Non-Standard Crank-Nicholson Method for Solving the Variable Order Fractional Cable Equation

Transcription

1 Appl. Mah. Inf. Sci. 9 No (5 943 Applied Mahemaics & Informaion Sciences An Inernaional Journal hp://dx.doi.org/.785/amis/944 Non-Sandard Crank-Nicholson Mehod for Solving he Variable Order Fracional Cable Equaion N. H. Sweilam and T. A. Assiri. Deparmen of Mahemaics Faculy of Science Cairo Universiy Giza Egyp. Deparmen of Mahemaics Faculy of Science Um-Alqura Universiy Saudi Arabia. Received: 5 Jun. 4 Revised: 3 Sep. 4 Acceped: 5 Sep. 4 Published online: Mar. 5 Absrac: In his paper a non-sandard Crank-Nicholson finie difference mehod (NSCN is presened. NSCN is used o sudy numerically he variable-order fracional Cable equaion where he variable order fracional derivaives are described in he Riemann- Liouville and he Grünwald-Lenikov sense. The sabiliy analysis of he proposed mehods is given by a recenly proposed procedure similar o he sandard John von Neumann sabiliy analysis. The reliabiliy and efficiency of he proposed approach are demonsraed by some numerical experimens. I is found ha NSCN is preferable han he sandard Crank-Nicholson finie difference mehod (SCN. Keywords: Non-sandard finie difference mehod Crank-Nicholson mehod; Variable order fracional Cable equaion; Von Neumann sabiliy analysis. Inroducion In modeling neuronal dynamics he Cable equaion is one of he mos fundamenal equaions. Due o is significan deviaion from he dynamics of Brownian moion he anomalous diffusion in biological sysems can no be adequaely described by he radiional Nerns-Planck equaion or is simplificaion he Cable equaion was inroduced for modeling he anomalous diffusion in spiny neuronal dendries (Henry e al. in. The resuling governing equaion he so-called fracional Cable equaion which is similar o he radiional cable equaion excep ha he order of derivaive wih respec o he space or ime is fracional 5. In recen years considerable ineres in fracional calculus has been simulaed by he applicaions ha finds in numerical analysis and differen areas of physics and engineering possibly including fracal phenomena (see The applicaions range from conrol. Variable-order differenial equaions have been considered in 74. In his sense he orders are funcion in any variable i.e. space variables ime variable or any oher variables. Samko and Ross 5 firs proposed he concep of variable order operaor and invesigaed he properies of variable order inegraion and differeniaion operaors of Riemann-Liouville ype. Differen auhors have inroduced differen definiions of variable order -differenial operaors each of hese wih a specific meaning o sui desired goals mos of hese definiions were exension o he fracional calculus definiions as Riemann-Liouville Grünwald Capuo and Riesz (45475 and some no as Coimbra definiion 7 8. The variable order differenials are imporan ool o sudy some sysems such as he conrol of nonlinear viscoelasiciy oscillaor for more deails see (7 8 and he references sied herein where he order changes wih respec o a parameer or more parameers. The main aim of his work is o presen he efficien numerical mehod; NSCN which is more accurae han SCN. Then we used NSCN o sudy numerically he variable order Cable equaion for more deails on non-sandard finie difference mehod see 8 3. The paper is organized as follows: In Secion some well-known mahemaical preliminaries on variable-order fracional differenial equaions and non-sandard discreizaion are given. in Secion 3 discreizaion of he variable order fracional Cable equaion using NSCN mehod wih shifed Grünwald formula is given. In Secion 4 we sudy he sabiliy of he presened mehod. Corresponding auhor nsweilam@sci.cu.edu.eg rieda8@gmail.com c 5 NSP

2 944 N. H. Sweilam T. A. Assiri: Non-Sandard Crank-Nicholson Mehod for Solving he... In Secion 5 numerical experimens of a ypical variable order fracional cable problem are presened. Finally in Secion 6 we give some conclusions. Preliminaries and Noaions In he following we presen some definiions of he variable-order fracional differenial operaors. Moreover some fundamenal conceps of NSFD for he soluion of parial differenial equaions are presened. Definiion The Riemann-Liouville variable order fracional derivaive is defined as 7: Dx α(x f(x= Γ(n α(x dx n d n x> where n < α(x<n. x f(xτ dτ (x τ α(x n+ ( Definiion The Grünwald-Lenikov variable order fracional derivaive is defined as: Dx α(x x/h f(x= lim h h α(x k= w (α(x k f(x hk x ( where x/h means he ineger par of x/h and w (α(x k are he normalized Grünwald weighs which are defined by w (α(x k =( k( α(x k. The Grünwald-Lenikov definiion is simply a generalizaion of he ordinary discreizaion formula for ineger order derivaives. The Riemann-Liouville and he Grünwald-Lenikov approaches coincide under relaively weak condiions; if f(x is coninuous and f (x is inegrable in he inerval x hen for every order < α(x < boh he Riemann-Liouville and he Grünwald-Lenikov derivaives exis and coincide for any value inside he inerval x.. Non-sandard Discreizaion The non-sandard finie difference (NSFD schemes were firsly proposed by Mickens (8-3 eiher for ordinary differenial equaions (ODEs or parial differenial equaions (PDEs. In his par we would like o inroduce several commens relaed o NSFD schemes. A scheme is called nonsandard if a leas one of he following condiions is saisfied: - Nonlocal approximaion is used. - Discreizaion of derivaive is no radiional and use a nonnegaive funcion. The forward Euler mehod is one of he simples discreizaion schemes. In his mehod he derivaive erm dy d is replaced by y(+h y( h where h is he sep size. However in he Mickens schemes his erm is replaced by where φ(h is a coninuous funcion of sep y(+h y( φ(h size h and he funcion φ(h saisfies he following condiions: φ(h=h+o(h <φ(h< h. Examples of funcions φ(h ha saisfy hese condiions are 7: φ(h=h sinhh e h e λh λ ec.... Noe ha in aking he lim h o obain he derivaive he use of any of hese φ(h will lead o he usual resul for he firs derivaive dy d = lim y+ φ (h y( y(+ h y( = lim. h φ (h h h In addiion o his replacemen if here are nonlinear erms in he differenial equaion 7 hese are replaced by { y yn y n+ y n y n. In dimensions wo and above nonlinear erms such as y(x( are eiher replaced by { yn x yx n+ y n+ x n. One can say ha here is no appropriae general mehod o choose he funcion φ(h or o choose which nonlinear erms are o be replaced some special echniques may be found in 8 and 9. 3 Discreizaion of he variable order fracional Cable equaion In his secion we will use NSCN mehod wih shifed Grünwald formula o obain he discreizaion finie of he iniial-boundary value problem of he variable order fracional Cable equaion 9: u(x = D β(x u(x x wih iniial and boundary condiions: µ D α(x u(x+ f(x (3 u(x= x L (4 u(= u(l= max (5 where < β(x α(x < µ > is a consan and D γ(x is he variable order fracional derivaive defined by he Riemann-Liouville operaor of order γ(x where γ(x is equal o β(x or α(x. Le us assume ha he coordinaes of he mesh poins are x n = nh n=...n; m = mτ m=...m c 5 NSP

3 Appl. Mah. Inf. Sci. 9 No (5 / where h = L/N τ = max /M and N M are he oal number of spaial nodes and he number of ime seps respecively. On he grid(x n m Eq. (3 can be wrien as u(x n m = D β(x n m u(x n m x µd α(x n m u(x n m + f(x n m. The Grünwald-Lenikov definiion is imporan for our purposes in his paper because i allows us o esimae numerically in a simple and efficien way. From he relaionship beween he Riemann-Liouville and he Grünwald-Lenikov hen Riemann-Liouville fracional parial derivaive of order γ can be expressed as follows: D γ /τ u(x= lim τ γ τ w (l u(x lτ. (6 According o I. Podlubny in 3 he righ-hand side of equaion (6 is approximaed by: m /τ lim τ τγ m /τ τ γ hence Eq. (6 yields o w (l u(x n m lτ= w (l u(x n m lτ+o(τ p (7 D γ u(x n m =τ γ m w (l u(x n m l +O(τ p. (8 This formula is no unique because here are many differen choices for w (l ha lead o approximaions of differen orders p (63. The definiion w (l =( l γ = l ( l( γ ( γ ( γ l+ l! provides order p=. Now we apply he Mickens discreizaion scheme o he equaion (3 by replacing he sep size h by a funcion of h φ(h and he sep size τ by a funcion of τ ψ(τ. The lef hand side of Eq. (3 is approximaed i.e. u(x n m = um n um n (9 ϕ(τ and he second derivaive in he righ hand side is approximaed wih he average of he cenral difference scheme evaluaed a he curren and he previous ime sep u(x n m = u m x n um n+u m n+ + um φ(h n um n +u m n+. φ(h ( Also if f(x has firs-order coninuous derivaive f(x hen f(x n m = f(x n m +O(ϕ(τ. ( where φ(h and ψ(τ have he properies: ψ(τ=τ+ O(τ and φ(h=h + O(h 4. Subsiuing equaions (8-( ino Eq. (3 we obain: u m n u m n ϕ(τ φ(h D β u m n um n + um n+ + um n um n + u m n+ µd α u m n + fn m +O(ϕ(τ. ( Now using he definiion of he Grünwald-Lenikov fracional variable-order given in (8 we ge u m n um n ϕ(τ = φ(h ϕ(τβ +ϕ(τ β m w (βl µϕ(τ α m w (αl where m (um l n u m l n w (βl = w (βl (u m l n um l n u m l n + u m l n+ +u m l n+ + fn m n=...n ; m=...m =( l β l =( l( β ( β ( β l+ l! w (αl =( l α l =( l( α ( α ( α l+. l! (3 u n = n=...n (4 u m = um N = m=...m. (5 Afer doing some algebraic manipulaion o equaion (3 we obain ϕ(τβ φ(h w(β u m n+ ++ ϕ(τβ φ(h w(β +µϕ(τ α w (α u m n c 5 NSP

4 946 N. H. Sweilam T. A. Assiri: Non-Sandard Crank-Nicholson Mehod for Solving he... ϕ(τβ φ(h w(β u m n = (6 φ(h ϕ(τβ w (β + ϕ(τ β w (β um n+ + ϕ(τβ φ(h w(β ϕ(τβ φ(h w(β µϕ(τα w (α + φ(h ϕ(τβ w (β + ϕ(τ β w (β um n + ϕ(τ β φ(h ϕ(τβ φ(h (w(βl+ + w (βl+ u l n+ (w(βl+ + w (βl+ u m n + µϕ(τα w (αl+ u l n + ϕ(τ β φ(h (w(βl+ + w (βl+ u l n +ϕ(τ fn m. Le U m =u m um...um N hen Eqs. (4-(6 can be ransferred ino a marix form as follows: AU m = BU m + C l U l + f m b m (7 where C C C ; A; B are marices of order(n (N. The srucure of marix A is where ( a (m n = ϕ(τβ φ(h A= + ϕ(τβ φ(h w (β a (m a (m w (β N a(m N ā(m N N a(m N + µϕ(τ α w (α and n = n = N. The marix A is a sricly diagonally dominan marix hence A is a nonsingular marix. Thus he linear equaion sysem (7 has a unique soluion and so he numerical mehod (4-(6 is uniquely solvable. 4 Sabiliy analysis In his secion we sudy he sabiliy analysis of he NSCN scheme (6 for he free force case. Theorem The variable order fracional NSCN discreizaion using he shifed Grünwald esimaes applied o he variable order fracional Cable Eq. (3 and defined by (6 is uncondiional sable for < β(x < and <α(x<. Proof. For simpliciy le us assume ha φ = ϕ(τβ φ(h and ψ = µϕ(τ α hen we can rewrie Eq. (6 φ w(β φ w(β u m n = u m n++(+φ w (β (φ w (β + φ w (β um n+ + ( φ w (β φ w (β ψ w (α (φ w (β + φ w (β um n + φ (φ ( (w(βl+ w (βl+ φ (w(βl+ + w (βl+ u l n+ + w (βl+ + w (βl+ u l n. + ψ w (α u m n u m n + + ψ w (αl+ u l n + Assume ha u m n = ξ me iqnh (as assumed in he von Neumann sabiliy procedure hen we ge φ w(β ξ m e iq(n+h + (+φ w (β + ψ w (α ξ m e iqnh φ (φ w (β + φ w (β ξ m e iq(n+h w(β ξ m e iq(n h ( φ w (β φ w (β ψ w (α ξ m e iqnh (φ w (β + φ w (β ξ m e iq(n h φ (w(βl+ + w (βl+ ξ le iq(n+h (φ w (βl+ + w (βl+ + ψ w (αl+ ξ l e iqnh + φ (w(βl+ + w (βl+ ξ le iq(n h = c 5 NSP

5 Appl. Mah. Inf. Sci. 9 No (5 / Divided by e iqnh φ w(β ξ m e iqnh +(+φ w (β + ψ w (α ξ m φ w(β ξ m e iqh (φ w (β + φ w (β ξ m e iqh ( φ w (β φ w (β ψ w (α ξ m (φ w (β + φ w (β ξ m e iqh φ (w(βl+ (φ ( w (βl+ + w (βl+ ξ le iqh + w (βl+ + ψ w (αl+ ξ l + φ (w(βl+ + w (βl+ ξ le iqh = Now using e iθ = cosθ+ isinθ we have φ w(β (cos(qh+isin(qh+(+φ w (β + ψ w (α φ w(β (cos(qh isin(qhξ m (φ w (β + φ w (β (cos(qh+isin(qh+ ( φ w (β φ w (β ψ w (α + (φ w (β + φ w (β (cos(qh isin(qhξ m φ (w(βl+ + w (βl+ (cos(qh+isin(qh (φ w (βl+ + w (βl+ + ψ w (αl+ + φ (w(βl+ + w (βl+ (cos(qh isin(qhξ l =. Under some simplificaions we ge φ (w (βl+ + w (βl+ ( sin (qh/+ψ w (αl+ ξ l =. Insering he expression ξ m = η ξ m in he above equaion +ψ w (α ψ w (α (sin (qh/η ξ m + φ w (β (sin (qh/ξ m (φ w (β + φ w (β φ (w (βl+ + w (βl+ ( sin (qh/+ψ w (αl+ η l ξ m = η = divided by ξ m o obain he following formula of η: ψ w (α +ψ w (α (φ w (β + φ w (β (sin (qh/ + + φ w (β (sin (qh/ φ (w (βl+ + w (βl+ ( sin (qh/+ψ w (αl+ η l. +ψ w (α + φ w (β (sin (qh/ The mode will be sable as long as η so ψ w (α (φ w (β + φ w (β (sin (qh/ + +ψ w (α + φ w (β (sin (qh/ φ (w (βl+ + w (βl+ ( sin (qh/+ψ w (αl+ η l. +ψ w (α + φ w (β (sin (qh/ By considering η = and since hen +ψ w (α + φ w (β (sin (qh/ > ( ψ w (α φ w (β (sin (qh/+ ( ψ w (α (φ w (β + φ w (β (sin (qh/ φ (w (βl+ + w (βl+ (sin (qh/( l + +ψ w (α ψ w (α + φ w (β (sin (qh/ξ m (φ w (β + φ w (β (sin (qh/ξ m ψ w (αl+ ( l c 5 NSP

6 948 N. H. Sweilam T. A. Assiri: Non-Sandard Crank-Nicholson Mehod for Solving he... w (β w (β w (β w (β (w (βl+ + w (βl+ ( l w (α + w (α + w(αl+ ( l φ ψ sin (qh/ le S mα = w(αl+ ( l and S mβ = Since w (α w (β w (β (w(βl+ + w (βl+ ( l S mβ w(α + w (α + S mα. φ ψ sin (qh/ = w (β = w (α + S mα > w (β + S mβ > and φ ψ sin (qh/ is always posiive we can conclude ha he scheme is uncondiional sable. he numerical soluions of he proposed mehod wih differen values of α(x β(x. Moreover in Figs. and 4 show 3D-drawing of he absolue error Error of he numerical soluions wih differen values of α(x β(x. From he resuls displayed in he Table and in all he figures i is obvious ha he proposed mehod is an efficien and able o give numerical soluions coincide closely wih he exac soluions. u(x Numerical experimens In his secion we presen numerical example o illusrae he efficiency and he validaion of he proposed numerical mehod when i applied o solve numerically he variable order fracional Cable equaion. Example Consider he following iniial-boundary problem of he variable-order fracional nonlinear Cable equaion... Exac Soluion Approximae Soluion x Fig. : The behavior of he numerical and exac soluions a β(x=5 x 3 α(x= cos (x. u (x= D β(x u xx (x D α(x u(x+ f(x on a finie domain <x< wih max. The source erm is given by: ( f(x= + π β(x+ Γ(+β(x + α(x+ sin(πx Γ(+α(x 6 5 x 3 he iniial and boundary condiions are: u(x=; u(=u(= Absolue error 4 3 le ψ(τ=sinh(τ and φ(h=e h. and he exac soluion is: u(x= sin(πx x.6.8 Table shows he maximum error Error of proposed mehod beween he exac soluion and he numerical soluion a M = using differen values of β(x and α(x. In Tables and a comparison beween he NSCN and he sandared Crank-Nicholson (SCN soluions where he accuracy of he NSCN is beer han he SCN. Figs. and 3 show he behavior of he exac soluions and Fig. : The absolue error of he numerical soluion a β(x=5 x 3 α(x= cos (x. c 5 NSP

7 Appl. Mah. Inf. Sci. 9 No (5 / Table : The maximum error Error of he numerical soluions using differen values of β(x and α(x. N = 5 M = N = M = 5 N = M = N = M = 3 β(x α(x Error Error Error Error 9 x sin 5 (x 6.4e- 8.9e e e (x 4 (x 5 6+(x 5.3e-.e- 5.8e-3.97e-4 8+x (x 4.5e-.3e- 6.7e-3.39e-4 +cos (x sin 3 (x.4e-.4e- 5.98e-3 8.e-5 5+(x 3 9 sin 4 (x 5 9 (x +sin 3 (x e-.e- 5.79e-3 6.5e-4 5 (x e-.5e- 8.74e-3.7e-3 5 x 3 cos (x 4.64e-3 5.e-3.8e-4.6e-5 Table : The maximum error Error of he SCN mehod using differen values of β(x and α(x. N = 5 M = N = M = 5 N = M = N = M = 3 β(x α(x Error Error Error Error 9 x sin 5 (x 6.7e-.73e-.4e- 7.97e (x 4 (x 5 6+(x 5.89e-.87e-.36e- 8.8e-3 8+x (x 4 3.6e-.7e-.5e- 9.99e-3 +cos (x sin 3 (x 3.5e-.98e-.44e- 9.4e-3 5+(x 3 9 sin 4 (x 5 9 (x +sin 3 (x e-.96e-.4e- 9.e-3 5 (x e-.33e-.7e-.e- 5 x 3 cos (x.5e-.37e- 9.68e-3 6.9e-.9 x u(x Exac Soluion Approximae Soluion x Absolue error x.6.8 Fig. 3: The behavior of he numerical and exac soluions a β(x= +cos (x α(x= sin3 (x. Fig. 4: The absolue error of he numerical soluion a β(x= +cos (x α(x= sin3 (x. 6 Conclusions In his paper NSCN mehod wih shifed Grünwald esimae applied for solving variable order fracional Cable equaion. Special aenion is given o sudy he sabiliy of he mehod. The obained numerical resuls are presened and compared wih he exac and he SCN soluions. The comparison beween NSCN and SCN show ha NSCN more accurae han SCN. From his comparison we can conclude ha he numerical soluions are in good agreemen wih he exac soluions. All compuaions in his paper are performed using Malab programming. c 5 NSP

8 95 N. H. Sweilam T. A. Assiri: Non-Sandard Crank-Nicholson Mehod for Solving he... References A. A. Kilbas M. H. Srivasava J. J. Trujillo Theory and applicaions of fracional differenial equaions Elsevier San Diego 6. A. D. Benson W. S.Wheacraf M. M. Meerschaer The fracional-order governing equaion of Lévy moion Waer Resour Res A. Reynolds On he anomalous diffusion characerisics of membrane bound proeins Phys. Le. A C. F. Lorenzo T. T. Harley Iniialized fracional calculus Applied Mahemaics C. F. Lorenzo T. T. Harley Variable order and disribued order fracional operaors Nonlinear Dynamics C. Lubich Discreized fracional calculus SIAM J. Mah. Anal C. F. M. Coimbra On he selecion and Meaning of variable order operaors for dynamic modeling Inernaional Journal of Differenial Equaions. 8 C. F. M. Coimbra L. E. S. Ramirez A variable order consiuive relaion for viscoelasiciy Annalen der Physik C. M. Chang F. Liu K. Burrage Numerical analysis for a variable-order nonlinear Cable equaion J. Compuaional Applied Mahemaics 9-4. D. Ingman and J. Suzdalnisky Conrol of damping oscillaions by fracional differenial operaor wih imedependen order Compuer Mehods in Applied Mechanics and Engineering D. Ingman J. Suzdalnisky M. Zeifman Consiuive dynamic-order model for nonlinear conac phenomena Journal of Applied Mechanics I. B. Henry M. A. T. Langlands L. S. Wearne Fracional cable models for spiny neuronal dendries Physical Review Leers I. Podlubny Fracional differenial equaions Academic Press San Diego I. Turner P. Zhuang F. Liu V. Anh A fracionalorder implici difference approximaion for he space-ime fracional diffusion equaion ANZIAM J. 47(EMAC J. Quinana- Murillo B. S. Yuse An explici numerical mehod for he fracional Cable equaion Inernaional Journal of Differenial Equaions Volume Aricle ID 39 pages. 6 M. A. T. Langlands I. B. Henry L. S. Wearne Fracional Cable equaion models for anomalous elecrodiffusion in nerve cells: infinie domain soluions Journal of Mahemaical Biology N. H. Sweilam M. M. Khader H. M. Almarwm Numerical sudies for he variable-order nonlinear fracional wave equaion Frac. Calc. Appl. Anal N. H. Sweilam M. M. Khader M. Adel On he sabiliy analysis of weighed average finie difference mehods for fracional wave equaion Journal of Fracional Differenial Calculus N. H. Sweilam M.M. Khader M. Adel Numerical simulaion of fracional Cable equaion of spiny neuronal dendries Journal of Advanced Research acceped 3. N. H. Sweilam M. M. Khader A. M. Nagy Numerical soluion of wo-sided space-fracional wave equaion using finie difference mehod. J. Compu. Appl. Mah N. H. Sweilam A. M. Nagy Numerical soluion of fracional wave equaion using Crank-Nicholson mehod World Applied Sciences Journal 3 (Special Issue of Applied Mah: 7-75 N. H. Sweilam M. M. Khader T. A. Assiri Efficien numerical reamen for fracional parial differenial Equaions Nonlinear science leers A N. H. Sweilam T. A. Assiri Error analysis of explici finie difference approximaion for he space fracional wave equaions SQU Journal for Science N. H. Sweilam T. A. Assiri Numerical simulaions for he space-ime variable order nonlinear fracional wave equaion Journal of Applied Mahemaics 3 Aricle ID pages 3. 5 S. G. Samko B. Ross Inegraion and differeniaion o a variable fracional order Inegral Transforms and Special Funcions R.E. Mickens Exac soluions o a finie-difference model of a nonlinear reacion-advecion equaion: Implicaions for numerical analysis Numer. Mehods Parial Diff. Eq R.E. Mickens Applicaion of nonsandard finie difference schemes World Scienific Publishing Co. Pe. Ld.. 8 R.E. Mickens Nonsandard finie difference model of differenial equaions World ScienificSingapore R. E. Mickens Nonsandard finie difference schemes for reacions-diffusion equaions Numer. Mehods Parial Differenial Equaions Fracals R. E. Mickens A nonsandard finie difference scheme for a fisher PDF having nonlinear diffusioncompu. Mah. Appl R. E. Mickens A nonsandard finie-difference scheme for he Loka-Volerra sysemappl. Numer. Mah Nasser H. Sweilam Professor of numerical analysis a he Deparmen of Mahemaics Faculy of Science Cairo Universiy. He was a channel sysem Ph.D. suden beween Cairo Universiy Egyp and TU-Munich Germany. He received his Ph.D. in Opimal Conrol of Variaional Inequaliies he Dam Problem. He is he Head of he Deparmen of Mahemaics Faculy of Science Cairo Universiy science May. He is referee and edior of several inernaional journals in he frame of pure and applied Mahemaics. His main research ineress are numerical analysis opimal conrol of differenial equaions fracional and variable order calculus bio-informaics and cluser compuing ill-posed problems. c 5 NSP

9 Appl. Mah. Inf. Sci. 9 No (5 / 95 Taghreed A. Assiri Ph.D. suden Faculy of Science Cairo Universiy. She received her M. Sc. degree in Mahemaics from Cairo Universiy. Her research ineres is Applied Mahemaics including Numerical Simulaions for Real-life Problems modeled by sysems of differenial equaions. She works as eaching assisance Faculy of Science Um-Alqura Universiy Saudi Arabia. c 5 NSP