1 Commun. Theor. Phys. (Beijing China) 5 (009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 5 No. 6 December Exact Solutions of Fractional-Order Biological Population Model A.M.A. El-Sayed 1 S.Z. Rida and A.A.M. Arafa 1 Department of Mathematics Faculty of Science Alexandria University Alexandria Egypt Department of Mathematics Faculty of Science South Valley University Qena Egypt (Received May ; Revised September 7 009) Abstract In this paper the Adomian s decomposition method (ADM) is presented for finding the exact solutions of a more general biological s. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method some examples are provided. PACS numbers: 0.30.Jr Key words: biological populations model fractional Calculus decomposition method Mittag Leffler function 1 Introduction Most scientific problems and phenomena such as the diffusion of biological populations occur nonlinearly and the fractional order differential equations appear more and more frequently in different research areas and engineering applications. Except in a limited number of these problems we have difficulty in finding their exact analytical solutions. An effective and easy-to-use method for solving such equations is needed. However known methods have certain disadvantages. Some methods for fractional differential equations of the rational order [1 3 do not work in the case of an arbitrary real order. For other methods there is an iteration method [4 which allows solution of fractional differential equations of an arbitrary real order but it works effectively only for relatively simple equations. Therefore methods to search for approximate analytical solutions were introduced among which the Adomian decomposition method (ADM) [56 is the most effective and convenient one for both weakly and strongly nonlinear equations. This method has been widely used in applied science and it does not need any transformation or linearization. We solve the degenerate parabolic equation arising in the spatial diffusion of biological populations of fractional order with given initial conditions. In this paper we implement the Adomian decomposition method (ADM) to this model with some initial conditions to find explicit solutions and numerical solutions rather than the traditional methods. The decomposition scheme is illustrated by studying the biological to compute explicit and numerical solutions. The method is useful for obtaining both approximate and numerical approximations of linear or nonlinear differential equations and it is also quite straightforward to write Computer codes in any symbolic languages. If the numerical solutions are necessary to compute the rapid convergence is obvious. The main aim of this paper is to solve the nonlinear fractional-order biological in the form y (u ) + f(u) (1) with given initial condition u(x y 0) where u denotes the population density and f represents the population supply due to births and deaths. For α 1 some properties of Eq. (1) such as Hölder estimates of its solutions are studied in [7. Three examples of constitutive equations for f are: f(u) = c u c = constant Malthusian Law [8 f(u) = c 1 u c u c 1 c =positive constants Verhulst Law [8 f(u) = c u p (c 0 0 < p < 1) Porous Media. [9 10 We consider a more general form of f as f(u) = h u a (1 r u b ) where h a r b are real numbers. If h = c a = 1 r = 0 and h = c 1 a = b = 1 r = c /c 1 this leads to Malthusian law and Verhulst law. We implement the Adomian s decomposition method (ADM) [5 6 to this model s u(x y 0) = f 0 (x y). () In this paper we present a solution of a more general model of fractional reaction-diffusion (FRD) model (1). The system is obtained from the standard diffusionreaction systems by replacing the first time derivative term by a fractional derivative of order. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional model (1) are of a more general nature. The resulting solutions spread faster than the
2 No. 6 Exact Solutions of Fractional-Order Biological Population Model 993 classical solutions and may exhibit asymmetry. The fundamental solutions of these equations of fractional order make them attractive for applications. We would like to put your attention that time fractional derivatives change also the solutions we usually get in standard reactiondiffusion systems. The concept of fractional or non-integer order derivation and integration can be traced back to the genesis of integer order calculus itself. [11 Almost most of the mathematical theory applicable to the study of noninteger order calculus was developed through the end of 19th century. However it is in the past hundred years that the most intriguing leaps in engineering and scientific application have been found. The calculation technique in some cases has to change to meet the requirement of physical reality. The use of fractional differentiation for the mathematical modeling of real world physical problems has been widespread in recent years e.g. the modeling of earthquake the fluid dynamic traffic model with fractional derivatives measurement of viscoelastic material properties etc. The derivatives are understood in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. The Adomian s decomposition method is applied for computing solutions to the systems of fractional partial differential equations considered in this paper. This method has been used to obtain approximate solutions of a large class of linear or nonlinear differential equations. It is also quite straightforward to write computer codes in any symbolic languages. The method provides solutions in the form of power series with easily computed terms. It has many advantages over the classical techniques mainly; it provides efficient numerical solutions with high accuracy minimal calculations. Cherruault [1 proposed a new definition of the method and he then insisted that it will become possible to prove the convergence of the decomposition method. Cherruault and Adomian [13 proposed anew convergence series. A new approach of the decomposition method was obtained in a more natural way than was given in the classical presentation. [14 Recently the application of the method is extended for fractional differential equations. [15 18 Fractional Calculus There are several approaches to the generalization of the notion of differentiation to fractional orders e.g. Riemann Liouville Gruönwald Letnikow Caputo and Generalized Functions approach. [19 Riemann Liouville fractional derivative is mostly used by mathematicians but this approach is not suitable for real world physical problems since it requires the definition of fractional order initial conditions which have no physically meaningful explanation yet. Caputo introduced an alternative definition which has the advantage of defining integer order initial conditions for fractional order differential equations. [0 Unlike the Riemann Liouville approach which derives its definition from repeated integration the Gruönwald Letnikow formulation approaches the problem from the derivative side. This approach is mostly used in numerical algorithms. Here we mention the basic definitions of the Caputo fractional-order integration and differentiation which are used in the up coming paper and play the most important role in the theory of differential and integral equation of fractional order. The main advantages of Caputo s approach are the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer order differential equations. Definition 1 The fractional derivative of f(x) in the Caputo sense is defined as: D α f(x) = J m α D m f(x) = 1 Γ(m α) x 0 (x t) m α+1 f (m) (t)dt for m 1 < α m m N x > 0. For the Caputo derivative we have D α C = 0 C is constant and 0 n α 1 D α t n = Γ(n + 1) Γ(n α + 1) tn α n > α 1. Definition For m to be the smallest integer that exceeds α the Caputo fractional derivatives of order α > 0 is defined as D α u(x t) = α u(x t) = 1 Γ(m α) t 0 (t τ)m α+1 m u(xτ) τ m dτ for m 1 < α < m m u(x t) for α = m N. tm 3 Numerical Solutions of Model We consider the generalized biological of the form y (u ) + hu a (1 ru b ) t > 0 x y R 0 < α 1 (3) with given initial condition u(x y 0) = f 0 (x y).
3 994 A.M.A. El-Sayed S.Z. Rida and A.A.M. Arafa Vol. 5 We can write the generalized biological population model in an operator form as D α t u(x y t) = D xx(u ) + D yy (u ) + hu a (1 ru b ) (4) where D xx = / x D yy = / y and D α t = α / are the fractional differential operators. Operating with J α in both sides of Eq. (4) we find u(x y t) = u(x y 0) + J α (D xx (u ) + D yy (u ) + hu a (1 ru b )). (5) The ADM assumes a series solution for u(x y t) given by u(x y t) = u n (x y t). (6) Substituting the decomposition series (6) into (5) yields u n (x y t) = u 0 (x y 0) + J α( ) A n. (7) Identifying the zeros components u 0 (x y t) by u 0 (x y 0) the remaining components where n 0 can be determined by using recurrence relation: u 0 (x y t) = u 0 (x y 0) u n+1 (x y t) = J α (A n ) n 0. (8) From this equation the iterates are defined by the following recursive way u 0 (x y t) = u 0 (x y 0) u 1 (x y t) = J α (A 0 ) u (x y t) = J α (A 1 ) u 3 (x y t) = J α (A ) (9) where A n are Adomain s polynomials which are derived as A 0 = (u 0 ) xx + (u 0 ) yy + h u a 0 rh ua+b 0 A 1 = (u 0 u 1 ) xx + (u 0 u 1 ) yy + a h u a 1 0 u 1 + r h (a + b)u a+b 1 0 u 1 (10) A = (u 1 + u 0u 1 ) xx + (u 1 + u 0u 1 ) yy ( + a h u a 1 0 u + 1 ) (a 1)ua 0 u 1 ( + r h(a + b) u a+b 1 0 u + 1 ) (a + b 1)ua+b 0 u 1 and so on. The rest of the polynomials can be constructed in a similar manner in Ref. [1. 4 Applications In this section we present some examples with analytical solution to show the efficiency of methods described in the previous section for solving Eq. (3). Example 1 Consider the following time-fractional biological y (u ) + hu 1 (1 ru) 0 < α 1 x y R t > 0 (11) h r u(x y 0) = 4 y + y + 5. (1) The exact solution for the special case α 1 is given by [1 u(x y t) = h r 4 y + y + h t + 5. (13) By using Eqs. (9) and (10) and let a = 1 b = 1 we could be able to calculate some of the terms of the decomposition u 0 = f 0 (x y) u 1 = f 1 (x y) u = f (x y) u 3 = f 3 (x y) Γ(3α + 1) where ( h r f 0 (x y) = f 1 (x y) = h ) 1/ 4 y + y + 5 ( h r 4 y + y + 5 ) 1/ f (x y) = h ( h r 4 y + y + 5 ) 3/ ) 5/ f 3 (x y) = 3h 3( h r 4 y + y + 5 and so on substituting u 0 u 1 u u 3... into Eq. (6) gives the solution u(x y t) = u 0 + u 1 + u + u 3 u(x y t) = f 0 + h tα n + 1 ( h t α) n. (14) f 0 Γ[(n + 1)α + 1 See Figs. 1. Example Consider the following time-fractional biological y (u ) + hu (15) f 0 u(x y 0) = xy. (16) By using Eqs. (9) and (10) and let a = 1 r = 0 we could be able to calculate some of the terms of the decomposition u 0 = xy u = h xy u 1 = h xy u 3 = h 3 xy
4 No. 6 Exact Solutions of Fractional-Order Biological Population Model 995 Fig. 1 The surface shows the numerical result for the absolute value of u(x yt) a 1 (a) comparison with the analytical solution (b); h = 0.01 r = 48 t = 10 Fig. The surface shows the approximate solution for the absolute value of u(x y t): (a) α = 0.95; (b) α = 0.99 Substituting u 0 u 1 u u 3... into Eq. (6) gives the solution u(x y t) = xy + h xy Γ(α + 1) + h xy Γ(α + 1) + h 3 xy = (h ) k xy u(x y t) = xye α (h ) (17) where E α (h ) is the Mittag Leffler function defined as [19 z k E α (z) = Γ(1 + kα). u(x y t) = (h t) k xy = xy e ht which is an exact solution to the standard form biological. [1 Example 3 Consider the following time-fractional biological y (u ) + u (18) u(x y 0) = sin x sinhy. (19) By using Eqs. (9) and (10) and let a = 1 r = 0 we could be able to calculate some of the terms of the decomposition u 0 = sin xsinhy u = sin xsinhy u 1 = sin xsinhy u 3 = sin xsinhy Substituting u 0 u 1 u u 3... into Eq. (6) gives the solution u(x y t) = sinxsinh y + sin xsinhy Γ(α + 1)
5 996 A.M.A. El-Sayed S.Z. Rida and A.A.M. Arafa Vol. 5 + sinxsinhy Γ(α + 1) + sinxsinhy = (t kα sin xsinhy u(x y t) = sin xsinhye α ( ). (0) u(x y t) = sin xsinhy (t) k = sin xsinhye t which is an exact solution. Example 4 Consider the following time-fractional biological y (u ) + u(1 ru) (1) u(x y 0) = exp (x + y) () and by using Eqs. (9) and (10) and let a = 1 b = 1 we could be able to calculate some of the terms of the decomposition u 0 = exp u = exp u 3 = exp (x+y) u 1 =exp (x+y) Γ(α+1) (x + y) (x + y) Substituting u 0 u 1 u u 3... into Eq. (6) gives the solution u(x y t) = exp (x + y) (x + y) Γ(α + 1) (x + y) Γ(α + 1) (x + y) = exp (x + y) t kα u(x y t) = exp (x + y) E α ( ). (3) u(x y t) = = exp (x + y) t k = exp (x + y) + t which is an exact solution to the standard form biological. [1 5 Conclusions We employ the ADM for finding the exact solutions of time fractional degenerate parabolic equations arising in the spatial diffusion of biological populations subject to some initial conditions. The ADM introduces a significant improvement in this field over existing techniques. The corresponding solutions are obtained according to the recurrence relation using Mathematica. References [1 F. Shakeri and M. Dehghan Comput. Math. Appl. 54 (007) [ M.J. Ablowitz and H. Segur Solitons and the Inverse Scattering Transformation SIAM Philadelphia (1981). [3 G.B. Whitham Linear and Nonlinear Waves John Wiley New York (1974). [4 M. Tajiri and Y. Murakami Phys. Lett. A 134 (1990) 17. [5 G. Adomian Solving Frontier Problems of Physics: The Decomposition Method Kluwer Academic Dordrecht (1994). [6 G. Adomain Math. Anal. Appl. 135 (1988) 501. [7 Y.G. Lu Appl. Math. Lett. 13 (000) 13. [8 M.E. Gurtin and R.C. MacCamy Math. Biosc. 33 (1977) 35. [9 J. Bear Dynamics of Fluids in Porous Media American Elsevier New York (197). [10 A. Okubo Diffusion and Ecological Problem Mathematical Models Biomathematics 10 Springer Berlin (1980). [11 J.D. Munkhammar Undergrad. J. Math. 6 (005) 1. [1 Y. Cherruault Convergence od Adomian s Method Kybernetes 18 (1989) 31. [13 Y. Cherruault and G. Adomian Math. Comput. Modell. 18 (1993) 103. [14 N. Nagarhasta B. Some K. Abbaoui and Y. Cherruault New Numerical Study of Adomian Method Applied to a Diffusion Model Kybernetes 31 (00) 61. [15 S. Momani Chaos Solitons and Fractals 8 (006) 930. [16 S. Momani and Z. Odibat Appl. Math. Comput. 177 (006) 488. [17 S. Momani and Z. Odibat Chaos Solitons and Fractals 31 (007) 148. [18 Z. Odibat and S. Momani Appl. Math. Comput. 181 (006) 767. [19 I. Podlubny Fractional Differential Equations: an Introduction to Fractional Derivatives Fractional Differential Equations to Methods of Their Solution and Some of Their Applications Academic Press New York (1999). [0 M. Caputo J. Roy. Austral. Soc. 13 (1967) 59. [1 A. Wazwaz Appl. Math. Comput. 111 (000) 53.