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

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1 Open Phys 16; 14: Research Article Open Access Ali Akgül* Dumitru Baleanu Mustafa Inc Fairouz Tchier On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method DOI 11515/phys Received Aug 16; accepted Nov 16 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 strongly support the theoretical results Keywords: kernel functions; electrohydrodynamic flow; approximate solutions PACS: Hq Gp Tb 1 Introduction The electrohydrodynamic flow of a fluid is governed by a non-linear ordinary differential equation The degree of non-linearity is stated by a nondimensional variable α 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 [1] The governing equations were *Corresponding Author: Ali Akgül: Siirt University Art Science Faculty Department of Mathematics TR-561 Siirt Turkey; aliakgul77@gmailcom Dumitru Baleanu: Çankaya University Faculty of Arts Sciences Department of Mathematics Computer Sciences 65 Ankara Turkey; Institute of Space Sciences Magurele Bucharest Romania Mustafa Inc: Fırat University Science Faculty Department of Mathematics 119 Elazığ Turkey Fairouz Tchier: Mathematics Department College of Science (Malaz) King Saud University PO Box 45 Riyadh Kingdom of Saudi Arabia turned to the following problem []: d γ φ dr γ + 1 r d β φ dr β + H with the boundary conditions ( ) φ 1 < r < 1 (1) 1 αφ φ () φ(1) () 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 γ β 1 Paullet [] showed the existence uniqueness of a solution to (1) () explored an error in the perturbative numerical solutions given in [1] for large values of α Fractional calculus is a years old has been enhanced progressively up to now The concept of differentiation to fractional order was described in 19th century by Rieman Liouville In several problems of physics mechanics engineering fractional differential equations have been demonstrated to be a 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 stard method for problems related fractional calculus has not been found Thus determining credible affirmative solution methods along with fast application techniques is beneficial worthy of further examination [] For more details see [ ] The goal of this paper is to give approximate solutions to (1) () 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 16 A Akgül et al published by De Gruyter Open This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs License Download Date 1/16/19 4:6 AM

2 686 A Akgül et al RKM Many problems such as population models complex dynamics have been solved in the reproducing kernel spaces [ ] For more details of this method see [ ] This study is arranged as follows Section presents useful reproducing kernel functions Solutions in W [ 1] a related linear operator are given in Section This section demonstrates the fundamental results The exact approximate solutions of (1) () are given in this section Examples are shown in Section 4 Some conclusions are given in the final section Definition 11 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 Construction of reproducing kernel space Definition 1 G 1 [ 1] is defined by: G 1 [ 1] {φ AC[ 1] : φ L [ 1]} φ ψ G 1 φ()ψ() + φ G 1 φ (r)ψ (r)dr φ ψ G 1 [ 1] φ φ G 1 φ G 1 [ 1] are the inner product the norm in G 1 [ 1] Lemma (See [6 page 17]) Reproducing kernel function Q of G 1 [ 1] is obtained as: { 1 + r r 1 Q (r) 1 + < r 1 Definition We denote the space W [ 1] by W[ 1] { φ AC[ 1] : φ φ AC[ 1] } φ () L [ 1] φ () φ(1) φ ψ W i φ (i) ()ψ (i) () + φ ψ W [ 1] φ W φ () (r)ψ () (r)dr φ φ W φ W [ 1] are the inner product the norm in W [ 1] Theorem 4 Reproducing kernel function B of W [ 1] is acquired as B (r) 5 64 r r 5 1 r r 187 r r r r r r r4 64 r4 5 4 r r r r r5 5 6 r 5 78 r 156 r r5 r r r 5 1 r r 187 r r r r r r r 64 4 r r r r r r r 5 78 r 156 r r < r 1 Proof Let φ W [ 1] 1 By using the definition integrating by parts we acquire φ B W i 1 φ (i) ()B (i) () + φ () (r)b () (r)dr φ()b () + φ ()B () + φ ()B () + φ (1)B () (1) φ ()B () () φ (1)B (4) (1) + φ ()B (4) () + 1 φ (r)b (5) (r)dr After substituting the values of B () B () B () B() () B (4) () B() (1) B(4) (1) into the above equation we get This completes the proof φ B W φ() Representation of the solutions The solution of (1) () is acquired in the W [ 1] We describe the linear operator T : W [ 1] G1 [ 1] by Tφ dγ φ dr γ + 1 r The problem (1) () alters to the problem { Tφ z(r φ) where z(r φ) H ( 1 () d β φ dr β φ W [ 1] (4) φ(1) φ () ) φ 1 αφ (5) Download Date 1/16/19 4:6 AM

3 Electrohydrodynamic flow with fractional differential equations 687 Theorem 1 T is a bounded linear operator Proof We will show Tφ G K 1 φ W We get Tφ G 1 Tφ Tφ G 1 [ Tφ() ] 1 + [ Tφ (r) ] dr by definition 1 By the reproducing property we obtain Thus φ(r) φ( ) B r ( ) W Tφ(r) φ( ) TB r ( ) W Tφ(r) φ W TB r W K 1 φ W where K 1 > Therefore Considering that then [ ] (Tφ) () dr K 1 φ W (Tφ) (r) φ( ) (TB r ) ( ) W (Tφ) (r) φ W (TB r ) W K φ W where K > Thus we acquire Therefore we get Tφ G 1 [ (Tφ) (r) ] K φ W [ (Tφ) (r) ] dr K φ W [ (Tφ) () ] 1 + [ (Tφ) (r) ] dr ( ) K1 + K φ W K φ W where K K 1 + K > This completes the proof We denote ϱ i (r) Q ri (r) η i (r) T * ϱ i (r) The orthonormal system { η i (r) } of i1 W [ 1] is obtained from Gram- Schmidt orthogonalization process of {η i (r)} i1 Theorem Let {r i } i1 be dense in [ 1] η i(r) T B r () ri Then the sequence { η i (r) } is a complete i1 system in W [ 1] Proof We obtain η i (r) (T * ϱ i )(r) T B r () ri (T * ϱ i )() B r () (ϱ i )() T B r () Therefore η i (r) W [ 1] For each fixed φ(r) W [ 1] let φ(r) η i (r) (i 1 ) ie φ(r) (T * ϱ i )(r) Tφ( ) ϱ i ( ) (Tφ)(r i ) Thus (Tφ)(x) φ This completes the proof Theorem If φ(r) is the exact solution of (5) then φ(r) T 1 z(r φ) i1 σ ik z(r k φ(r k )) η i (r) (7) k1 where {(r i )} i1 is dense in [ 1] Proof We acquire φ(r) i1 i1 i1 i1 φ(r) ηi (r) η W i (r) σ ik φ(r) ηk (r) η W i (r) k1 k1 σ ik φ(r) T * ϱ k (r) η i (r) W σ ik Tφ(r) ϱk (r) η G 1 i (r) k1 from (6) By uniqueness of the solution of (5) we acquire φ(r) i1 i1 σ ik z(r φ) Qrk k1 G 1 σ ik z(r k φ(r k )) η i (r) k1 η i (r) The approximate solution φ n (r) is achieved as φ n (r) n i1 σ ik z(r k φ(r k )) η i (r) (8) k1 η i (r) σ ik η k (r) (σ ii > i 1 ) (6) k1 Theorem 4 Let φ be any solution of (1) in W [ 1] Then φ n φ W n Download Date 1/16/19 4:6 AM

4 688 A Akgül et al is monotonically de- Moreover the sequence φ n φ W creasing in n Proof We obtain φ n φ W by (7) (8) Therefore in+1 k1 σ ik z(r k φ(r k )) η i (r) φ n φ W n W Furthermore φ n φ W in+1 k1 σ ik z(r k φ(r k )) η i (r) ( ) β ik Ψi in+1 k1 Obviously φ n φ W is monotonically decreasing in n 4 Numerical experiments We solve (1) () numerically in this section Tables 1 present the approximate solutions of the problem (1) () for different values of γ β α Figures 1 show the approximate solutions for several values of the intimate variables The results depend on both H α We use MAPLE to solve the BVP In figures 1 we give numerical solutions of the BVP for values of α 5 1 H 5 1 Remark 41 A Spectral Method [] Homotopy analysis method [] 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 W Figure : Graph of numerical results for γ β 1 α 1 several values of H Table 1: Approximate solutions of (1) () when α 5 r γ 19 β 9 γ 19 β Table : Approximate solutions of (1) () when α 1 r γ 19 β 9 γ 19 β Conclusion Figure 1: Graph of numerical results for α 5 γ β 1 several values of H 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 Download Date 1/16/19 4:6 AM

5 Electrohydrodynamic flow with fractional differential equations 689 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 γ in (1) () Reproducing kernel functions were found to be very useful to get these results they prove that the RKM is very effective Competing interests: The authors declare that they have no competing interests References [1] Abbasby S B Azarnavid M S Alhuthali A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems J Comput Appl Math [] Akgül A New reproducing kernel functions Math Probl Eng pages Art ID [] Akgül A Inc M Karatas E Baleanu D Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique Adv Difference Equ 15 [4] Aronszajn N Trans Amer Math Soc [5] Bushnaq S Maayah B Momani S Alsaedi A A reproducing kernel Hilbert space method for solving systems of fractional integro differential Equations Abstr Appl Anal pages Art ID [6] Cui M Lin Y Nova Science Publishers Inc New York 9 [7] Du J Cui M Solving the forced Duflng equation with integral boundary conditions in the reproducing kernel space Int J Comput Math 1 87(9) 88 1 [8] Geng F Z Qian S P Solving singularly perturbed multi pantograph delay equations based on the reproducing kernel method Abstr Appl Anal pages Art ID [9] Geng F Cui M A reproducing kernel method for solving nonlocal fractional boundary value problems Appl Math Lett 1 5(5) [1] Hashemi M S Constructing a new geometric numerical integration method to the nonlinear heat transfer equations Commun Nonlinear Sci Numer Simul 15 (1-) [11] Hashemi M S Abbasby S A geometric approach for solving troesch s problem Bulletin of the Malaysian Mathematical Sciences Society [1] Hashemi M S Baleanu D Numerical approximation of higherorder time fractional telegraph equation by using a combination of a geometric approach method of line J Comput Phys [1] Hashemi M S Baleanu D Haghighi M P A lie group approach to solve the fractional poisson equation Romanian Journal of Physics [14] Hashemi M S Baleanu D Haghighi M P Darvishi E Solving the timefractional diffusion equation using a lie group integrator Therm Sci [15] Inc M Akgül A Kilicman A Numerical solutions of the second order One-dimensional telegraph equation based on reproducing kernel Hilbert space method Abstr Appl Anal pages Art ID [16] Javadi S Babolian E Moradi E New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations Commun Numer Anal pages Art ID [17] Jiang W Cui M Solving nonlinear singular pseudoparabolic equations with nonlocal mixed conditions in the reproducing kernel space Int J Comput Math 1 87(15) 4-44 [18] Jiang W Tian T Numerical solution of nonlinear Volterra integro differential equations of fractional order by the reproducing kernel method Appl Math Model 15 9(16) [19] Komashynska I Smadi M A Iterative reproducing kernel method for solving second-order integrodifferential equations of Fredholm type J Appl Math pages Art ID [] Mastroberardino A Homotopy analysis method applied to electrohydrodynamic flow Commun Nonlinear Sci Numer Simul 11 16(7) 7-76 [1] McKee S Watson R Cuminato J A Caldwell J Chen M S Calculation of electrohydrodynamic flow in a circular cylindrical conduit Z Angew Math Mech (6) [] Moghtadaei M Nik H S Abbasby S A spectral method for the electrohydrodynamic flow in a circular cylindrical conduit Chin Ann Math Ser B 15 6() 7- [] Paullet J E On the solutions of electrohydrodynamic flow in a circular cylindrical Conduit ZAMM Z Angew Math Mech (5) 57-6 [4] Sakar M G Iterative reproducing kernel Hilbert spaces method for Riccati differential equations J Comput Appl Math [5] Shawagfeh N Arqub O A Momani S Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method J Comput Anal Appl 14 16(4) [6] Kumar B Vangeepuram S Reproducing Kernel Space Embeddings Metrics on Probability Measures ProQuest LLC Ann Arbor MI Thesis (PhD) University of California San Diego 1 [7] Wu B Li X Application of reproducing kernel method to third order three-point boundary value problems Appl Math Comput 1 17(7) [8] Xu L Luo B Tang Y Ma X An eflcient multiple kernel learning in reproducing kernel Hilbert spaces (RKHS) Int J Wavelets Multiresolut Inf Process 1(): [9] Zayed A I Solution of the energy concentration problem in reproducingkernel Hilbert space SIAM J Appl Math 15 75(1) 1 7 [] Zhang R Lin Y A novel method for nonlinear boundary value problems J Comput Appl Math Download Date 1/16/19 4:6 AM