Applied Mathematical Scieces, Vol. 8, 2014, o. 66, 3255-3261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.45347 A Taylor Series Based Method for Solvig a Two-dimesioal Secod-order Equatio Samaeh Afshar Departmet of Mathematics, Islamic Azad Uiversity Cetral Tehra Brach, Tehra, Ira Copyright c 2014 Samaeh Afshar. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, we focus o the two-dimesioal liear Telegraph equatio with some iitial ad boudary coditios. We trasform the model of partial differetial equatio (PDE) ito a system of first order, liear, ordiary differetial equatios (ODEs). Our method is based o fidig a solutio i the form of a polyomial i three variables U (x, y, t) = i=0 k=0 U(i, j, k)xi y j t k with udetermied coefficiets U(i, j, k). The mai idea of our process is based o the differetial trasformatio method (DTM). Keywords: Liear hyperbolic equatio; two-dimesioal Telegraph equatio; Differetial Trasformatio method 1 Itroductio I this paper we focus o the followig diffusio equatio i two space variables 2 u t +2α u 2 t + β2 u = δ 2 u x + γ 2 u + f(x, y, t), 0 <x<1, 0 <y<1, t>0, 2 y2 (1) while the iitial ad boudary coditios are u(x, y, 0) = p(x, y), u(x,y,0) = m(x, y),u(0,y,t)=q(y, t),u(x, 0,t)=h(x, t), t u(1,y,t)=r(y, t),u(x, 1,t)=g(x, t), 0 t T, 0 x 1, (2)
3256 Samaeh Afshar The fuctios f(x, y, t), p(x, y), q(y, t),h(x, t), r(y, t), g(x, t) ad m(x, y) are kow fuctios ad the costats α, β, δ ad γ are kow costats. For α>0, β = 0 ad δ = γ = 1, Eq. (1) represets a damped wave equatio ad for α > β > 0 ad δ = γ = 1, is called telegraph equatio. Dig ad Zhag[9], preseted a three level compact differece scheme of O(τ 4 + h 4 ) for the differece solutio of problem (1) ad (2). A efficiet approach for solvig the two dimesioal liear hyperbolic telegraph equatio by the compact fiite differece approximatio of fourth order ad the collocatio method have bee developed i [18]. Authors of [10, 11, 12] also studied Eq. (1) with other methods. For solvig these kids of equatios there are several methods, such as differetial trasformatio method [2, 3, 6, 4, 12, 11, 7, 8, 10, 21, 22, 23, 24, 25], Tau method [19, 20] ad homotopy perturbatio method [13]. A ew matrix formulatio techique with arbitrary polyomial bases has bee proposed for the umerical/aalytical solutio of the heat equatio with olocal boudary coditio [1] ad Two matrix formulatio techiques based o the shifted stadard ad shifted Chebyshev bases are proposed for the umerical solutio of the wave equatio with the o-local boudary coditio [5]. 2 Three-dimesioal differetial trasform Cosider a fuctio of two variables w(x, y, t), ad suppose that it ca be represeted as a product of two sigle-variable fuctios, i.e., w(x, y, t) = ϕ(x)φ(y)ψ(t). The the fuctio w(x, y, t) ca be represeted as w(x, y, t) = W (i, j, k)x i y j t k. (3) i=0 k=0 where W (i, j, k) is called the spectrum of w(x, y, t). Now we itroduce the basic defiitios ad operatios of three-dimesioal DT as follows[8]. Defiitio 2.1. Give a w fuctio which has three compoets such as x, y, t. Three-dimesioal differetial trasform of w(x, y, t) is defied [ ] W (i, j, k) = 1 i+j+k w(x, y, t), i!j!k! x i y j t (4) k x=x 0,y=y 0,t=t 0 where the spectrum fuctio W (i, j, k) is the trasformed fuctio, which is also called the T-fuctio. let w(x, y, t) as the origial fuctio while the uppercase W (i, j, k) stads for the trasformed fuctio. Now we defie The
A Taylor series based method 3257 differetial iverse trasform of W (i, j, k) as followig: w(x, y, t) = W (i, j, k)(x x 0 ) i (y y 0 ) j (t t 0 ) k. (5) i=0 k=0 Usig Eq. (4) i (5), we have w(x, y, t) = i=0 k=0 1 i!j!k! = i=0 k=0 W (i, j, k)xi y j t k. [ ] i+j+k w(x, t) x i y j t k x i y j t k x=x 0,y=y 0,t=t 0 Now we give the fudametal theorems for the three-dimesioal case of DTM by usig the followig theorem Theorem 2.1. Assume that W (i, j, k), U(i, j, k) ad V (i, j, k) are the differetial trasforms of the fuctios w(x, y, t), u(x, y, t) ad v(x, y, t), respectively; the: 1 If w(x, y, t) =u(x, y, t)±v(x, y, t), the W (i, j, k) =U(i, j, k)±v (i, j, k), 2 If w(x, y, t) =cu(x, y, t), where c R, the W (i, j, k) =cu(i, j, k) 3 If w(x, y, t) = u(x, y, t), the W (i, j, k) =(k +1)U(i, j, k +1) t 4 If w(x, y, t) = r+s+m u(x, y, t), the x r y s t m W (i, j, k) = (i+r)!(j+s)!(k+m)! U(i + r, j + s, k + m) i!j!k! Proof. See [8]. 3 Reformulatio of the problem I this sectio, we covert the problem (1) ad (2) ito a system of first order, liear, ordiary differetial equatio. I Eqs. (1) ad (2), the fuctios f(x, y, t), p(x, y), g(x, y), h(x, t), r(y, t) ad m(t) geerally are ot polyomials. We assume that these fuctios are polyomial or they ca be approximated by polyomials to ay degree of accuracy. The if we suppose that f(x, y, t) i=0 k=0 F (i, j, k)xi y j t k,p(x, y) i=0 P (i, j)xi y j, g(x, y) i=0 G(i, j)xi y j,h(x, t) i=0 k=0 H(i, k)xi t k, r(y, t) k=0 R(i, k)yj t k,m(x, y) i=0 M(i, j)xi y j, (7) Therefore we cosider approximate solutio of the form (6) U (x, y, t) = U(i, j, k)x i y j t k, (8) i=0 k=0
3258 Samaeh Afshar where U (x, y, t) is the approximatio of u(x, y, t). If we fid the values of U(i, j, k), for i, j, k =0, 1, 2,...,, the U (x, y, t) ca be foud by usig Eq. (7). To fid these ukows, we proceed as follows. By utilizig Theorem 2.1 ad Eqs. (7),(8) ito Eqs. (1) ad (2), we get (k + 1)(k +2)U(i, j, k +2)+α(k +1)U(i, j, k +1)+βU(i, j, k) =δ(i + 1)(i +2) U(i +2,j,k)+γ(j + 1)(j +2)U(i, j +2,k)+F (i, j, k) i, j, k =0, 1, 2,...,. i=0 U(i, j, 0)xi y j = i=0 P (i, j)xi y j,i,, 1,...,. k=0 U(0,j,k)yj t k = k=0 Q(j, k)yj t k,j,k=0, 1,...,. i=0 k=0 U(i, 0,k)xi t k = i=0 k=0 H(i, k)xi t k, i,k =0, 1,...,. i=0 k=0 U(i, j, k)yj t k = k=0 R(j, k)yj t k,j,k=0, 1,...,. i=0 k=0 U(i, j, k)xi t k = i=0 k=0 G(i, k)xi t k,i,k=0, 1,...,. i=0 U(i, j, 1)xi y j = M(i, j), i,j =0, 1,...,. (9) Therefore, to fidig the values of U(i, j, k), for i, j, k =0, 1, 2,...,, we must costruct a system of ( + 1)( + 1)( + 1) equatios. To this ed we arrage the obtaied liear equatios from the previous sectio. Because the problems (1) ad (2) have bee ot defied for etire of the rage, we select oly the equatios that satisfy i the defied rage. Firstly, the ( +1) 2 values of U ca be obtaied from as follows: U(i, j, 0) = P (i, j), i,, 1,...,. U(0,j,k)=Q(j, k), j =0, 1,...,, k =1, 2,...,. U(i, 0,k)=H(i, k), i,k=1, 2,...,. U(i, j, 1) = M(i, j), i,j =0, 1,...,. (10) So, we ca obtai 4 2 +5 + 2 values of U idepedetly. The the 3 2 2 1 remaider values of U must be obtaied from the other equatios. We choose 2 2 9 equatios from he followig equatios i=0 U(i, j, k) =R(j, k), j =1,..., 1, k=2,..., 1, i=0 U(i, 1,k)=R(1,k), i=0 U(i,, k) =R(, k),k =2,..., 1, i=0 U(i, j, ) =R(j, ),j =2,..., 1, U(i, j, k) =G(i, k), i=1,..., 1, k=2,..., 1, U(1,j,k)=G(1,k), U(, j, k) =G(, k), k=2,..., 1, U(i, j, ) =G(i, ), i=2,..., 2, (11)
A Taylor series based method 3259 Fially, the remaider 3 3 2 2 + 8 equatios ca be foud i the form of (k + 1)(k +2)U(i, j, k +2)+α(k +1)U(i, j, k +1)+βU(i, j, k) = δ(i + 1)(i +2)U(i +2,j,k)+γ(j + 1)(j +2)U(i, j +2,k) +F (i, j, k), i,j =1, 2,..., 2, k=0, 1,..., 2, (k + 1)(k +2)U(0,j,k+2)+α(k +1)U(0,j,k+1)+βU(0,j,k) =2δU(2,j,k)+γ(j + 1)(j +2)U(0,j+2,k) +F (0,j,k), j =1, 2,..., 2, k=2,..., 2 (k + 1)(k +2)U(i, 0,k+2)+α(k +1)U(i, 0,k+1)+βU(i, 0,k) = δ(i + 1)(i +2)U(i +2, 0,k)+2γU(i, 2,k) +F (i, 0,k), i =1, 2,...,, k =2,..., 2. (12) Eqs. (10)-(12) give a liear algebraic system of 3 2 2 1 equatios that by solvig this system, the remaider values of U will be obtaied ad the from Eq. (13), we ca obtaie U (x, y, t) that it is the approximatio of u(x, y, t). 4 Coclusios I this article, the solutio of the secod order two space dimesioal hyperbolic telegraph equatio has bee discussed by usig differetial trasformatio method. Covertig the model of PDE to a system of liear equatios, is the mai part of this paper. The computatioal difficulties of the other methods ca be reduced by applyig this process. Refereces [1] B. Soltaalizadeh, Numerical aalysis of the oe-dimesioal Heat equatio subject to a boudary itegral specificatio, Optics Commuicatios, 284 (2011), 2109-2112 [2] B. Soltaalizadeh, A. Yildirim, Applicatio of Differetial Trasformatio method for umerical computatio of Regularized Log Wave equatio, Z. fuer Naturforschug A., 67a (2012), 160-166. [3] B. Soltaalizadeh, Applicatio of Differetial Trasformatio Method for Numerical Aalysis of Kawahara Equatio, Australia Joural of Basic ad Applied Scieces, 5(12) (2011), 490-495. [4] B. Soltaalizadeh, Applicatio of Differetial Trasformatio method for solvig a fourth-order parabolic partial differetial equatios, Iteratioal joural of pure ad applied mathematics, 78 (3) (2012), 299-308.
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