Conseguences of soil crude oil pollution on some wood properties of olive trees Mathematics |192 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Algorithm to Solve Linear Volterra Fractional Integro- Differential Equation via Elzaki Transform Amaal A. Mohammed Yasmin H. Abd Dept. of ways & Transportation / College of Engineering / Al-Mustansiriya University Received in :28/ June /2016 , Accepted in : 4 /January/ 2017 Abstract In this work, Elzaki transform (ET) introduced by Tarig Elzaki is applied to solve linear Volterra fractional integro-differential equations (LVFIDE). The fractional derivative is considered in the Riemman-Liouville sense. The procedure is based on the application of (ET) to (LVFIDE) and using properties of (ET) and its inverse. Finally, some examples are solved to show that this is computationally efficient and accurate. Key word: integral, differential, Volterra, fractional, Elzaki. Mathematics |193 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Introduction Many fields of science and engineering are described by integro-differential equation of fractional order such as, fluid mechanics, vis co-elasticity, diffusion processes, biology and so on [1-4]. Several methods to solve VFIDEs have been proposed such as, expansion methods and spline method [5], variation iteration method [6], Laplace decomposition method [7] and Legender pseudo spectral method [8]. In recent years, Elzaki transform (ET) was introduced by Tarig Elzaki (2010) has been used to find solutions of a wide class of differential, integral, partial differential and the integro- differential equations, see [9-12]. In this paper, ET and some of its fundamental properties are used to solve LVFIDEs with initial value problems:  t 0 (1)dτ)τ(y)τx,(k)t(g)t(y D α Subject to the initial conditions Nn,nα1-n,n...,2,1,z,)t(yD zc z-α  where zc is a specified constant, α is a parameter describing the order of the time fractional, g and k are known functions, y(t) is the unknown function and αD is the fractional differential operator of order α . The present paper has been organized as follows. In section 2, some definitions, theorems and properties of the fraction calculus and ET are presented. In section 3, the solution steps to solve LVFIDE by using ET are described. Finally, in section 4, some test examples are solved. Preliminaries In this section, we present some basic definitions, some theorems and useful properties of the fractional calculus and Elzaki transform, as well as we proved some theorems which are used in this work. Definition (1): [1, 13] The Riemann–Liouville fractional derivative of order α , is defined to be for α > 0 : dt)t(ft)(x dx d α)Γ(n 1 (x)fDD(x)fD x 0 1α n n n n-αnα      or for α < 0 : dt)t(ft)(x )αΓ( 1 (x)fD x 0 1αα      where n – 1 < α ≤ n Definition (2): [6, 13, 14] The Riemann – Liouville fractional integral operator of order α ≥ 0 , of a function 1,Cf  μ is defined as: dt)t(ft)(x )αΓ( 1 (x)fJ(x)fD x 0 1ααα    It has the following properties: For 0 γand0β,α  Mathematics |194 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 1) (x)f(x)fJ 0  2) (x)fJ(x)fJJ βαβα  3) (x)fJJ(x)fJJ αβ βα  4) 0α,(x)f(x)fJD αα  5) γαγα x 1)γΓ(α 1)γΓ( xJ     6) ...,10,γ,x 1)γαΓ(- 1)γΓ( xD γα-γα      k k C(0)fDandnα1nwhere, 1)kαΓ( xC (x)f(x)fDJ7) k-α n 1k k-α αα      Definition (3): [11, 15] Elzaki transform is defined by: ),(u, 0 t 21 2 dt)t(ufu(u)T])t(f[ e kkE     Where the function f (t) in te set A defined by:          ),[0x1)(tifM)t(f0k,k,M)t(fA jjkt 21 e Definition (4): [15, 12] Defined for Re(s) > 0 , the Laplace transform (LT) is given by: dt)t(f(s)F})t(f{ 0 t e    s L Some properties of ET are given by following theorems: Theorem (1): [9, 15] Let          ),[0x1)(tifM)t(f0k,k,M)t(fA)t(f jt jk 21 e with LT , F(s) then the ET, T(u) of f (t) is given by: ) s 1 (Ts(s)Fand) u 1 (Fu(u)T  Theorem (2): [9] Let T(u) denote the ET of the f(t), the ET of the definite integral of f(t), that is T(u)u)]t(h[Ethendτ)τ(f)t(h t 0   Theorem (3): [9, 11, 15] Let the ET, F(u) and G(u) of the functions f(t) and g(t) respectively, then the ET of the convolution of f and g is: G(u)F(u) u 1 )]t(g)*[(fEbygivenisdτ)τ-t(g)τ(f)t(g)*(f t 0   Theorem (4): [1, 5, 13] The LT of the Riemann – Liouville is defined as: Nn,nα1nandαallfor , xd f(0)d s(s)Fs(s)F}(t)fD{L 1-n 0k 1-k-α 1-k-α kααα    Theorem (5): If F(s) and T(u) are LT and ET of f(t) respectively, then Mathematics |195 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 }(t)fD{L(s)Fwhere) u 1 (Fu(u)T}(t)fD{ ααααα E Proof:- From definition(3) we have: dt])t(uf[Du(u)T])t(fD[E 0 tααα e2     Now, let w = ut then we get: ) u 1 (Fudw])(wf[Du(u)T α 0 uwαα e     Theorem (6): The ET of the Riemann – Liouville fractional derivative is defined:    1-n 0k 1-k-α 1-k-α k-1 α αα xd f(0)d u u (u)T (u)T}(t)fD{E Proof:- At first from the above theorem we have: ) u 1 (Fu(u)T])t(fD[E ααα  And using theorem (4) with u 1 s  we obtain:           1-n 0k 1-k-α 1-k-α kαα xd f(0)d ) u 1 () u 1 F() u 1 (u(u)T Then by theorem (1), we get:    1-n 0k 1-k-α 1-k-α k-1 α α xd f(0)d u u (u)T (u)T Theorem (7): Let T(u) be ET of the f(x), then ET for f(x) = )x(erfxe is given by: u-1 uu T(u) 2  Proof:- The proof of this equation is easy, by using table of LT, [16] and theorem (1) as follows:   1)(ss 1 )x(erfeL x   u1 uu 1) u 1 ( u 1 1 u) u 1 F(u(u)T 2 and     Applications to (LVFIDE) Recall equation (1), the LVFIDE with initial value problems: Mathematics |196 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017  t 0 α dτ)τ(y)τx,(k)t(g)t(y D Subject to the initial conditions Nn,nα1-n,n...,2,1,z,)t(yD zc z-α  The first step, we apply ET on both sides we have:             t 0 α dτ)τ(y)τx,(k)t(g)t(y D EEE Can easily be transformed into its ET using theorem (6), equivalent to:        t 0 α dτ)τ(y)τx,(k)u(GC u (u)T E where    1-n 0k 1-k-α 1-k-α k-1 xd y(0)d uC , T(u) and G(u) are ET for y(t) and g(t) respectively. The general Elzaki solution is: )2(dτ)τ(y)τx,(ku)u(GuCu(u)T t 0 ααα        E The second step is to find ET of integral in above equation which depends on the type function )τx,(k (kernel of the integral equation) as follows: A. If the kernel is constant that is, )τ,x(k = R , then by using theorem (2) we have: (3)T(u)uRdτ)τ(yR t 0       E where T(u) is the ET of y(t). B. If the kernel is deference that is, )τx(k)τ,x(k  , by theorem (3) we get: (4)T(u)*K(u) u 1 dτ)τ(y)τx(k t 0        E where K(u) and T(u) are ET of the functions k and y respectively. C. If the kernel is any function, we represent the solution as an infinite series, that is:     0i (t)yy(t) i The ET of the integral is become: Mathematics |197 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 )5(dτ)τ(y)τx,(kdτ)τ(y)τx,(k t 0 0i i t 0                  EE The third step, we substitute either eq.(3) or eq.(4) or eq.(5) into eq.(2), we will get: either, T(u)uR)u(uu(u)T 1ααα GC  Then the general Elzaki solution: )6( uR1 ))u(G(Cu (u)T 1α α     or T(u)*K(u)u))u(G(Cu(u)T 1-αα  Then )7( K(u)u1 ))u(G(Cu (u)T 1-α α    or                      t 0 0i i αα 0i i dτ)τ(y)τx,(ku))u(G(Cu(t)y EE Matching both sides of this equation yields the following iterative relations:   )8())u(G(Cu(t)y α 0 E          t 0 α dτ)τ(y)τx,(ku(t)y 01 EE          t 0 α dτ)τ(y)τx,(ku(t)y 12 EE In general,   )9(...1,0,i,dτ)τ(y)τx,(ku(t)y t 0 i α 1i         EE The fourth step, we apply inverse ET of either eq.(6) or eq.(7) or (eq.(8) and eq.(9)) give the general solution for (LVFIDE): either )10( uR1 ))u(G(Cu T(t)y 1α α 1              or )11( K(u)u1 ))u(G(Cu T(t)y 1-α α 1             or   )12(P(t)))u(G(Cu(t)y α10  E Mathematics |198 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 with )13(...1,0,i,dτ)τ(y)τx,(ku(t)y t 0 α i 1 1i                      EE The function P(t) defined in eq.(12) can be decomposed into two parts: P(t) = P1(t) + P2(t) So, the modified recursion relation is obtained )14( ...2,1,i,dτ)τ(y)τx,(ku(t)y dτ)τ(y)τx,(ku(t)P(t)y (t)P(t)y t 0 α t 0 α i 1 1i 0 1 21 10                                                       EE EE The solution depends on the choice of P1(t) and P2(t). We will show how to suitably choose P1(t) and P2(t) as well as testing the method by some examples in the next section. Test Examples In this section, we present some test examples to show the effectiveness of the ET method for solving LVFIDE's. Example (1): Consider the following linear LVFIDE [5]  x 0 dt)t(y1)xerf( xπ 1 )x(y D x0.5 eex with 0)0(yD 1 0.5- c  since 1nthen,nα1-nand0.5α  Applying (ET) of the above equation and since k(x, t) = 1, then by using eq.(10) we have the solution: )15( u1 ))u(G(Cu T(x)y 1.5 0.5 1             where 0)0(yDuC 0.5-  and        1)xerf( xπ 1 G(u) xx eeE from appendix given in [11] ( ET of some functions) and theorem (7) we have: Mathematics |199 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 u1 uuu u u1 u u1 uu uuG(u) 3 2 22        Substituting in eq.(15), we obtain:                       u1 u T )uu(1u)(1 uuu T(x)y 2 1 32 1 Again from appendix given in [11], we get the exact solution: y(x) = e x . Example (2): Let us consider the following LVFIDE with difference kernel )16(dt)t(y tx )x(g)x(y D x 0 e 56    where x66x6x3xx 5)14Γ( 6 g(x) e23 59  with initial conditions: 0)0(yDand0)0(yD 2 54 1 51 cc   Take the (ET) of eq.(16) and since k(x, t) is difference kernel, so by eq.(11) yields: )17( K(u)u1 ))u(G(Cu T(x)y 51 56 1             where C = u c1 + c2 = 0 .                        u1 uuu 6 u1 u uuuuu6g(x)G(u) 6524519 2 2345519 E and   u1 u K(u) 2 xe   E Now, eq.(17) can be written in the form:  51 511 53665 1 u6T ) u1 u (1u)(1 uuu T6(x)y                     Upon inverting, we get the exact solution: y(x) = x 3 . Example (3): Consider the following LVFIDE Mathematics |200 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017     t 0 dτ)τ(yτ)(t 15 65 220.5 tt)t(y D with initial conditions: πc1  . Applying the ET to both sides of above equation as well as using eq.(12) and eq.(13) give:       ))u(G(Cu(t)y 0.51 0 E ...1,0,i, t 0 dτ)τ(y)τt((t)y i 20.51 1i u                         EE where πucu)0(yDuC 1 0.5-  292 u 2)7Γ( ttG(u) 15 65 15 65 and         E (t)P(t)PP(t) t 1 u 15 2)7Γ(65 uπE 21 5231 0 3 t 45 2)7Γ(28 (t)yThen          So we have the following relations: )18(u t 1 t 0 dτ)τ(y)τt(t 45 2)7Γ(82 (t)y (t)y 0 20.51 1 0 3                        EE and ...2,1,i, t 0 dτ)τ(y)τt((t)y i 20.51 1i u                         EE Now, we find the ET of integral in eq.(18) we have: 2922 u 65 tt 15 2)7Γ( 15 56 t 0 dτ τ 1 )τt(                  EE Then       51 1 u 15 2)7Γ(65 t 2)7Γ(28 (t)y 3 45 E That is ...2,1,i0and0 (t)y(t)y 1i1   Therefore, the solution is obtained to be and its exact solution: t 1 (t)yy(t) 0i i     . Mathematics |201 2102( عام 2( العدد ) 30المجلد ) مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Conclusions The properties of the Elzaki transform are used to solve and to get the general solution of linear Volterra fractional integro-differentioal equations. The fractional derivative is considered in the Riemman-Liouville sense. The results of illustrate examples as well as the simplicity of the algorithm and obtained exact solution show efficiency and accuracy of the method also show that it is a special case of the analysis methods. Finally, the proposed approach is very powerful to find analytical solution of linear problems in fractional calculus field. References 1. Oldham, K. B. and Spanier, J. (1974) The Fractional Calculus, London, Academic Press. Inc. 2. Nishimoto, K. (1984) Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Descartes Press Co. Koriyama, Japan. 3. Schmidt, A. and Gaul, L. (2000) FE Implementation of Viscoelastic Constitutive Stress- Strain Relations Involving Fractional Time Derivatives, A Report, Institute A Fur Mechanic, Universitat Stuttgart, Germany. 4. Adolfsson, K. and Enelund, M. (2004), Dicretization of Integro-Differential Equations Modeling Dynamic Fractional Order Viscoelasticity, Appl. Mech. 5. Al-Rahhal, D. M. (2005) Numerical Solution for Fractional Integro-Differential Equations, Ph. D. Thesis, University of Baghdad, College of Ibn Al-Haitham. 6. Kurulay, M. and Secer, A. (2011), Variational Iteration Method For Nonlinear Fractional Integro-Differential Equations, International Journal of Computer Science & Emerging Technologies, Issue 1, 2, 18-20. 7. Yang, C. and Hou, J. (2013), '' Numerical Solution of Integro-Differential Equations of Fractional Order By Laplace Decomposition Method'', Wseas Transactions on Mathematics, Issue 12,12,1173-1183. 8. Khader, M. M.; Sweilam, N. H. and Assiri, T. A. (2013), On The Numerical Solution For The Fractional Wave Equation Using Legender Pseudo Spectral Method, International Journal of Pure and Applied Mathematics, 84, 4, 307-319. 9. Elzaki, T. M. and Elzaki, S. M. (2011), Solution of Integro-Differential Equations By Using Elzaki Transform, Global Journal of Mathematical Sciences: Theory and Practical, 3, 1,1-11. 10. Elzaki, T. M. and Elzaki, S. M. (2011), The New Integral Transform ''Tarig Transform'' Properties and Applications to Differential Equations, Elixir International Journal, 38, 4239-4242. 11. Elzaki, T. M.; Elzaki, S. M. and Elnour, E. A. (2012), On Some Applications of New Integral Transform ''Elzaki Transform'', Global Journal of Mathematical Sciences: Theory and Practical, 4, 1,15-23. 12. Chopade, P. P.and Devi, S. B. (2015), Applications of Elzaki Transform to Ordinary Differential and Partial Differential Equations, International Journal of Advanced Research in Computer Science and Software Engineering, Issue. 3, 5, 38-41. 13. Mohammed, A. A. (2006), Approximate Solutions for a System of Fractional Order Integro-Differential Equations of Volterra Types, Ph. D. Thesis, University of Al- Mustansiriy, College of science. 14. Asgari, M. (2015) Numerical Solution for Solving a System of Fractional Integro- Differential Equations, IAENG Inter. Journal of Applied Mathematics, 45, 2. 15. Elzaki, T. M.; Elzaki, S. M. and Elnour, E. A. (2012) On The New Integral Transform ''Elzaki Transform'' Fundamental Properties Investigations and Applications, Global Journal of Mathematical Sciences: Theory and Practical, 4, 1, 1-13.