Approximate Analytical Solution Of One-Dimensional Beam .

HindawiJournal of Applied MathematicsVolume 2020, Article ID 7627385, 13 pageshttps://doi.org/10.1155/2020/7627385Research ArticleApproximate Analytical Solution of One-Dimensional BeamEquations by Using Time-Fractional Reduced DifferentialTransform MethodDessalegn Mekonnen Yadeta , Ademe Kebede Gizaw , and Yesuf Obsie MussaDepartment of Mathematics, College of Natural Sciences, Jimma University, Jimma, EthiopiaCorrespondence should be addressed to Ademe Kebede Gizaw; kebedeademe2020@gmail.comReceived 23 July 2020; Revised 4 November 2020; Accepted 4 December 2020; Published 22 December 2020Academic Editor: Ying HuCopyright 2020 Dessalegn Mekonnen Yadeta et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.In this paper, a recent and reliable method, named the fractional reduced differential transform method (FRDTM) is employed tosolve one-dimensional time-fractional Beam equation subject to the appropriate initial conditions. This method provides thesolutions very accurately and efficiently in convergent series form with easily computable coefficients. The efficacy and accuracyof this method are verified by means of three illustrative examples which indicate that the present method is very effective,simple, and easy to implement. Finally, it is observed that the FRDTM is the prevailing and convergent method for the solutionsof linear and nonlinear fractional-order partial differential equations.1. IntroductionIn last few decades, fractional calculus has been attractedmuch attention due to its enormous numbers of applicationsin almost all disciplines of applied sciences and engineering.The fractional calculus became an aspirant to find out thesolution of complex systems that exist in numerous fields ofsciences (for detail see [1–4]). This branch of mathematicalanalysis, extensively investigated in the recent years, hasemerged as an effective and powerful tool for the mathematical modeling of several engineering and scientific phenomena. One of the key factors for the popularity of the subjectis the nonlocal nature of fractional-order operators. In thefield of mathematical modeling, having partial derivativesof fractional order naturally seems to be dealing with the generality of the current traditional models [5]. In the field ofmodern science and engineering, the fourth-order parabolictime-fractional beam equation plays an important role inmodern science and engineering. For example, airplanewings and transverse vibrations of sustained tensile beamscan be modeled as plates with initial/different boundary supports which are successfully governed by superdiffusionfourth-order differential equations [6].The fourth-order parabolic PDEs are of great importance. These PDEs describe various physical phenomena,including deformation of beams, viscoelastic and inelasticflows, transverse vibrations of a homogeneous beam, platedeflection theory, engineering, and applied sciences [7–13].In this work, we concentrate our discussion on the following classes of time-fractional nonlinear Beam equation(fourth-order time-fractional nonlinear parabolic PDEs) ofthe form [14]. 2α u 4 u Ax,t f ðx, t, u, ut , ux , uxx , uxxx Þ, ðx, t Þ Ω,ðÞ t 2α x4ð1Þwith initial conditionsuðx, 0Þ f ðxÞut ðx, 0Þ gðxÞ, a x b,ð2Þ

2finding the solution of this equation has been the subject ofmany investigators in the recent years.Before the nineteenth century, there was no scheme available for the analytical solutions of the fractional differentialequations. In the beginning of the twentieth century,researchers started to pay attention to find the robust andstable analytical approaches for the exact (approximate)solution of the fractional differential equations [15]. Subsequently, several schemes such as the Adomian decomposition methods [16–18], differential transform method [19–21], Homotopy perturbation method [22–25], Local fractional variation iteration method [26], Variation iterationmethod [27, 28], and Shifted Chebyshev polynomials basedmethod [29] have been developed for the analytical solutionsof fractional differential equations.Most of these methods sometimes require complex andhuge calculation in order to obtain approximate solutions.To overcome such difficulties and drawbacks, an alternativemethod, the so-called fractional reduced differential transform method (FRDTM), has been developed by Keskin andOturanc [30]. FRDTM plays a vital role among all the listedmethods because it takes small size computation, easy toimplement as compared to other techniques [31].The basic motivation of this paper is to propose FRDTMto find an approximate analytical solution of the timefractional Beam equation (the governing equation) given in(1). Using this method, it is possible to find both exact andapproximate solutions in a rapidly convergent power seriesform.It is worth mentioning that the FRDTM is applied without any linearization or discretization or restrictive assumptions. FRDTM is a very reliable, efficient, and effectivepowerful computational technique for solving physical problems [32–37].The rest of this paper is organized as follows: in Section 2,we give some fundamental definitions and lemmas associatedwith fractional calculus. In Section 3, some basic definitionsand properties related to one-dimensional fractional reduceddifferential transformation in the Caputo sense are presented;some lemmas are proved. In Section 4, we present the formulation of the method. Section 6 is devoted to apply themethod to solve linear and nonlinear time-fractional beamequation in one dimension and present graphs to show theeffectiveness, validity, and performance of the FRDTM forsome values of α. Finally, the conclusion is presented in Section 6.2. Fractional CalculusIn this section, some basic definitions and lemmas ofFRDTM associated with fractional calculus are presented.Some of these definitions are due to Riemann Liouville andCaputo sense; for details, see [34, 36, 37].Definition 1. The Gamma function. The Gamma function ΓðzÞ is simply a generalization of the factorial real arguments.The Gamma function can be defined as [3]Journal of Applied MathematicsΓðz Þ ð e t t z 1 dt, z 0:ð3Þ0Definition 2. Let μ ℝ, m ℕ. A function f : ℝ ℝbelongs to the space C μ if there exists a real number k μsuch thatf ðt Þ t p gðt Þ, t 0,ð4ÞðmÞ Cμ :gðt Þ C½0, Þ and f ℂmμ if fð5ÞwhereDefinition 3. Let J αx be Riemann-Liouville fractional integraloperator of order α 0 and let f ℂμ , μ 1, thenJ t α f ðt Þ 1ΓðαÞðtðt τÞα 1 f ðτÞdτ,ð6Þ00J t f ðt Þ f ðt Þ, t 0:Definition 4. If m 1 α m, m ℕ, t 0, then the Caputo’sfractional derivative of f Cμ is defined asDx α f ðxÞ J x m α Dx m f ðxÞ 1Γðm αÞðxðx t Þm α 1 f ðmÞ ðt Þdt:0ð7ÞThe fundamental properties of the Caputo fractionalderivative are given in the following lemma.Lemma 5. If m 1 α m, m ℕ and f ðxÞ C mμ , μ 1,thenβα βðiÞDαt Dt f ðt Þ Dtβf ðt Þ Dt Dαt f ðt ÞðiiÞDαt J αt f ðt Þ f ðt Þ, t 0ðiiiÞJ αt Dαt f ðt Þ m 1f ðt Þ fk 0ðkÞð8Þtkð0 Þ , t 0:k! 3. Fractional Reduced Differential TransformMethod (FRDTM)Definition 6 (see [34, 36, 37]). If uðx, tÞ is analytic and continuously differentiable with respect to space variable x and timevariable t in the domain of interest, then time-fractionalreduced differential transform (or the spectrum function) is"#1 αkU k ðx Þ uðx, t ÞΓðkα 1Þ t αk,ð9Þt t 0where α is a parameter which describes the order of timefractional derivative in the Caputo sense, and k is an integerðk 0Þ.

Journal of Applied Mathematics3Remark 7. In this study, the lowercase uðx, tÞ represents theoriginal function, while the uppercase U k ðxÞ stands for thetransformed function.Definition 8 (see [34, 36, 37]). The fractional reduced differential inverse transform of U k ðxÞ is defined as uðx, t Þ U k ðxÞðt t o Þ :ð10Þkαk 0Substituting Eq. (9) into Eq. (10), we obtain"# 1 kαuðx, t Þuðx, t Þ ðt t o Þkα ,kαΓkα 1ðÞ tk 0ð11Þt t owhich in practical application can be approximated by afinite seriesnun ðx, t Þ U k ðxÞðt t o Þkα ,Lemma 10. If f ðx, tÞ xm cos ðηx θtÞ , then the fractionalreduced differential transform of f isF k ðxÞ xm ðθk /k!Þ cos ðηx ðπk/2ÞÞ , where η and θ areconstants.Proof. From Definition 6 and FRDTM properties, we have."#1 kαf ðx, t ÞF k ðx Þ Γðkα 1Þ t kαt t 0"#1 kα mF k ðx Þ x cos ðηx θt ÞΓðkα 1Þ t kαt t 0"#kα1 F k ðx Þ x mcos ðηx θt ÞΓðkα 1Þ t kα t t 0 θkπkmθ, for k 0, 1, 2, 3, cos ηx F k ðx Þ x2k!ð12Þð16Þk 0where n is the order of this approximate solution. Therefore,the exact solution can be obtained as uðx, t Þ lim un ðx, t Þ U k ðxÞðt t o Þkα :n ð13Þk 0If t 0 0, then Eq. (13) reduces to the form uðx, t Þ lim un ðx, t Þ U k ðxÞðt Þkα :n !kΓðkα 1Þ 4 Ar ðxÞ 4 U k r ðxÞ F ðU k ðxÞÞ ,U k 2 ðxÞ Γðkα 2α 1Þ xr 0ð14Þand using FRDTM to the initial conditions (2), we getk 0Lemma 9. If f ðx, tÞ xm sin ðηx θtÞ , then the fractionalreduced differential transform of f is F k ðxÞ ðθk /k!Þxm sinðηx ðπk/2ÞÞ , where η and θ are constants.Proof. From Definition 6 and FRDTM properties, we have"#1 kαF k ðx Þ f ðx, t ÞΓðkα 1Þ t kαt t 0"#1 kα mF k ðx Þ x sin ðηx θt ÞΓðkα 1Þ t kαt t 0"#kα1 F k ðx Þ x msin ðηx θt ÞΓðkα 1Þ t kα Applying properties of FRDTM to Eq. (1), we obtain thefollowing recurrence relation:ð17ÞSome of the basic properties of one-dimensional fractional reduced differential transform function that are constructed based on Definitions 6 and 8 are given below.t t 0 θ mπk, f ork 0, 1, 2, 3, F k ðxÞ x sin ηx 2k!k4. Solution of the Problem by FRDTMU 0 ðxÞ f ðxÞ, U 1 ðxÞ gðxÞ:ð18ÞUsing Eqs. (17) and (18) and k 0, 1, 2, 3, and by iterativecalculation, the following results are obtained:!1 4 A0 ðxÞ 4 U 0 ðxÞ F ðU 0 ðxÞÞ ,U 2 ðx Þ Γð2α 1Þ x"#!Γðα 1Þ 4 4U 3 ðxÞ A0 ðxÞ 4 U 1 ðxÞ A1 ðxÞ 4 U 0 ðxÞ F ðU 1 ðxÞÞ , x xΓð3α 1Þ2 233 4 4A0 ðxÞ 4 U 2 ðxÞ A1 ðxÞ 4 U 1 ðxÞ 7667Γð2α 1Þ 6 6 x x7 F U x 7,U 4 ðxÞ ð 2 ð Þ Þ76 67Γð4α 1Þ 4 455 4A2 ðxÞ 4 U 0 ðxÞ x130 2 4 4A0 ðxÞ 4 U 3 ðxÞ A1 ðxÞ 4 U 2 ðxÞ 7CB6Γð3α 1Þ B 6 x x7 F U x C:U 5 ðxÞ ð 3 ð ÞÞC7B 6Γð5α 1Þ @ 4A5 4 4A2 ðxÞ 4 U 1 ðxÞ A3 ðxÞ 4 U 0 ðxÞ x xð19Þð15ÞContinuing in a similar fashion the remaining successiveterms of the FRDTM can be obtained.

4Journal of Applied Mathematics30–2201u (x, t)u (x, t)230–120tt1x155x10010(a)(b)1.0u (x, t)320tu (x, x105010(c)0(d)Figure 1: The physical behavior of FRDTM solution of Eq. (21) for (a) α 0:25, (b) α 0:5, (c) α 0:75, and (d) α 1.Then, the fractional reduced differential inverse transform of the set of values of ½U k ðxÞ k 0 giving the series solution of Eq. (1) asuðx, t Þ f ðxÞ gðxÞt α !1 A0 ðxÞ 4 U 0 ðxÞ F ðU 0 ðxÞÞ t 2α :Γð2α 1Þ x4ð20ÞIf α 1, the FRDTM solution (20) gives the exact solutionof Eq. (1).5. Illustrative examplesExample 11. Consider one-dimensional homogeneous timefractional beam equation: 2α 4uðx, t Þ 4 uðx, t Þ 0, x ℝ, t 0, 0 α 1:2α x tð21ÞSubjected to the initial conditions:uðx, 0Þ cos x, ut ðx, 0Þ sin x:ð22ÞApplying properties of FRDTM to Eq. (21), we obtain thefollowing recurrence relation:U k 2 ðxÞ Γðkα 1Þ 4U ðx Þ:Γðkα 2α 1Þ x4 kð23Þand using FRDTM to the initial conditions (22), we getU 0 ðxÞ cos x, U 1 ðxÞ sin x:ð24Þ