Differential transform method to solve two-dimensional Volterra integral equations with proportional delays

Transcription

1 NTMSCI 5, No. 4, 65-7 (27) 65 New Treds i Mathematical Scieces Differetial trasform method to solve two-dimesioal Volterra itegral equatios with proportioal delays Şuayip Yüzbaşi ad Nurbol Ismailov Departmet of Mathematics, Faculty of Sciece, Akdeiz Uiversity, TR 758 Atalya, Turkey Received: 7 March 26, Accepted: 4 May 26 Published olie: 25 October 27. Abstract: I this paper, the differetial trasform method is exteded by providig a ew theorem to two-dimesioal Volterra itegral equatios with proportioal delays. The method is useful for both liear ad oliear equatios. If solutios of goverig equatios ca be expaded for Taylor series, the the method gives opportuity determie coefficiets Taylor series, i.e. the exact solutios are obtaied i series form. I illustrate examples the method applyig to a few type equatios. Keywords: Two-dimesioal Volterra itegral equatios with proportioal delays, partial differetial equatios, differetial trasform method. Itroductio I 897 by Vito Volterra [] cosidered itegral equatios which limits of itegratio variable ad limits represets a proportioal delays vaishig at t =. Volterra preceded the aalysis of the existece ad uiqueess of the solutio. I 927 ad 937 papers, o populatio dyamics Volterra studied itegro-differetial equatios with delays. We ca see plety of moographs ad papers devoted for Volterra fuctioal equatios ad their applicatios. For example, the oliear Volterra itegral ad itegro-differetial equatios with delays are described models i epidemiology ad populatio growth [2,3,4,5,6,7,8]. There are may authors has studied umerical aalysis of Volterra itegral ad itegro-differetial equatios, for example, the collocatio methods for Volterra itegral ad itegro-differetial equatios with proportioal delays were first studied i detail i Bruer [9], Zhag [], Takama [], Belle [2]. Yüzbaşı [3] has applied Laguerre polyomials for patograph-type Volterra itegro-differetial equatios. The systems of Volterra itegral equatios with variable coefficiets has bee solvig by Bessel poliomials i [4]. I additio, the homotopy perturbatio method [5], the variatioal iteratio method [6], the Galerki method [7], the Adomia decompositio method [8] ad theirs various modified methods has bee used for solvig above metioed equatios. I this paper, we cosider the two-dimesioal Volterra itegral equatios with proportioal delays the followig forms: u(x,t)= f (x,t)+g(x,t), () Correspodig author syuzbasi@akdeiz.edu.tr

2 66 S. Yuzbasi ad N. Ismailov: Extesio of the differetial trasform method to two-dimesioal Volterra... where f (x,t)= where r,r 2, p,q (,], h, f, v are give fuctios. t r t v(x,t) h(y,z)u(py,qz)dydz u(py,qz) v(y,z) dydz r t u(py,qz)dydz, The rest of this paper is arraged as follows. I sectio 2, the fudametal relatios ad two theorems are give for two-dimesioal differetial trasform method. I Sectio 3, we exted the differetial trasform method by the ew theorem for two-dimesioal Volterra itegral equatios with proportioal delays. We apply this method to some two-dimesioal Volterra itegral equatios with proportioal delays i Sectio 4. Sectio 5 cocludes this study with a brief summary. 2 Two-dimesioal differetial trasform The differetial trasform method is preseted by Pukhov [9] ad Zhou [2] i study of electric circuits. The mai idea of method is trasformed the give fuctioal equatios to differece equatios, ad by usig iitial coditios calculate the values of derivatives of fuctios at give poit. I recet years, the method have bee applyig a large class of problems, i particular, Tari et al. [2, 22] ad Jag [23] are applied for two-dimesioal Volterra itegro-differetial equatios. Suppose a fuctio u(x,t) is aalytic i the give domai D ad(x,t ) D. Defiitio. The two-dimesioal differetial trasform of fuctio u(x,t) at(x,t ) is defied as followig U(,m)=!m! [ +m ] u(x,t) x t m Defiitio 2. Differetial iverse trasform of U(,m) is defied as u(x,t)= = m=, (2) x=x t=t U(,m)(x x ) (t t ) m. (3) From the defiitios differetial trasform ad differetial iverse trasform it is easy to obtai the followig theorems. Theorem. Assume that U(,m) ad U i (,m)(i=,2) are the two-dimesioal differetial trasforms of the fuctios u(x,t) ad u i (x,t) at(,) respectively, the If u(x,t)=au (x,t)±bu 2 (x,t), the U(,m)=aU (,m)±bu 2 (,m), a ad b are real umbers. If u(x,t)=u (x,t)u 2 (x,t), the U(,m)= m l= k= U (k,l)u 2 ( k,m l). If u(x,t)= k+l v(x,t), the U(,m)=(+)(+2)...(+k)(m+)(m+2)...(m+l)V(+k,m+l). k x l t If u(x,t)= t v(y,z)dydz, the U(,)= U(,m)=, U(,m)= m V(,m ). Theorem 2. Assume that U(,m) ad U i (,m)(i=,2) are the two-dimesioal differetial trasforms of the fuctios u(x,t) ad u i (x,t) at(,) respectively, p,q, p i,q i (,], the

3 NTMSCI 5, No. 4, 65-7 (27) / 67 If u(x,t)=v(px,qt), the U(,m)= p q m V(,m). If u(x,t)=u (p x,q t)u 2 (p,q 2 t), the U(,m)= m l= k= p k pl 2 q l q m l 2 U (k,l)u 2 ( k,m l). If u(x,t)= k+l v(px,qt), the U(,m)=(+)(+2)...(+k)(m+)(m+2)...(m+l)p +k q m+l V(+k,m+l). k x l t The proofs of Theorems -2 ca be foud i [2,24]. 3 Mai results I this sectio, we preset the differetial trasform relatios that ca be used for solvig two-dimesioal Volterra itegral equatios with proportioal delays. Theorem 3. Assume that F(, m),u(, m) ad V(, m) are the two-dimesioal differetial trasforms of the fuctios f(x,t),u(x,t) ad v(x,t) at (,) respectively, p,q,r,r 2 (,], the: (a) If f(x,t)= r t (b) If f(x,t)= l+ ). t (c) If f(x,t)= v(x,t) u(py,qz)dydz, the F(,)=F(,m)=, F(,m)= m qm p r2 rm U(,m ). u(py,qz) v(y,z) dydz, the U(,m)= p q m r t u(py,qz)dydz, the m l= k= m l= k= r k 2 r l m ( k+)(m l+)v(k,l)f( k+,m V(k,l)F( k,m l)= m r (r q) m r 2 (r 2 p) U(,m ) Proof. (a) From defiitio differetial trasform we have F(, ) = F(, m) =, (, m =,, 2,...). Sice from Theorem -2 we have 2 f(x,t) (r ) (r t) = u(pr 2x,qr t), r r 2 (+)(m+)f(+,m+)=(pr 2 ) (qr ) m U(,m). (b) Aalogously to part(a), F(,)=F(,m)=,(,m=,,2,...). Sice u(r 2 px,r qt)= v(r,r t) 2 f(x,t), r r 2 x t usig differetial trasform of multiplicatio of fuctios ad Theorem 2, we have the followig: (r 2 p) (r q) m U(,m)= r r 2 (c) Sise v(x,t) f(x,t) = r t m l= k= r2 k rl ( k+)(m l+ )V(k,l)F( k+,m l+ ), (,m=,2,...). u(py, qz) dydz, usig differetial trasform for multiplicatio of fuctios from Theorem ad two-dimesioal itegral with proportioal delays from Theorem 3(a), we get ecessary equatio. By usig this theorem proved for solvig two-dimesioal itegral equatios, two-dimesioal itegral equatios will be solve usefully.

4 68 S. Yuzbasi ad N. Ismailov: Extesio of the differetial trasform method to two-dimesioal Volterra... 4 Illustrate examples I this sectio, usig differetial trasform ad relatios i Theorem 3, we get solutios i series form of itegral equatios (). Example. Let us cosider liear two-dimesioal Volterra itegral equatio with proportioal delays give by For this problem, f(x,t)= t u(x,t)=xt+ 2xt 2 8 x2 t 2 t 6 x2 t 3 + u( y 2,z)dydz. u( y 2,z)dydz ad g(x,t)=xt+ 2xt2 8 x2 t 2 6 x2 t 3. Usig differetial trasform of equatio, we have the followig U(,m)= m2 U(,m )+δ(,m )+2δ(,m 2) 8 δ( 2,m 2) 6 δ( 2,m 3), where δ is Kroeker symbol ad δ(,m) = δ()δ(m). U(,) = U(,m) = (,m =,,2,...), U(,) =, U(,2)=2, I other cases U(,m)=. Usig equatio(3) we get the exact solutio u(x,t)=xt+ 2xt 2. Example 2. We cosider the followig two-dimesioal Volterra itegral equatio with proportioal delays where This problem is give by f(x,t) = u(x,t)=e x+t 3t 4 ( e 3 2x )+ 2 t trasform to give equatio ad usig Theorem 3, we have Solvig recurrece equatios(4), we obtai 2 t u( y 3,z) e y+z dydz. u( y 3,z) e y+z dydz ad g(x,t) = e x+t 3t 4 ( e 3 ) i (). Now, applyig differetial U(,m)=F(,m)+!m! 3 4 δ(,m )+ 3 4 ( 2 3 ) δ(m ). (4)! U(,)=F(,)+++ U(,2)= F(,2)+ 2! + U(,)=F(,)++ U(,2)= F(,2)+ 2! U(,)=F(,) U(2,)= F(2,)+ 4 2! U(,)=F(,)++ U(2,2)= F(2,2)+ 2 2! 2! U(2,)= F(2,)+ + U(3,)= F(3,)+ 2! 3!2! + 3 ( 2 ) !...

5 NTMSCI 5, No. 4, 65-7 (27) / 69 where F(, m) defie from Theorem 3 F(,)=F(,m)= ad U(,m )= 3 2 m m Usig equatio(3), we get l= k= 2 l l!k! ( k)(m l)f( k,m l). u(x,t)=+x+ t+ xt+ 2! t2 + 2! x2 +!2! xt2 + 2!! x2 t+ 2!2! x2 t which is the Taylor series of fuctio u(x,t)=e x+t ad exact solutio of Example 2. Example 3. Cosider the followig two-dimesioal itegral equatio with proportioal delays u(x,t)=cos(x+ t) 8si x 2 si t 4 )+ cos( x 2 + t 4 ) The exact solutio of the problem is u(x, t) = cos(x + t). t/2 x/3 u( y 2,z)dzdy. Now, usig differetial trasform of last equatio, we have U(,m)=F(,m)+!m! cosπ 2 (+m) 8!m!2 4 m si π mπ si 2 2 (5) where F(, m) defie from followig relatios: m l= k= Usig(5) we have the followig relatios: cos π 2 (+m) k!l!2 k 4 l F( k,m l)= F(,)=F(,m)= (6) m3 2+m U(,m ). (7) U(,)=F(,)+ U(2,)=F(2,) 2 U(,)=F(,)+ U(,)=F(,)+ U(,)=F(,) U(,2)=F(,2) 2... Usig relatios(6 7) from Theorem 3 ad equatio(3), we gai U(,)=F(,2)+ U(2,)=F(2,)+ U(2,2)=F(2,2)+ 4 u(x,t)= xt 2 t x2 t which is the Taylor series of exact solutio of Example 3.

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