature proﬁle. It is interesting to note that the dimensionless temperature distribution increases as the radia- tion parameter R increases. Likewise, some qualitative behaviors for the thin ﬁlm ﬂow and Khader Boundary Value Problems (2018) 2018:77 Page 9 of 11 Figure 5 The velocity distribution with diﬀerent values of δ Figure 6 The temperature distribution with diﬀerent values of δ Figure 7 The temperature distribution with diﬀerent values of R heat transfer characteristics are demonstrated in the same ﬁgure, in which the thickness for the ﬁlm is ﬁxed for changing the values of the radiation parameter R. 4Conclusionandremarks This numerical study highlights the lineaments of radiation parameter, velocity slip pa- rameter, magnetic number, and the unsteadiness parameter in the ﬂow and heat transfer for a Newtonian thin liquid ﬁlm. We use the Chebyshev collocation method to investi- gate the approximate solution of the resulting nonlinear system of ODEs for the proposed problem. Numerical evaluations of the closed-form results are performed, and graphical results are obtained to illustrate the details of the thin ﬁlm ﬂow and heat transfer charac- teristics and their dependence on some physical parameters. On the basis of the obtained results, we can see the following observations: (1) The eﬀect of increasing both the values of the unsteadiness parameter and the magnetic parameter increases both the dimensionless velocity and the dimensionless temperature throughout the ﬁlm layer. Khader Boundary Value Problems (2018) 2018:77 Page 10 of 11 (2) The temperature distribution can be aﬀected by changing the values of the velocity slip parameter and the thermal radiation parameter. (3) Because of the presence of slip velocity parameter, there may be a lower velocity distribution near the stretching sheet and also thinning the ﬁlm thickness. Acknowledgements The author is very grateful to the editor and referees for carefully reading the paper and for their comments and suggestions, which have improved the paper. Funding Not applicable. Availability of data and materials Not applicable. Competing interests The author declares that there is no conﬂict of interests regarding the publication of this paper. Authors’ contributions The paper by one author. Author read and approved the ﬁnal manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 31 December 2017 Accepted: 7 May 2018 References 1. Wang, C.Y.: Liquid ﬁlm on an unsteady stretching sheet. Q. Appl. Math. XLVIII, 601–610 (1990) 2. Usha, R., Sridharan, R.: The axisymmetric motion of a liquid ﬁlm on an unsteady stretching surface. ASME Fluids Eng. 117, 81–85 (1995) 3. Andersson, H.I., Aarseth, J.B., Dandapat, B.S.: Heat transfer in a liquid ﬁlm on an unsteady stretching surface. Int. J. Heat Mass Transf. 43, 69–74 (2000) 4. Dandapat, B.S., Santra, B., Andersson, H.I.: Thermocapillarity in a liquid ﬁlm on an unsteady stretching surface. Int. J. Heat Mass Transf. 46, 3009–3015 (2003) 5. Dandapat, B.S., Maity, S.: Flow of a thin liquid ﬁlm on an unsteady stretching sheet. Phys. Fluids 18, 102101 (2006) 6. Dandapat, B.S., Santra, B., Vajravelu, K.: The eﬀects of variable ﬂuid properties and thermocapillarity on the ﬂow of a thin ﬁlm on an unsteady stretching sheet. Int. J. Heat Mass Transf. 50, 991–996 (2007) 7. Liu, I.-C., Andersson, H.I.: Heat transfer in a liquid ﬁlm on an unsteady stretching sheet. Int. J. Therm. Sci. 47, 766–772 (2008) 8. Santra, B., Dandapat, B.S.: Unsteady thin-ﬁlm ﬂow over a heated stretching sheet. Int. J. Heat Mass Transf. 52, 1965–1970 (2009) 9. Noor, N.F.M., Abdulaziz, O., Hashim, I.: MHD ﬂow and heat transfer in a thin liquid ﬁlm on an unsteady stretching sheet by the HAM. Int. J. Numer. Methods Fluids 63, 357–373 (2009) 10. Conte, S.D.: The numerical solution of linear boundary value problems. SIAM Rev. 8(3), 309–321 (1966) 11. Häcker, T., Uecker, H.: An integral boundary layer equation for ﬁlm ﬂow over inclined wavy bottoms. Phys. Fluids 21(9), 1–10 (2009) 12. Marin, M.: Harmonic vibrations in thermoelasticity of microstretch materials. J. Vib. Acoust. 132(4), 1–6 (2010) 13. Scott, M.R., Watts, H.A.: Computational solution of linear two-point boundary value problems via orthonormalization. SIAM J. Numer. Anal. 14(1), 40–70 (1977) 14. Just, A., Stempien, Z.: Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation. Opusc. Math. 36(2), 239–252 (2016) 15. Kumar, S., Kumar, D., Singh, J.: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 5(4), 383–394 (2016) 16. Molica Bisci, J., Repovs, D.: Multiple solutions of p-biharmonic equations with Navier boundary conditions. Complex Var. Elliptic Equ. 59(2), 271–284 (2014) 17. Consiglieri, L.: Radiative eﬀects for some bidimensional thermoelectric problems. Adv. Nonlinear Anal. 5(4), 347–366 (2016) 18. Khader, M.M.: On the numerical solution and convergence study for system of non-linear fractional diﬀusion equations. Can. J. Phys. 92(12), 1658–1666 (2014) 19. Khader, M.M., Hendy, A.S.: A new Chebyshev computational matrix method: an application for solving the high order fractional diﬀerential equations. Walailak J. Sci. Technol. 11(4), 289–305 (2014) 20. Khader, M.M.: Shifted Chebyshev collocation method for the ﬂow and heat transfer due to a stretching sheet embedded in a porous medium with variable thickness, variable thermal conductivity and thermal radiation. Mediterr. J. Math. 13(4), 2319–2336 (2015) 21. Wang, C.: Analytic solutions for a liquid thin ﬁlm on an unsteady stretching surface. Heat Mass Transf. 42, 759–766 (2006) 22. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, New York (2003) Khader Boundary Value Problems (2018) 2018:77 Page 11 of 11 23. Sweilam, N.H., Nagy, A.M., El-Sayed, A.: On the numerical solution of space fractional order diﬀusion equation via shifted Chebyshev polynomials of the third kind. J. King Saud Univ., Sci. 28, 41–47 (2016) 24. Handan, C.Y.: Numerical solution of fractional Riccati diﬀerential equation via shifted Chebyshev polynomials of the third kind. J. Eng. Technol. Appl. Sci. 28, 1–11 (2017) 25. 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Boundary Value Problems – Springer Journals
Published: May 16, 2018
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