Solution of Integro-Differential Equations by Using ELzaki Transform

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1 Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp:// Soluion of Inegro-Differenial Equaions by Using ELzaki Transform Tarig. M. Elzaki and Salih M. Elzaki, Mah. Dep., Sudan Uniersiy of Science and Technology, ( arig.alzaki@gmail.com, salih.alzaki@gmail.com Absrac In his paper, we sole he inegro-differenial equaion by using new inegral ransform called ELzaki ransform. New heorems for he ransform of inegrals are inroduced and proed. Keywords: ELzaki ransform-inegro-differenial equaions. Inroducion Many problems of physical ineres are described by differenial and inegral equaions wih appropriae or boundary condiions. These problems are usually formulaed as iniial alue problem, boundary alue problems, or iniial boundary alue problem ha seem o be mahemaically more igorous and physically realisic in applied and engineering sciences. ELzaki ransform mehod is ery effecie for soluion of he response of differenial and inegral equaions and a linear sysem of differenial and inegral equaions. The echnique ha we used is ELzaki ransform mehod which is based on Fourier ransform. I inroduced by Tarig ELzaki () In his sudy, ELzaki ransform is applied o inegral and inegrao-differenial equaions which he soluion of hese equaions hae a major role in he fields of science and engineering. When a physical sysem is modeled under he differenial sense, if finally gies a differenial equaion, an inegral equaion or an inegrodifferenial equaion. Recenly.Tarig ELzaki inroduced a new ransform and named as ELzaki ransform which is defined by: f (), T e f () d, ( k, k) () Ε = =

2 Tarig. M. Elzaki and Salih M. Elzaki Or for a funcion f ( ) which is of exponenial order, f ( ) Me k < k Me,, () ELzaki ransform, henceforh designaed by he operaor Ε [.], is defined by he inegral equaion. f () T f ( ) e d, k k () Ε = = Where M is a real finie number and k, k can be finie or finie. Theorem (-): Le T ( ) is he ELzaki ransform of f () f ( τ ), τ ( f () ) T Ε = and g ( ) =, < τ τ Then: Ε g () = e T ( τ ) Ε g = e f d τ. Le = λ + τ we find ha: λ+ τ τ τ λ ( λ) λ = ( λ) λ = e f d e e f d e T Which is he desired resul ELzaki ransform can cerainly rea all problems ha are usually reaed by he well- known and exensiely used Laplace ransform. Indeed as he nex heorem shows ELzaki ransform is closely conneced wih he Laplace ransform F ( s ). Theorem (-): Le { } / k j i () = (),, >,, () <, ( ) [, ) f A f M k k such ha f Me if

3 Soluion of Inegro-Differenial Equaions Wih Laplace ransform F ( s ), Then ELzaki ransform T ( ) of f T = F is gien by (4) Le: f () A. Then for k < < k T () e fd = Le w= hen we hae: w w dw T = e f ( w) = e f ( w) dw= F. Also we hae ha T () F ( ) = so ha boh ELzaki and Laplace ransforms mus coincide a = s=. In fac he connecion of ELzaki ransform wih Laplace ransform goes much deeper, herefore he rules of F and T in (4) can be inerchanged by he following corollary. Corollary (-): Le f () haing F and T for Laplace and ELzaki ransforms respeciely, hen: F( s) = st (5) s This relaion can be obained from ( 4 ) by aking = s The equaions (4) and (5) form he dualiy relaion goerning hese wo ransforms and may sere as a mean o ge one from he oher when needed. ELzaki Transform of Deriaies and Inegrals Being resaemen of he relaion ( 4 ) will sere as our working definiion, since he Laplace ransform of sin is hen iew of (4), is. ELzaki ransform is + s Ε [ sin ] = his exemplifies he dualiy beween hese wo ransforms. + Theorem (-): Le F ( s) and T () be he Laplace and ELzaki ransforms of he deriaie of f ().

4 4 Tarig. M. Elzaki and Salih M. Elzaki Then: T () () i T = f() (6) ( n ) T ( n ) n+ k ( k ii T = f ) ( n ), n (7) k = n n Where ( n f ) ( ) of he funcion f ( ). deriaie T and F s are ELzaki and Laplace ransforms of he nh (i) Since he Laplace ransform of he deriaies of f () is F () s = sf() s f() hen: T T = F = F f F f f = = (ii) By definiion, he Laplace ransform for n n n k+ k n F s S F s S f = k = ( ) f ( n ) () is gien by Therefore: F n f F = ( k ) ( ) n n k + k = Now, since ( k ) ( k ) ( n ) T k T = F for k m, we hae T = f n k = n n+ k Theorem (-) Le T ( ) and F ( s) denoe ELzaki and he Laplace ransforms of he definie inegral of f ( ). () = ( τ) τ. = () = h f d Then T E h T = = By he definiion of Laplace ransform F ( s) L h( ) F ( s) s Hence:

5 Soluion of Inegro-Differenial Equaions 5 T F F F T = = = = Theorem (-) (shif): Le f ( ) A wih ELzaki ransform T ( ).Then: a E e f () = T a a From definiion of ELzaki ransform we hae: a ( a) E e f = f e d w = a dw = a d, () Le Then: w w f e dw = T a a a a Theorem (-4) (conoluion): Le () F s and Gs () and ELzaki ransforms M ( ) and N(). Then ELzaki ransform of he Conoluion f and g be defined in A haing Laplace ransforms of f and g ( f * g) ( ) = f ( ) g( τ ) dτ Is gien by: E ( f * g) ( ) = M N The Laplace ransform of ( f * g ) is gien by: L ( f * g) = F( s) G( s) By he dualiy relaion ( 4 ) we hae: E ( f g) = L ( f g) * *, and since M = F, N = G Then E ( f * g) ( ) = F. G = M N. = M N Applicaions o Inegral Equaions and Inegro-Differenial Equaions Inegral equaion is an equaion haing he form. b y ( ) F( ) k ( u, ) y ( u) du = + a

6 6 Tarig. M. Elzaki and Salih M. Elzaki Where F ( ) and K ( u, ) are known, abare, eiher gien consans or funcion of, and he funcion y ( ) which appears under he inegral sign is o be deermined. The funcion k ( u, ) is ofen called he kernel of he inegral equaion. If a and b are consans, he equaion is ofen called a Fredholm inegral equaion. If a is consan while b =, i is called a Volerra inegral equaion. In his work, we show how ELzaki ransform mehod can be applied successfully o sole, he condiion inegral equaions. This mehod is simple and sraigh for ward, and can be illusraed by examples: To sole he conoluion inegral equaion of he form: = + ( ) f h λ g τ f τ dτ (8) We ake he ELzaki ransform of his equaion o obain. T = h + λ Ε g( τ) f ( τ) dτ Which is, by he conoluion heorem we hae: λ T = h + T g Where ha T () is Elzaki ransform of he funcions f (),hen: h ( ) T ( ) = λ g Inersion gies he formal soluion in he form: h ( ) f ( ) = T λg ( ) In many simple cases, he righ-hand side can be inered by using parial fracion or any mehod.hence, he soluion can readily be found. Example () Consider he following linear inegral equaion, which is soled by ELzaki ransform. y ( ) = + ( u) y ( u) du 6 (9) Applying ELzaki ransform of his equaion we hae

7 Soluion of Inegro-Differenial Equaions 7 T 6 T T 6 = = + Or T = = () By using inerse ELzaki ransform of equaion () we obain he soluion y ( ) = sin + sin h Example () Consider he following boundary alue ype inegro-differenial equaion y ( u) y ( u) du = 4, y = () Being in a similar way wih he firs example, we obain T 5. T = T = 44 or T = ± () a a Inering equaion () and using he relaion Ε =Γ a+ + ± 6 We obain he soluion in he form y() = = ±.. π π Example () Sole he inegro-differenial equaion () δ () ( τ) ( τ) τ f = + f cos d, f = () Where δ () is a direc dela funcion. On using ELzaki ransform for eq () we hae T f = + T + 4 T = T = + +

8 8 Tarig. M. Elzaki and Salih M. Elzaki Apply he inerse ELzaki ransform o find he Soluion of () in he form f = + Example (4) Find he soluion of he inegral equaion: c ( τ ) f = a e c f e dτ (4) n b () ( τ ) Where abc,, are consans. Soluion: Taking ELzaki ransform, we obain n+ c T = an! T. + b c T n+ n+ n+ c = an! And T = an! can! + c + b + b c c n+ n+ c T = an! can! b + b b c Inersion yields he soluion of (4) can! c c f () = a + e n+! b b n n + b Example (5) Sole he following linear inegro-differenial equaion. x i x x y = x+ xe + e + y x y d (5) Wih he iniial Condiions y =, y =, y =, y = (6) By using ELzaki ransform of Eq (5) we hae: T y y 4 y y = T T (7) Subsiuing Eq (6) ino Eq(5) we ge:

9 Soluion of Inegro-Differenial Equaions T = T = + ( ) ( ) Then by using inerse ELzaki ransform we obain he Soluion in he form of x y x = + xe Example (6) Consider he following inegro-differenial equaion. () f = asin + f τ sin τ dτ, f() = (8) Taking ELzaki ransform of Eq (8) we ge: a T ( ) T ( ) = + f + + Using he condiion f = o obain: T a = + + Or T ( ) = a ( ) Inersion yields he soluion of (8) in he form: f = ae Conclusion In his sudy we inroduced new inegral ransform o sole he Inegro-differenial equaions. I is shown ha ELzaki ransform is a ery efficien ool for soling Inegro-differenial equaion in he bounded domains. References [] Tarig M. Elzaki,The New Inegral Transform Elzaki Transform Global Journal of Pure and Applied Mahemaics, ISSN ,Number (), pp [] Tarig M. Elzaki & Salih M. Elzaki, Applicaion of New Transform Elzaki Transform o Parial Differenial Equaions, Global Journal of Pure and Applied Mahemaics, ISSN ,Number (), pp

10 Tarig. M. Elzaki and Salih M. Elzaki [] Tarig M. Elzaki & Salih M. Elzaki,On he Connecions Beween Laplace and Elzaki ransforms, Adances in Theoreical and Applied Mahemaics, ISSN Volume 6, Number (),pp. -. [4] Tarig M. Elzaki & Salih M. Elzaki,On he Elzaki Transform and Ordinary Differenial Equaion Wih Variable Coefficiens, Adances in Theoreical and Applied Mahemaics. ISSN Volume 6, Number (), pp [5] Tarig M. Elzaki, Adem Kilicman, Hassan Elayeb. On Exisence and Uniqueness of Generalized Soluions for a Mixed-Type Differenial Equaion, Journal of Mahemaics Research, Vol., No. 4 () pp [6] Tarig M. Elzaki, Exisence and Uniqueness of Soluions for Composie Type Equaion, Journal of Science and Technology, (9). pp [7] Lokenah Debnah and D. Bhaa. Inegral Transform and Their Applicaion Second Ediion, Chapman & Hall /CRC (6). [8] A.Kilicman and H.E.Gadain. An applicaion of double Laplace ransform and Sumudu ransform, Lobacheskii J. Mah. () (9), pp.4-. [9] J. Zhang, A Sumudu based algorihm for soling differenial equaions, Comp. Sci. J. Moldoa 5() (7), pp -. [] Hassan Elayeb and Adem kilicman, A Noe on he Sumudu Transforms and Differenial Equaions, Applied Mahemaical Sciences, VOL, 4,, no.,89-98 [] Kilicman A. & H. ELayeb. A Noe on Inegral Transform and Parial Differenial Equaion, Applied Mahemaical Sciences, 4() (), PP.9-8. [] Hassan ELayeh and Adem kilicman, on Some Applicaions of a New Inegral Transform, In. Journal of Mah. Analysis, Vol, 4,, no., -.