Homotopy perturbation method for solving viral dynamical model

Transcription

1 C.Ü. Fen-Edebiyat Fakültesi Fen Bilimleri Dergisi ()Cilt Say Homotopy perturbation method for solving viral dynamical model ehmet ERDA¹ ve Tahir KHAYEV² ¹Gümühane University Engineering Faculty Civil Engineering 9 Gümühane Turkey ²TOBB University of economics and technology Faculty of Engineering Department of Industrial Engineering 656 Ankara Turkey ¹merdan@ktu.edu.tr ²khaniyevtahir@yahoo.com Received:..8 Accepted:..9 Abstract: In this article homotopy perturbation method is implemented to give approximate and analytical solutions of nonlinear ordinary differential equation systems such as viral dynamical model. The proposed scheme is based on homotopy perturbation method (HP) Laplace transform and Padé approximants. Some plots are presented to show the reliability and simplicity of the methods. Keywords: Padé approximants; Homotopy perturbation method; viral dynamical model. Viral Dinamik odel Çözümü için Homotopy Pertürbation Yöntemi Özet: Bu makalede viral dinamik model gibi lineer olmayan adi diferensiyel denklem sisteminin yaklak analitik çözümünü bulmak için homotopy perturbation yöntemi uyguland. Homotopy perturbation yöntemi temel alnarak Laplace dönüümü ve Padé yaklamlar uyguland. Yöntemleri doruluunu ve basitliini göstermek için baz grafikler sunuldu. Anahtar kelimeler: : Padé yakla; Homotopy perturbation yöntemi; viral dinamik model 65

2 . Introduction On the behavior of solution of viral dynamic model is examined at the study []. The components of the basic three-component model are uninfected CD4+ T-cells infected cells and free virus particles are denoted respectively by xt () yt () and vt (). These quantities satisfy dx s -x -xv dt dy xv-y dt dv cy-v dt with initial conditions: x() y() v(). (.) The motivation of this paper is to extend the application of the analytic homotopy-perturbation method (HP) and variational iteration method [ 5] to solve the a three-species food chain model (.). The homotopy perturbation method (HP) was first proposed by Chinese mathematician He [8-9-5]. The first connection between series solution methods such as an Adomian decomposition method and Padé approximants was established in. The transmission and dynamics of HTLV-I feature several biological characteristics that are of interest to epidemiologists mathematicians and biologists see for example [-6] etc. Like HIV HTLV-I targets CD4+ T-cells the most abundant white cells in the immune system decreasing the body s ability to fight infection. Padé approximaton A rational approximation to f( x ) on ab is the quotient of two polynomials P ( x) and Q ( x ) of degrees and respectively. We use the notation R ( x ) to denote this quotient. The R ( x ) Padé approximations to a function f( x ) are given by [] 66

3 R P( x) ( x) for a x b. Q ( x) (.) The method of Padé requires that f( x ) and its derivative be continuous at x. The polynomials used in (.) are P x p p x p x p x ( )... (.) ( )... (.) Q x q x q x q x The polynomials in (.) and (.) are constructed so that f( x ) and R ( x) agree at x and their derivatives up to agree at x. In the case Q ( x) the approximation is just the aclaurin expansion for f( x ). For a fixed value of the error is smallest when P ( x) and Q ( x ) have the same degree or when P ( x ) has degree one higher then Q ( x ). otice that the constant coefficient of Q is q. This is permissible because it notice be and R ( x ) is not changed when both P ( x) and Q ( x ) are divided by the same constant. Hence the rational function R ( x ) has unknown coefficients. Assume that f( x ) is analytic and has the aclaurin expansion f x a axax ax (.4) k ( )... k... and from the difference f( x) Q ( x) P ( x) Z( x): i i i i ax i qx i px i cx i i i i i (.5) The lower index j in the summation on the right side of (.5) is chosen because the first derivatives of f( x ) and R ( x ) are to agree at x. When the left side of (.5) is multiplied out and the coefficients of the powers of i x are set equal to zero for... linear equations: k the result is a system of 67

4 a q a a p q a q a a p q a q a q a a p q a q a a p and p q a q a... q a + a q a q a... q a + a..... q a q a... q a + a. (.6) (.7) otice that in each equation the sum of the subscripts on the factors of each product is the same and this sum increases consecutively from to. The equations in (.7) involve only the unknowns q q q... q and must be solved first. Then the equations in (.6) are used successively to find p p p... p []..Homotopy perturbation method To illustrate the homotopy perturbation method (HP) for solving non-linear differential equations He [8 9] considered the following non-linear differential equation: A( u) f( r) r (.) subject to the boundary condition u Bu r n (.) where A is a general differential operator B is a boundary operator f(r) is a known analytic function is the boundary of the domain and n denotes differentiation along the normal vector drawn outwards from. The operator A can generally be divided into two parts and. Therefore (.) can be rewritten as follows: ( u) ( u) f( r) r (.) He [8 9] constructed a homotopy v( r p): x which satisfies 68

5 H( v p) ( p) () v ( u ) p A() v f() r (.4) which is equivalent to H( v p) () v ( u ) p( v ) p () v f() r (.5) where p is an embedding parameter and u is an initial approximation of (.5). Obviously we have Hv ( ) v ( ) u ( ) Hv ( ) Av ( ) fr ( ). (.6). The changing process of p from zero to unity is just that of H(vp) from () v ( v )to A() v f() r. In topology this is called deformation and () v ( v )and A() v f() r are called homotopic. According to the homotopy perturbation method the parameter p is used as a small parameter and the solution of Eq. (.4) can be expressed as a series in p in the form v v pv p v p v (.7)... When p Eq. (.4) corresponds to the original one Eqs. (.) and (.7) become the approximate solution of Eq. (.) i.e. u lim v v v v v... (.8) p The convergence of the series in Eq. (.8) is discussed by He in [8 9]. 4. Applications In this section we will apply the homotopy perturbation method to nonlinear ordinary differential equation systems (.). 4. Homotopy perturbation method to viral dynamic model According to homotopy perturbation method we derive a correct functional as follows: pv x pv s v vv pv y pv vv v pv v pv cv v (4.) where dot denotes differentation with respect to t and the initial approximations are as follows: 69

6 v ( t) x ( t) x() v ( t) y ( t) y() v () t v () t v(). (4.) and v v pv pv pv... v v pv pv pv... v v pv pv pv... (4.) Where vi j i j... are functions yet to be determined. Substituting Eqs.(4.) and (4.) into Eq. (4.) and arranging the coefficients of p powers we have v v ( v v v v ) p... v pv v v v p v ( v v v v ) v p... v c pv cv v p v cv v p... v s p v v v v p (4.4) In order to obtain the unknowns vi j( t) i j we must construct and solve the following system which includes nine equations with nine unknowns considering the initial conditions v () i j i j v s v v v v v v ( v v v v ) v v v v v v ( v v v v ) v v c v cv v v cv v. From Eq. (.8) if the three terms approximations are sufficient we will obtain: (4.5) 7

7 x() t lim v () t v () t k p k y() t lim v () t v () t p p k k v() t lim v () t v () t k k (4.6) therefore x() t s t s c s t + t 6 y() t t s c s c c s c s s c c s t c c s 6 v() t c t c s c s c s c c t ccc s c t 6 c c Table Variables and parameters for contagion s the (assumed constant) rate of production of CD4+ T-cells.7 their per capita death rate.6 xy the rate of infection of CD4+ T-cells by virus.7 the per capita rate of disappearance of infected cells. c the rate of production of virions by infected cells 5 the death rate of virus particles t (4.7) 7

8 This was done with the standard parameter values given above and initial values and for the three-component model. A few first approximations for xt () yt () and vt () are calculated and presented below: Three terms approximations: x( t) -.9t.6965t -.477t y t t t t ( ) v t t t t ( ) (4.8) Four terms approximations: x( t) -.9t.6965t t t 4 y t t t t t 4 ( ) v t t t t t 4 ( ) Five terms approximations: x( t) -.9t.6965t t t t t 4 y( t).7t-.4485t.7566t t v( t) - t.675t -.788t t t Six terms approximations: t.58 t x( t) -.9t.6965t t t 4 y( t).7t-.4485t.7566t t t t t 4 v( t) - t.675t -.788t t t 4 5 (4.9) (4.) (4.) In this section we apply Laplace transformation to (4.) which yields Lxs () 4 5 s s s s s s s 7

9 L ys () 4 s s s s s s s s s s s s s () 4 L vs (4.) For simplicity let s ; then t L( x( t)) t-.9 t +.58t t t t t 6 7 L( y( t)).7t t t t t t 6 7 (4.) L( v( t)) t t +5.5t t t t +4.77t 6 7 Padé approximant 4/4of (4.) and substituting t By using the inverse Laplace transformation we obtain s we obtain 4/4 in terms of s. x( t) e.787e t t e e t 4.579t y( t) e e * t.66584t e t (4.4) v( t) e e t t e e.6645t t These results obtained by Padé approximations for xt () yt () and vt () are calculated and presented follow. 7

10 5 x uninfected CD4+ T-cells x 4 x -4 infected cells y 5.5. x 57 - free virus particles -4 v Figure.. Plots of Padé approximations for viral dynamical model.5. These results obtained by homotopy perturbation method three four five and six terms approximations for xt () yt () and vt () are calculated and presented follow. 74

11 5 Three terms approximations 5 Four terms approximations 5 x y v 5 x y v -5 Five terms approximations - x y v - -5 Six terms approximations 5 5 x y v -5 Figure.. Plots of three four five and six terms approximations for viral dynamic model 5. Conclusions In this paper homotopy perturbation method was used for finding the solutions of nonlinear ordinary differential equation systems such as viral dynamical model. We demonstrated the accuracy and efficiency of these methods by solving some ordinary differential equation systems. We use Laplace transformation and Padé approximant to obtain an analytic solution and to improve the accuracy of homotopy perturbation method. We apply He s homotopy perturbation method to calculate certain integrals. It is easy and very beneficial tool for calculating certain difficult integrals or in deriving new integration formula. The computations associated with the examples in this paper were performed using aple 7 and atlab 7 75

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