In the present work the modified differential transform method, incorporating the Adomian polynomials into the differential transform method(DTM), is used to solve the nonlinear and singular Lane-Emden equations in integer and fractional order. Numerical examples with different types are solved. The results show that this method is very effective and simple.Publisher's Versio

Marasi, Hamidreza

Piri, Hossein

Sharif, Negar

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.5, N.1, 2015, pp. 124-131.MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR SINGULARLANE-EMDEN EQUATIONS IN INTEGER AND FRACTIONALORDERH. R. MARASI1, N. SHARIFI1, H. PIRI1, §Abstract. In the present work the modified differential transform method, incorporat-ing the Adomian polynomials into the differential transform method(DTM), is used tosolve the nonlinear and singular Lane-Emden equations in integer and fractional order.Numerical examples with different types are solved. The results show that this methodis very effective and simple.Keywords: Differential transform method, Adomian polynomials, singular Lane-Emdenequation.AMS Subject Classification: 35A15, 35A201. IntroductionIn recent years, the study of singular initial value problems modelled by second ordernonlinear ordinary differential equations[1, 2] has attracted many mathematicians andphysicists. One of the important equations in this literature is the Lane-Emden typeequation of the form∂2u∂t2+2t∂u∂t+ f(u) = 0 (1)with initial conditionsu(0) = a, u′(0) = b, (2)where f is a given function of u and a,b are constants. Lane-Emden type equations, firstpublished by Janathan Homer Lane in 1870 [3] and further explored in detail by Emden[4].Lane-Emden type equation was used the model the thermal behavior of a spherical cloudof gas acting under the mutual attraction of its molecules and subject to the classicallows of thermodynamics. So, a substantial amount of work has been done on this type ofproblems. In astrophysics, the Lane-Emden equation is Poisson’s equation for the grav-itational potential of a self-gravitating spherically symmetric polytropic fluid. Also, thisis a type of equations have many applications in the fields of radioactiviely colling and inthe mean-field treatment of a phase transition in critical adsorption or in the modelling ofclusters of galaxies.In most differential equations with variable coefficients it is impossible to obtain an exactsolution, so one must resort to various approximation methods of solution, such as as-ymptotic techniques[11, 12, 13], analytical[14, 15] and numerical methods[16]. Analytical1 Department of Mathematics, University of Bonab, Bonab 5551761167, Iran.e-mail: marasi@bonabu.ac.ir;§ Manuscript received: July 31, 2014.TWMS Journal of Applied and Engineering Mathematics, Vol.5, No.1; c⃝ Işık University, Departmentof Mathematics, 2015; all rights reserved.124H. R. MARASI, N. SHARIFI, H. PIRI: MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR... 125techniques have been dominated by perturbation methods and have found many applica-tions in science, engineering and technology. However, like other analytical techniques,perturbation methods require the presence of a small parameter in the nonlinear equation.Selection of small parameter requires a special skill and very important. Therefore, analyt-ical methods which do not require a small parameter are welcome. Because of singularitybehavior at the origin and nonlinearity, the Lane-Emden equations may not be solved bystandard semi-analytical methods, such as the homotopy perturbation method[17], vari-ational iteration method[7, 8] and Adomian decomposition method[6, 5]. The basic ideaof differential transform method was initially introduced by Zhou[18]. The DTM is analternative procedure for getting Taylor series solution of the equation. This method re-duces the size of computations of taylor coefficients. The motivation for presenting thiswork comes from the aim of introducing a reliable framework that combines the powerfuldifferential transform method and Adomian decomposition method.Many physical problems can be described by mathematical models that involves fractionaldifferential equations. There are several mathematical definitions about the generalizing ofthe notion of differential to fractional orders e.g. Riemann-Lioville, Grunuald-Letinikow,Caputo and generalized functions approach.Definition 1.1. A real function f(t), t > 0, is said to be in the space Cµ, µ ∈ R, if thereexist a real number p ≥ µ such that f(t) = tpf1(t) where f1(t) ∈ C(0,∞) and it is said tobe in the space Cmµ if f(m) ∈ Cµ,m ∈ N .Definition 1.2. For a continuous function f : [0,∞) → R, the Caputo derivative offractional order α is defined bycDαf(t) =1Γ(n− α)∫ t0(t− s)n−α−1f (n)(s)ds,where n− 1 < α < n, n = [α] + 1 and [α] denotes the integer part of α.Definition 1.3. The Riemann- Liouville fractional derivative of order α for a continuousfunction f is defined byDαf(t) =1Γ(n− α)(ddt)n∫ t0f(s)(t− s)α−n−1ds, n = [α] + 1,where the right-hand side is pointwise defined on (0,∞).Definition 1.4. Let [a, b] be an interval in R and α > 0. The Riemann-Liouville fractionalorder integral of a function f ∈ L1([a, b], R) is defined byIαa f(t) =1Γ(α)∫ taf(s)(t− s)1−αds,whenever the integral exists.2. The fractional differential transform method(FDTM)In the FDTM the analytic function u(t) is expanded in terms of a fractional power seriesin the formu(t) =∞∑k=0U(k)(t− t0)kθ ,where θ is the order of the fraction to be selected and U(k) is the kth fractional differentialtransform of u(t). Since the initial conditions are implemented by integer-order derivatives126 TWMS J. APP. ENG. MATH. V.5, N.1, 2015for practical applications, the transformation of the initial conditions is defined as followsU(k) =1(kθ )![∂kθ∂tkθu(t)]t=t0 ,kθ∈ Z+,0,kθ̸∈ Z+,(3)where k = 0, 1, · · · , (αθ − 1) and α is the order of the fractional diffential equation beingconsidered. Thus θ should be chosen such that αθ is a positive integer. Here we presentsome basic properties of the FDTM. Let u(t), v(t) and w(t) be functions of time t andU(k), V (k) and W (k) are their corresponding fractional differential transforms with orderof fraction θ. Then for constant c and p the followings hold,i: If u(t) = v(t)± w(t), then U(k) = V (k)±W (k).ii: If u(t) = cv(t), then U(k) = cV (k).iii: If u(t) = Dαt v(t), then U(k) =Γ(α+1+ kθ)Γ(1+ kθ)V (k + αθ).We consider the case of a nonlinear function f(u) that is approximated by the seriesf(u) =∞∑n=0An,where the An are the Adomian polynomials determined by the definitional formulaAn =1n![dndλn[f(∞∑i=0λiui)]]λ=0, n = 0, 1, · · · .For the nonlinear fractional differential equation of the formDαu = f(u),where f(u) denotes a nonlinear function, we apply recurrence scheme of the formΓ(α+ 1 + kθ )γ(1 + kθ )U(k + αθ) = Ãk,where Ãk are obtained from the Adomian polynomials of f(u) by replacing each u(k) andDβuk by U(k) andΓ(α+1+ kθ)γ(1+ kθ)U(k + αθ), respectively [20].3. Numerical ApplicationsThe Lane-Emden equation has been used to formulate several phenomena in mathe-matical physics and astrophysics. This equation encounters wide applications in modelingof the thermal behaviour of a spherical cloud of gas acting under a mutual attraction ofits molecules and subject to the classical laws of thermodynamics. In this section, wesolve three differential equations, the so-called Lane-Emden equation, to demonstrate theeffectiveness and the validity of the present method.Example 3.1. Consider the fractional Lane-Emden equationDβu+2tut = −u, 1 < β ≤ 2, (4)subject to the initial conditionsu(0) = 1, ut(0) = 0. (5)H. R. MARASI, N. SHARIFI, H. PIRI: MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR... 127Taking the fractional differential transform and choosing θ = 2β , we write the recurrenceschemeΓ(1 + (k + 1)β2 )Γ(1 + (k − 1)β2U(k + 1) + 2(k + 1)U(k + 1) = −U(k − 1), k = 1, 2, · · · . (6)So, we will get the approximationsU(0) = 1, U(1) = 0, U(2) = − 14 + Γ(1 + β), U(3) = 0,U(4) =14 + Γ(1 + β)· Γ(1 + β)Γ(1 + 2β) + 8Γ(1 + β), U(5) = 0, · · · .Here, if we put β = 2 then we will get the following solutionu(t) = 1− 13!t2 +15!t4 − 17!t6 + · · ·which is the Taylor expansion of the exact solution u(t) = sin tt of the problem in this case.Example 3.2. Consider the fractional non-linear Lane-Emden equationDβu+2tut = −u5, 1 < β ≤ 2, (7)subject to the initial conditionsu(0) = 1, ut(0) = 0. (8)By taking the fractional differential transform and choosing θ = 2β , we obtain therecurrence schemeΓ(1 + (k+12 )β)Γ(1 + k−12 β)U(k + 1) + 2(k + 1)U(k + 1) = −Ãk−1, k = 1, 2, · · · (9)where Ãk are DTM’s of the Adomian polynomials, Ak, for the non-linear term, u5, asfollowsA0 = (u0)5, Ã0 = (U(0))5,A1 = 5(u0)4u1, Ã1 = 5(U(0))4U(1),A2 = 5(u0)4u2 + 10(u0)3(u1)2, Ã2 = 5(U(0))4U(2) + 10(U(0))3(U(1))2,A3 = 5(u0)4u3 + 20(u0)3u1u2 + 10(u0)2(u1)3, Ã3 = 5(U(0))4U(3) + 20(U(0))3U(1)U(2) +10(U(0))2(U(1))3,...Then, we have the following approximationsU(0) = 1, U(1) = 0, U(2) = − 1Γ(1 + β) + 4, U(3) = 0,U(4) =5Γ(1 + β)6(Γ(1 + 2β) + 8Γ(1 + β)), U(5) = 0, U(6) = −420864Γ(1 + 2β)Γ(1 + 3β) + 12Γ(1 + 2β), · · ·Sou(t) = 1− 1Γ(1 + β) + 4t2+56Γ(1 + β)Γ(1 + 2β) + 8Γ(1 + β)t4−420864Γ(1 + 2β)Γ(1 + 3β) + 12Γ(1 + 2β)t6+...When the limit of the series solution as β approaches 2 is evaluated, we obtainu(t) = 1− 16t2 +124t4 − 5432t6 + · · · = 1√1 + t23,128 TWMS J. APP. ENG. MATH. V.5, N.1, 2015which is the exact solution of the problem in this case.Example 3.3. Now, we consider the following fractional non-linear Lane-Emden equationDβu+2tut = −eu, 1 < β ≤ 2, (10)subject to conditionsu(0) = 0, u(1) = 0. (11)Taking the fractional differential transform and choosing θ = 2β we write the recurrenceschemeΓ(1 + (k+12 )β)Γ(1 + k−12 β)U(k + 1) + 2(k + 1)U(k + 1) = −Ãk−1, k = 1, 2, · · · (12)where Ãk are obtained from the Adomian polynomials for the nonlinear term, eu, as followsA0 = eu0 Ã0 = eU(0),A1 = u1eu0 , Ã1 = U(1)eU(0),A2 = (u2 +(u1)22 )eu0 , Ã2 = (U(2) +(U(1))22 )eU(0),A3 = (u3 + u1u2 +(u1)33! )eu0 Ã3 = (U(3) + U(1)U(2) +(U(1))33! )eU(0),...Then, we can obtain the following valuesU(0) = U(1) = U(3) = 0, U(2) = − 1Γ(1 + β) + 4, U(4) =16Γ(1 + β)Γ(1 + 2β) + 8Γ(1 + β),U(5) = 0, U(6) = − 85!× 3× Γ(1 + 2β)Γ(1 + 3β) + 12Γ(1 + 2β), · · ·This yields the series solutionu(t) =−1Γ(1 + β) + 4t2 +16Γ(1 + β)Γ(1 + 2β) + 8Γ(1 + β)t4 − 8360Γ(1 + 2β)Γ(1 + 3β) + 12Γ(1 + 2β)t6 + ...When the limit of the series solution as β approaches 2 is evaluated, we obtainu(t) = − 13× 2!t2 +15× 4!x4 − 821× 6!t6 +12281× 8!t8 − · · · (13)which is the same solution obtained in [19] by modified homotopy perturbation methodfor the case β = 2. Using the Matlab symbolic code, we plot the Pade approximationsof[2,2], [3,3] and [5,5] in fig.1. This shows the trend of convergence of the solution for thesethree approximants. Shown in fig.2 are the modified FDTM solution of the problem with5 components as parametrized by β.4. ConclusionInstead of computing the differential transform of the nonlinear term, it is replacedin the recurrence relation by its Adomian polynomial of index k. The nonlinear partialdifferential equations related to the Lane-Emden problem are solved. The validity andthe effectiveness of the method are shown in a systematic fashion. The proposed solutionsof the three singular linear and nonlinear differential equations of Lane-Emden problemusing the method have verified the physical properties of the equilibrium of Lane-Emdenproblem, as x → ∞ the solution monotonically approaches a constant. This is indeed animportant physical property of the Lane-Emden problem known as the equilibrium state.H. R. MARASI, N. SHARIFI, H. PIRI: MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR... 129−5 −4 −3 −2 −1 0 1 2 3 4 5−2−1.8−1.6−1.4−1.2−1−0.8−0.6−0.4−0.20x  [2,2][3,3][5,5]Figure 1. Plots of the [2,2], [3,3], [5,5] Pade approximants for Example(3.3).0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1t(6234590327124965 t27/10)/288230376151711744 −...− (2561942702991429 t9/2)/2305843009213693952 β=2β=1.95β=1.9β=1.85Figure 2. Plots of the FDTM Solution with 5 components for Example(3.3) asparametrized by β.References[1] Agarwal, R. P., Regan, D. O. and Lakshmikanthamr, V., (2001), Quadratic forms and nonlinear non-resonant singular second order boundary value problems of limit circle type, Zeitschrift fur Analysisund ihre Anwendungen, 20, pp. 727-737.[2] Agarwal, R. P. and Regan, D. O., (2001), Existence theory for single and multiple solutions to singularpositone boundary value problems, J. Diff. Equ., 175(2), pp. 393-414.[3] Lane, J. H., (1870), On the theoretical temperature of the sun under the hypothesis of a gaseous massmaintaining its volume by its internal heat and depending on the laws of gases known to terrestrialexperiment, The American Journal of Science and Arts, 50, pp. 57-74.130 TWMS J. APP. ENG. MATH. V.5, N.1, 2015[4] Emden, R., (1907), Gaskugeln, Teubner, Leipzig and Berlin.[5] Marasi, H. R. and Nikbakht, M., (2011), Adomian decompositiom method for boundary value prob-lems, Aus. J. Basic. Appl. Sci., 5, pp. 2106-2111.[6] Adomian, G., (1994), Solving frontier problems of physics: The decomposition method, Kluwer Aca-demic, Dordrecht.[7] Marasi, H.R. and Karimi, S., (2014), Convergence of variational iteration method for solving fractionalKlein-Gordon equation, J. Math. Comp. Sci., 4, pp. 257-266.[8] Assas, L. M. B., (2008), Variational iteration method for solving coupled-KdV equations, ChaosSolitons Fractals., 38(4), pp. 1225-1228.[9] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., (2006), Theory and applications of fractionaldifferential equations, North-Holland Mathematics Studies., 204, pp. 7-10.[10] Podlubny, I., (1999), Fractional differential equations, Academic Press, New york.[11] Marasi, H. R. and Jodayree Akbarfam, A., (2007), On the canonical solution of indefinite problemwith m turning points of even order, J. Math. Anal. Appl., 332, pp. 1071-1086[12] Marasi, H. R., (2011), Asymptotic form and infinite product representation of solution of a secondorder initial value problem with a complex parameter and a finite number of turning points, J. Cont.Math. Anal., 4, pp. 57-76[13] Marasi, H. R. and Jodayree Akbarfam, A., (2012), Dual equation and inverse problem for an indefniteSturm-Liouville problem with m turning points of even order, Math. Modell. Anal., 17(5), pp. 618-629.[14] Chowdhury, M. and Hashim, I., (2009), Solutions of Emden-Fowler equations by homotopy perturba-tion method, Non-Linear Analysis: Real World Application., 101, pp. 104-115.[15] Yildirim, A. and Ozi, T., (2009), Solutions of singular IVPs of Lane-Emden type by the variationaliteration method, Nonlinear Analysis: Theory, Methods Applications, 70(6), pp. 2480-2484.[16] Parand, K. and Pirkhedri, A., (2010), Sinc-collocation method for solving astrophysics equations, NewAstronomy, 15(6), pp. 533-537.[17] He, J., (2006), Homotopy perturbation method for solving boundary value problems, Phys. Lett. A.,350, pp. 87-88.[18] Zhou, J. K., (1986), Deferential transformation and its application for electrical circuits, HuazhongUniversity Press, Wuhan China.[19] Nazari-Golshan, A., Nourazar, S. S., Ghafoori-Fard, H., Yildirim, A. and Campo, A., (2013), Amodified homotopy perturbation method coupled with the Fourier transform for nonlinear and singularLane-Emden equations, Appl. Math. Lett., http://dx.doi.org/10.1016/j.aml., pp. 2013.05.010.[20] Elsaid, A., (2012), Fractional differential transform method combined with the Adomian polynomials,Apll. Math. Comput., 218, pp. 6899-6911.Hamidreza Marasi was born in 1972 in Leilan-Azerbaijan, Iran. He received the B.Sc.,M.Sc. and Ph.D. degrees from Tabriz University, Tabriz, Iran, in 1998, 2000 and 2008respectively. He is currently working as an Associate Professor at the Department ofApplied Mathematics, University of Bonab, Bonab, Iran. His major research areas areAsymptotic Theory, Eigenvalue Problems, Analytical and Numerical methods.Negar Sharifi was born in 23th August 1980 Bonab-Iran. She graduated from MaraghePayamnoor University in 2007 and she received her M.Sc. in 2013 in Department ofApplied Mathematics, University of Bonab, Bonab, Iran. Her research interests focus inAnalytical and Numerical solutions of ODE and PDE.H. R. MARASI, N. SHARIFI, H. PIRI: MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR... 131Hossein Piri was born in 1979 in Shahindezh, Azerbaijan-Iran. He received the B.Sc.,M.Sc. and Ph.D. degrees from Tabriz University, Tabriz, Iran, in 2002, 2006 and 2010respectively. He is currently working as an Assistant Professor at the Department of PureMathematics, University of Bonab, Bonab, Iran. He has done research in Non-LinearFunctional Analysis, Complex Analysis, Semi-Group Theory and Ergodic retraction.

Modified differential transform method for singular lane-emden equations in integer and fractional order

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Modified differential transform method for singular lane-emden equations in integer and fractional order

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