In this paper we propose a new solution technique for numerical solution of fractional Benney equation, a fourth degree nonlinear fractional partial differential equation with broad range of applications. The method could be described as a hybrid technique which uses advantages of both wavelets and operational matrices. Having applied the present method, fractional Benney equation is converted into a matrix equation, which is easy to solve. To the best of our knowledge, the fractional Benney equation has not been solved with any numerical or analytical method in the literature. Solving this equation numerically and investigating the applicability of the wavelets on this problem is the main goal of this paper. Haar wavelets and Caputo type fractional derivatives are employed in the calculations. Computational results point the strength of Haar wavelets and feasibility of the present solution algorithm

Ali Akinlar, Mehmet

Bayram, Mustafa

Secer, Aydin

Arab Journals Platform

Appl. Math. Inf. Sci.8, No. 4, 1633-1637 (2014) 1633Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/080418Numerical Solution of Fractional Benney EquationMehmet Ali Akinlar, Aydin Secer and Mustafa Bayram∗Yildiz Technical University, Faculty of Chemical and Metallurgical Engineering, Department of Mathematical Engineering, 34210-Davutpasa-Istanbul, TurkeyReceived: 19 Jul. 2013, Revised: 21 Oct. 2013, Accepted: 22 Oct. 2013Published online: 1 Jul. 2014Abstract: In this paper we propose a new solution technique for numerical solution of fractional Benney equation, a fourth degreenonlinear fractional partial differential equation with broad range of applications. The method could be described as a hybrid techniquewhich uses advantages of both wavelets and operational matrices. Having applied the present method, fractional Benney equation isconverted into a matrix equation, which is easy to solve. To the best of our knowledge, the fractional Benney equation has not beensolved with any numerical or analytical method in the literature. Solving this equation numerically and investigating the applicabilityof the wavelets on this problem is the main goal of this paper. Haar wavelets and Caputo type fractional derivatives are employed in thecalculations. Computational results point the strength of Haar wavelets andfe sibility of the present solution algorithm.Keywords: Fractional Benney equation, Haar wavelets, numerical solution, Caputofrac ional derivative.1. IntroductionFractional calculus is one of the most popular calculustypes having a vast range of applications in manydifferent scientific and engineering disciplines. Order ofthe derivatives in the fractional calculus might be any reelnumber which separates the fractional calculus from theordinary calculus where the derivatives are allowed to beonly natural numbers. Fractional calculus is a highlyefficient and useful tool in the modeling of many sorts ofscientific phenomena including image processing,earthquake engineering, biomedical engineering,computational fluid mechanics and Physics. Fundamentalconcepts of fractional calculus and applications of it todifferent research areas can be seen in the references, [1],[2], [8], [10], [13] and [14] (amongst many others).In this paper we employ a hybrid technique whichuses advantages of both wavelets and fractional calculusfor the numerical solution of the fractional Benneyequation. The present numerical method could bedescribed as unification of wavelets with fractionalcalculus which obtains the solution in a highly simple andefficient manner. To the best of our knowledge, thefractional Benney partial differential equation has notbeen solved with the present method, solution of thisequation with a new method is the main goal of thispaper. Another important point to mention inhere is thatapplicability of the wavelet techniques to the fractionaldifferential equations have not been studied in detail inthe literature. Therefore, investigating the applications fwavelet based methods to the numerical solution of such ahigh order and nonlinear partial differential equation shallbe a significant contribution to the subject.Fractional calculus is a quite useful and significantlyimportant subject which usually involves challengingproblems, efficient solution algorithms, fast solvers andhigh storage requirements. Nevertheless, a generalsolution algorithm or technique which could be employedat almost every sorts of problems has not yet beenestablished. Each solution method has been developed forparticular types of problems. As a consequence, a singlestandard method for problems regarding fractionalcalculus has not emerged. Therefore, finding reliable andefficient solution techniques along with fastimplementation methods are useful and active researchareas. Some well known methods for the analytical andnumerical solutions of fractional differential and integralequations might be listed as power series method [11],differential transform method [3] and [9], homotopyanalysis method [4], variational iteration method [5] andhomotopy perturbation method [6]. Typical numericalmethods including collocation, finite differences andelements are among the most popular numericaltechniques and detailed information about most of which∗ Corresponding author e-mail:msbayram@yildiz.edu.trc© 2014 NSPNatural Sciences Publishing Cor.1634 M. A. Akinlar et. al. : Numerical Solution of Fractional...can be obtained from the aforementioned references andtheir own references.Deterministic Benney partial differential equation isdefined asut + (un)x + uxx + kuxxx + uxxxx = 0, (1)where k is a constant real number andn is a positiveinteger number. Consideringn = 3, we can write1 asut + 3u2ux + uxx + kuxxx + uxxxx = 0. (2)Benney equation has a broad range of applications atsolitons theory, computational fluid dynamics and Physics.Some applications regarding the Benney equation can beseen, for example, at [?].In this paper, we are particularly concerning with thenumerical solution of fractional Benney equation definedas a perturbation of the time derivative at (2). FractionalBenney equation that we consider in this paper isDα∗tu+ 3u2ux + uxx + kuxxx + uxxxx = 0. (3)This paper consists of four major parts. After thepresent introduction section, in the second and thirdsections we briefly overview some basic definitions andtheorems regarding wavelets and fractional calculus,respectively. In section 4 we solve the equation (3)numerically in an efficient manner. We complete thepaper with a discussion part where we summarize thepresent paper and mention some future extensions.2. WaveletsWavelets developed in the beginning of the last centuryis a highly useful and efficient method in signalprocessing and computational mathematics. Wavelettheory involves representing functions via simple buildingblocks at different scales and positions. The fundamentalidea behind wavelets is to analyze according to scale.Some of the most popular wavelet types are Haar,Daubechies, Shannon, Coiflet, spline, Battle-Lemaries,some orthogonal wavelets such as Legendre, Hermite andChebyshev.In the last decade wavelets have been employed in thesolutions of differential equations. The structure ofwavelets makes them quite appropriate for solving thepartial differential equations. Typical wavelet basedcomputational techniques employ some basis functionswhich are differentiable and do not vanish on the compactsupport. Some other related methods might be listed asfinite difference methods (FDM), finite volume methods(FVM), finite elements methods (FEM), spectral methods.FDM and FVM are an approximation to the differentialequation while other methods are an approximation to itssolution. Even though the spectral bases are infinitelydifferentiable, they have global support which could beconsidered as a negative point. Wavelets have thecapability of representing the solutions at different scales,which make them particularly useful for developingefficient solutions to the approximate solutions of partialdifferential equations and fractional differential equations.A functionψ ∈ L2(R) is said to be a wavelet if∫∞−∞ψ(t)dt = 0. (4)Notice that ifψ ∈ L2(R)⋂L1(R) that satisfiescψ = 2π∫∞−∞|ψ̂(w)|2|w|dw ≤ ∞ (5)whereψ̂(w) is the Fourier transform ofψ, then (4) holds.Lemma: Let ψ ∈ L2(R) have a compact support. The thefollowing two statements are equivalent.(i.)ψ is a wavelet.(ii.)The equality (5) is satisfied.In the literature there are so many different types ofwavelets. In this paper we are interested in an orthogonalwavelet having a compact support, namely, Haar waveletthat could be defined as follows:Defineψ(x) =1, 0 ≤ x < 1/2;−1, 1/2 ≤ x < 1;0, otherwise.Is is clear thatψ(x) satisfies (4) and ψ(x) has acompact support on[0, 1]. It is not hard to prove thatψ(x)also satisfies (5). ψ(x) defined in this way is known asHaar wavelet developed by A. Haar in 1909.Discrete versions of continuous wavelets might bedefined as follows. A functionψ ∈ L2(R) is a discretewavelet if the family of functionsψj,k(t) defined byψj,k(t) = 2j/2ψ(2jt− k) (6)where j and k are arbitrary integers, is an orthonormalbasis in the Hilbert spaceL2(R).Wavelet coefficients of a functionf ∈ L2(R), denotedbydj,k =∫Rf(t)ψj,k(t)dt. (7)The series∑j,k∈Zψj,k(t)∫Rf(t)ψj,k(t)dt (8)is called the wavelet series off . The expressionf =∑j,k∈Zψj,k(t)∫Rf(t)ψj,k(t)dt (9)c© 2014 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.8, No. 4, 1633-1637 (2014) /www.naturalspublishing.com/Journals.asp 1635is called the wavelet representation off .In the application section we shall illustrate anapproximation to au(x, t) ∈ L2([0, 1) × [0, 1)) functionwith Haar wavelets, therefore we leave the informationregarding the applications by Haar wavelets to functionsto the main section. For detailed and further properties ofwavelets and particularly Haar wavelets, interested readercan read, for instance, [7] in conjunction with the presentpaper.3. Fractional calculusIn this section, we briefly overview the somefundamental concepts of fractional calculus. As wementioned in the introductory part, orders of derivativesand integrals in fractional calculus might be at any realnumber. The most popular definitions of a fractionalderivative of a function are Riemann–Liouville,Grunwald–Letnikow, Caputo and Generalized functions.In this paper Caputo’s definition of fractionaldifferentiation will be employed.Definition. A real functionf(x), x > 0, is said to be inthe spaceCρ, ρ ∈ R if there exists a real number(p >ρ), such thatf(x) = xpf1(x) for a continuous functionf1(x) ∈ C[0,∞).Definition. The Riemann–Liouville fractional integraloperator of orderα ≥ 0 of a functionf ∈ Cρ, ρ ≥ −1, isdefined asJv0 f(x) =1Γ (v)∫ x0(x− t)v−1f(t)dt, v > 0,J0f(x) = f(x).It has the following properties:Forf ∈ Cρ, ρ ≥ −1, α, β ≥ 0 andγ > 1 :i.)JαJβf(x) = Jα+βf(x),ii.)JαJβf(x) = JβJαf(x),iii.)Jαxγ = Γ (γ+1)Γ (α+γ+1)xα+γ .Next we present the Caputo sense derivative.Definition. The fractional derivative off(x) in the Caputosense is defined asDv∗f(x) =1Γ (m− v)∫ x0(x− t)m−v−1f (m)(t)dt,for m− 1 < v < m, m ∈ N, x > 0, f ∈ Cm−1.Definition. For m to be the smallest integer that exceedsα, the Caputo time-fractional derivative operator of orderα > 0 is defined asDα∗tu(x, t) =1Γ (m−α)∫ t0(t− ξ)m−α−1 ∂mu(x,ξ)∂ξm dξ,m− 1 < α < m,∂mu(x,t)∂tm , α = m ∈ Nand the space-fractional derivative operator of orderβ > 0is defined asDα∗xu(x, t) =1Γ (m−β)∫ x0(x− θ)m−β−1 ∂mu(θ,t)∂θm dθ,m− 1 < β < m,∂mu(x,t)∂xm , β = m ∈ N.Lemma. If m − 1 < α < m, m ∈ N and f ∈ Cmρ , ρ ≥−1, thenDα∗Jαf(x) = f(x),JαDv∗f(x) = f(x)−∑m−1k=0 fk(0+)xkk! , x > 0.The Caputo fractional derivative is considered herebecause it allows traditional initial and boundaryconditions to be included in the formulation of theproblem. In the next section we solve (3).4. Main results: Numerical solution offractional Benney equationConsider an orthonormal matrixΨ(τ), approximatesolution ofn−times integral ofΨ(τ) is defined by∫ t0· · ·∫ t0Ψ(τ)(dτ)n ∼= ΥnΦΨ(t), n ∈ NwhereΨ(t) = [Φ0(t), · · · , Φm−1(t)]T. Here,Φi(t), i =0, · · ·m− 1 are orthogonal basis functions on the interval[0, 1), ΥΦ is matrix representation for integral ofΨ(t).Let u(x, t) ∈ L2([0, 1) × [0, 1)). u(x, t) can berepresented by Haar wavelet basis asu(x, t) =m−1∑i=0m−1∑j=0wijhi(x)hj(t), (10)where the coefficients are defined aswij :=∫ 10u(x, t)hi(x)dx∫ 10u(x, t)hj(t)dt,i, j = 0, 1, · · · ,m− 1.By taking step∆ = 1m of x, t, (10) can be written indiscrete form asU(x, t) = HT (x)WH(t),whereW = [wij ]m−1,m−1 is the coefficient matrix ofU(x, t) andhi(x) is the Haar wavelet basis consisting ofthe Haar wavelet matrixH. The correspondingoperational matrix is defined in terms of the well-knownand so useful block matrix asΥαH = HΥαBHT ,c© 2014 NSPNatural Sciences Publishing Cor.1636 M. A. Akinlar et. al. : Numerical Solution of Fractional...whereΥαB is the Block pulse function. The operationalmatrix for integration of the block pulse function is givenbyΥαBm :=1m1/2 1 · · · 10 · · · · · · · · ·· · · 0 1/2 10 · · · 0 1/2m×m.Equipped with these machinery, we can write theapproximations of each term in the (3) in the followingway.Dα∗tu =∂αu∂tα≈∂αU∂tα= HT (x)W∂α∂tαH(t)= HT (x)WΥ−αH H(t),ux(x, t) =∂u(x, t)∂x≈ HT (x)(Υ−1H )TWH(t),uxx(x, t) =∂2u(x, t)∂x2≈ HT (x)(Υ−2H )TWH(t),uxxx(x, t) =∂3u(x, t)∂x3≈ HT (x)(Υ−3H )TWH(t),uxxxx(x, t) =∂4u(x, t)∂x4≈ HT (x)(Υ−4H )TWH(t),(11)Letα = 23 . Then, using the identities given at (11), wecan express the fractional Benney equation (3) as follows:HTWΥ−2/3H H + 3HTWHHTWHHT (Υ−1H )TWH +HT (Υ−2H )TWH + kHT (Υ−3H )TWH +HT (Υ−4H )TWH = 0.Multiplying this equation withH from left andHT fromright, we obtainWΥ−2/3H + 3W2(Υ−1H )TW + (Υ−2H )TW +kH(Υ−3H )TW + (Υ−4H )TW = 0. (8)It is clear that (9) is a matrix equation which can besolved by any convenient matrix solver softwaresincluding Mathematica, Maple and Matlab. In order toobtain W, one can obtain experimental results fordifferent cases of the size of the operation matrixm (suchasm = 8, 16, 32, 64, 128), and different values oft andα.5. Conclusion and future plansIn this paper we presented a new solution technique fornumerical solution of fractional Benney equation. Themethod could be described as a hybrid technique whichuses advantages of both wavelets and fractional calculus.In fact, the present method could be described as awavelet Galerkin method that employs the techniques onmulti resolution analysis and use the advantage ofmultiscale methods, which make them more powerfulmethods than traditional numerical techniques. Haarwavelets and Caputo type fractional derivatives areemployed in the calculations. Computational results pointthe strength of Haar wavelets and the present solutionalgorithm. Employing some other orthogonal waveletsand investigating the applicability of the present methodto some other nonlinear fractional (stochastic) differentialand integral equations might be useful problem as a futureextensions of the present method.References[1] I. Podlubny, Fractional differential equations,Anintroduction to fractional derivatives, fractional differentialequations, some methods of their solutionand some of theirapplications.San Diego: Academic Press, (1999).[2] R. Hilfer (Ed.), Applications of fractional calculus inphysics,Academic pres, Orlando, (1999).[3] A. Secer, M. A. Akinlar and A. Cevikel,EfficientSolutions of Systems Of Fractional PDEs By DifferentialTransform Method,Advances in Difference EquationsDOI:10.1186/1687-1847-2012-188,2012, (2012).[4] S. J. Liao, The proposed homotopy analysis techniquesfor the solution of nonlinear problems.Ph.D. dissertation,Shanghai Jiao Tong University, Shanghai, (1992).[5] Z. Odibat and S. Momani,Application of variationaliteration method to nonlinear differential equations offractional order, Int. J. Nonlin. Sci. Numer. Simul.,7, 15-27 (2006).[6] J. Biazar and H. Ghazvini,Exact solutions for non-linearSchr̈odinger equations by He’s homotopy perturbationmethod,Physics Letters A,366, 79–84 (2007).[7] S. Mallat, A wavelet tour of signal processing,AcademicPress, (2008).[8] Y. Feng, X. Lin and S. Zhou and H. Li,Chaos ina fractional-order neutral differential system,Appliedmathematics and information sciences (AMIS),7, 233–238(2013).[9] M. Kurulay and M. Bayram, Approximate analyticalsolution for the fractional modified KdV by differentialtransform method,Communications in nonlinear scienceand numerical simulation,15, 1777–1782 (2010).[10] M. Bayram,Automatic analysis of the control of metabolicnetworks,Computers in biology and medicine,26, 401–408(1996).[11] E. Celik, E. Karaduman and M. Bayram,Numericalsolutions of chemical differential-algebraic equations,Applied mathematics and computation,139, 259–264(2003).[12] H. Konno and P.S. Lomdahl,The Birth-Death StochasticProcesses of Solitons in the 1D Benney Equation,J. Phys.Soc. Japan,69, 1629–1641 (2000).[13] B. P. Moghaddam and A. Aghili,A numerical methodfor solving Linear Non-homogenous Fractional OrdinaryDifferential Equation by using the operational matricesof the Bernstein polynomials,Applied mathematics andinformation sciences (AMIS),6, 441–445 (2012).c© 2014 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.8, No. 4, 1633-1637 (2014) /www.naturalspublishing.com/Journals.asp 1637[14] N. Guzel, I. Emiroglu, C. Guler, F. Tasci and M. Sivri,ASolution Proposal to the Interval Fractional TransportationProblem, Applied mathematics and information sciences(AMIS), 6, 567–571 (2012).MehmetAli Akinlar is an assistantprofessor of mathematicsat Yildiz TechnicalUniversity. He completed hisM.Sc. at McMaster universityand Ph.D. at UT, Arlington in2005 and 2009, respectively.Upon completion of his Ph.D.he worked as a post-doctoralresearch associate atdepartment of scientific computing at Florida StateUniversity under the supervision of Max Gunzburger. Hisresearch interests lie at applied and computationalmathematics as well as solutions of all kinds ofdifferential equations.Aydin Seceris an assistant professor ofmathematics at department ofmathematical engineering atYildiz Technical University,Istanbul, Turkey. He is inthe organizing committee ofthe International Conference:ICAAA2012. He is interestedin scientific computing,computer programming,numerical methods as well as differential and integralequations.Mustafa Bayram is a fullprofessor of mathematics atYildiz Technical University.Currently he acts as the deanof the faculty of chemical andmetallurgical engineering.His research interestsinclude applied mathematics,solutions of differentialequations, enzyme kineticsand mathematical biology. Heraised many masters and Ph.D. students and is the authorof many efficient research articles at prestigious researchjournals. He is the chief and founder editor of new trendsin mathematical sciences journal. He also serves as aneditor and reviewer for many outstanding mathematicsjournals.c© 2014 NSPNatural Sciences Publishing Cor.

Numerical Solution of Fractional Benney Equation

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