Using Laplace Transform Method For Obtaining The Exact .

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 5021-5035 Research India Publicationshttp://www.ripublication.comUsing Laplace transform method for obtaining theexact analytic solutions of some ordinary fractionaldifferential equationsGihan E. H. Ali1,2, A. A. Asaad 3, S. K. Elagan3, 4 ,El Amin Mawaheb1 and M. Saif AlDien31Khurma Branch, Faculty of Sciences, Taif University, Taif, Kingdom of Saudi Arabia.234Faculty of Petroleum and Mining Engineering, Seuz University Department ofEngineering, Egypt.Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif,El-Haweiah, P. O. Box 888, Zip Code 21974, Kingdom of Saudi Arabia.Department of Mathematics, Faculty of Science, Menoufia University, Shibin Elkom,Egypt.AbstractIn this paper, we applied the Laplace transform to obtain an exact analyticsolution of some ordinary fractional differential equations. We used the Cauchyresidue theorem and the Jordan Lemma to obtain the inverse Laplacetransform for some complicated functions and this implied to obtain an exactanalytic solution of some ordinary fractional differential equations. Thefractional derivatives would described in the Caputo sense which obtained byRiemann-Liouville fractional integral operator. We showed that the Laplacetransform method was a powerful and efficient techniques for obtaining anexact analytic solution of some ordinary fractional differential equations.Keywords:Fractional-order differential equations; Laplace Transform; InverseLaplace Transform.1. INTRODUCTIONIn the past two decades, the widely investigated subject of fractional calculus hasremark ablygained importance and popularity due to its demonstrated applications in

Gihan E. H. Ali, et al5022numerous diverse fields of science and engineering. These contributions to the fields ofscience and engineering are based on the mathematical analysis. It covers the widelyknown classical fields such as Abel’s integral equation and viscoelasticity. Also,including the analysis of feedback amplifiers, capacitor theory, generalized voltagedividers, fractional-order Chua-Hartley systems, electrode-electrolyte interfacemodels, electric conductance of biological systems, fractional-order models of neurons,fitting of experimental data, and the fields of special functions [1-6].Several methods have been used to solve fractional differential equations, fractionalpartial differential equations, fractional integro-differential equations and dynamicsystems containing fractional derivatives, such as Adomian’s decomposition method[7–11], He’s variational iteration method [12–16], homotopy perturbation method[17–19], homotopy analysis method [20], spectral methods [21–24], and other methods[25–28].This paper is organized as follows;we begin by introducing some necessary definitionsand mathematical preliminaries of the fractional calculus theory. In section 3, theLaplace transform and the inverse Laplace transform for some functions isdemonstrated. In section 4, the proposed method is applied to several examples. Alsoconclusions given in the last section.2. PRELIMINARIES AND NOTATIONSIn this section, we give some basic definitions and properties of fractional calculustheory which are further used in this article.Definition 2.1.A real function f ( x), x 0 is said to be in space C , if thereexists a real number p , such that f (t ) t p f1 (t ) , where f1 (t ) C (0, ) , and it issaid to be in the space C n if and only if f n C , n .Definition 2.2.The Riemann-Liouville fractional integral operator of order 0 , of afunction f C , 1 , is defined asJ f (t ) 1 t(t s) 1 f ( s)ds , 0 0 ( )J 0 f (t ) f (t )Some properties of the operator J , which are needed here, are as follows:for f C , 1 , , 0 and 1 :(1)

Using Laplace transform method for obtaining the exact analytic solutions 5023(1) J J f (t ) J f (t )(2) J t ( 1) t ( 1)(2)Definition 2.3.The fractional derivative of f (t ) in the Caputo sense is defined asD f (t ) J m D m f (t )(3)for m 1 m , m , t 0 and f C m1 .Caputo fractional derivative first computes an ordinary derivative followed by afractional integral to achieve the desired order of fractional derivative.Similar to theinteger-order integration, the Riemann-Liouville fractional integral operator is a linearoperation:nni 1i 1J ( ci f i (t )) ci J f i (t )(4)where {ci }in 1 are constants.In the present paper, the fractional derivatives are considered in the Caputo sense. Thereason for adopting the Caputo definition, as pointed by [10], is as follows: to solvedifferential equations (both classical and fractional), we need to specify additionalconditions in order to produce a unique solution. For the case of the Caputo fractionaldifferential equations, these additional conditions are just the traditionalconditions,which are akin to those of classical differential equations, and are thereforefamiliar to us. In contrast, for the Riemann-Liouville fractional differential equations,these additional conditions constitute certain fractional derivatives (and/or integrals) ofthe unknown solution at the initial point x 0 , which are functions of x . These initialconditions are not physical; furthermore, it is not clear how such quantities are to bemeasured from experiment, say, so that they can be appropriately assigned in ananalysis. For more details see [2].3. LAPLACE OPERATIONThe Laplace transform is a powerful tool in applied mathematics and engineering.Virtually every beginning course in differential equations at the undergraduate levelintroduces this technique for solving linear differential equations. The Laplacetransform is indispensable in certain areas of control theory.

Gihan E. H. Ali, et al50243.1. Laplace TransformGiven a function f (x) defined for 0 x , the Laplace transform F (s) is defined as F ( s) L[ f ( x)] f ( x)e sx dx0(5)at least for those s for which the integral converges.Let f (x) be a continuous function on the interval [0, ) which is of exponentialorder, that is, for some c and x 0sup f ( x) .ecxIn this case the Laplace transform exists for all s c .Some of the useful Laplace transforms which are applied in this paper, are as follows:For L[ f ( x)] F (s) and L[ g ( x)] G(s)L[ f ( x) g ( x)] F ( s ) G ( s ), ( 1), 1,s 1L[ f ( n ) ( x)] s n F ( s ) s n 1 f (0) s n 2 f ' ( n 1) (0),L[ x ] L[ x n f ( x)] ( 1) n F ( n ) ( s ) ,xL[ f (t )dt ] 0(6)F (s),sxL[ f ( x t ) g (t )dt ] F ( s )G ( s ) .0Lemma 3.1.1.The Laplace transform of Riemann-Liouville fractional integraloperator of order 0 can be obtained in the form of:L [ J f ( x)] F ( s)s (7)Proof. The Laplace transform of Riemann-Liouville fractional integral operator oforder 0 is :L[ j f ( x)] L[1 x1( x t ) 1 f (t )dt ] F ( s)G ( s) , 0 ( ) ( )(8)whereG( s) L[ x 1 ] ( )s (9)

Using Laplace transform method for obtaining the exact analytic solutions 5025and the lemma can be proved.Lemma 3.1.2.The Laplace transform of Caputo fractional derivative form 1 m , m , can be obtained in the form of:L[ D f ( x)] s m F ( s) s m 1 f (0) s m 2 f ' ( m 1) (0).s m (10)Proof. The Laplace transform of Caputo fractional derivative of order 0 is :L[ D f ( x)] L[ J m f ( m) ( x)] L[ f ( m) ( x)].s m (11)Using Eq.(6), the lemma can be proved.Now, we can transform fractional differential equations into algebraic equations andthen by solving this algebraic equations, we can obtain the unknown Laplace functionF (s) .3.2.Inverse Laplace TransformThe function f (x) in ((5)) is called the inverse Laplace transform of F (s) and willbe denoted by f ( x) L 1[ F (s)] in the paper. In practice when one uses the Laplacetransform to, for example, solve a differential equation, one has to at some point invertthe Laplace transform by finding the function f (x) which corresponds to somespecified F (s) .The Inverse Laplace Transform of F (s) is defined as:f ( x) L 1[ F ( s)] iT sx1lim iT e F ( s) ds,2 i T (12)where is large enough that F (s) is defined for the real part of s . surprisingly, thisformula isn't really useful. Therefore, in this section some useful function f (x) isobtained from their Laplace Transform. In the first we define the most importantspecial functions used in fractional calculus the Mittag-Leffler functions and thegeneralized Mittag-Leffler functionsFor , 0 and z

Gihan E. H. Ali, et al5026zn E ( z ) n 0 (n 1),zn.E , ( z ) n 0 ( n ) (13)Now, we prove some Lemmas which are useful for finding the function f (x) from itsLaplace transform.Lemma 3.2.1.For , 0 , a and s a we have the following inverseLaplace transform formula sx 1 e x E , ax dx 0s .s as i.e. L [ ] x 1 E , (ax )s a(14) 1Proof. ake st x 1 k dx k 0 k 0 sx 1 e x E , ax dx 0ak k s k k 0 k s as k(15)k 0s 1 as s .s aSo the inverse Laplace Transform of above function isx 1 E , ( ax ) .(16)The following two lemmas are known see [29], but we include the proof forconvenience of the reader.Lemma 3.2.2.For 0 , a and s a we have

Using Laplace transform method for obtaining the exact analytic solutions 5027 ( a) k ( nk k )1 ( n 1) 1] xL [ x k ( ) . (17) n 1( s as )k 0 (k ( ) (n 1) ) 1Proof. Using the series expansion of (1 x) n 1 of the from 1 ( nk k )( x) k n 1(1 x)k 0(18)we have:1111 n 1 n 1 n 1( s as )( s ) (1 a ) n 1 ( s )s (k 0n kk)( a k) (19)s Giving the inverse Laplace Transform of above function can prove the Lemma.Lemma 3.2.3.For , , a , s a and s as b we haveL 1[ ( b) n ( a) k ( nk k )s 1] xx k ( ) n . s as bn 0 k 0 ( k ( ) ( n 1) )(20)Proof. s /( s as b) by using the series expansion can be rewritten ass s s as b s as 1 s ( b) n n 1n 0 ( s as ) 1b s as (21)Now by using Lemma 3.2.2 the Lemma3.2.3 can be proved.Titchmarsh Theorem [28]:Let F p be an analytic function having no singularitiesin the cut plane. Assuming that F p F p and the limiting valuesl F t lim F t e i , F t F (t ) . Exist for almost all (i) F p o 1 for p and F p o p 1 for p 0 uniformly in anysectorarg p , 0,(ii) There exists 0 such that for everyF rei1 rL1, F reiWhere a r doesn't depend on and a r ea r ,rL1,for any 0 . Then

Gihan E. H. Ali, et al5028f t L 1 1Im F e t d F s 04. ILLUSTRATIVE EXAMPLESThis section is applied the method presented in the paper and give an exact solution ofsome linerar fractional differential equations.Example 4.1. Consider the composite fractional oscillation equationD 2 y x aD y x by ( x) 8 ,0 1'y (0) y (0) 0.Hence, we have two cases for 0 (22)11and 122For case one using the Laplace transform, F (s) is obtained as followss 2 F ( s ) a s F ( s) bF ( s) F ( s) s 2 8,s(23)8s 1. as bUsing the Lemma 3.2.3, the exact solution of this problem can be obtained as:b n a k ( nk k ) x k 2 n.n 0 k 0 ( k 2 n 1 1) y ( x) 8 x 2 (24)For case two, alpplying the Laplace transform, one hass 2 F ( s ) a s F ( s) bF ( s) F ( s) s 2 8,s8s 1. as bWhich the solution is similar to case 0 1when y(0) y' (0) 02Example 4.2. Consider the following system of fractional algebraic-differentialequationsD x(t ) tD y(t ) x(t ) (1 t ) y(t ) 0,y(t ) sin t 0 ,0 1(25)