He's Variational Iteration Method for Solving

19 a 21 de Outubro de 2016 ˜ Pessoa - PB Universidade Federal da Para´ıba – Joao

He’s Variational Iteration Method for Solving Fractional Bertalanffy Differential Equation Fernando S. Silva1 - [email protected] Marcelo A. Moret2 - [email protected] 1 Universidade 2 Programa

Estadual do Sudoeste da Bahia, DCET - Vitória da Conquista, BA, Brazil de Modelagem Computational - SENAI CIMATEC, Salvador, BA, Brazil

Abstract. We will consider He’s variational iteration method (VIM) is used to obtain approximate analytical solutions of the fractional Bertalanffy differential equation. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This technique provides a sequence of functions which converges to the exact solution of the problem. The present method performs extremely well in terms of efficiency and simplicity. Keywords: Bertalanffy, VIM, multipliers, Caputo derivative. 1.

INTRODUCTION

Most fractional differential equations do not have exact analytic solutions. In recent years, much attention has been paid to the application of the Variational iteration method (He, 1999) to various problems due to its simplicity. Von Bertalanffy (1938) introduced his growth equation to model fish weight growth. He proposed the form given below which can be seen to be a special case of the Bernoulli differential equation. We consider here the following nonlinear fractional Bertalanffy differential equation: Dαt y(t) = A(t)y µ − B(t)y q ,

(1)

subject to the initial conditions y (k) (0) = ck ,

k = 0, 1, · · · , n − 1,

(2)

where α is fractional derivative order, n is an integer, y (k) denotes the k-th order derivative of y, A(t) and B(t) are known real functions, and ck is a constant. In this paper, we adopt the two usually used definitions: the Caputo and its reverse operator Riemann-Liouville. As for further detailed account of fractional calculus, see Kilbas, Srivastava and Trujillo (2006), Podlubny (1999), Miller and Ross (1993), Samko, Kilbas and Marichev (1993), Oldham and Spanier (1974). Proceedings of XIX ENMC - National Meeting on Computational Modeling and VII ECTM - Meeting on Materials Science and Technology João Pessoa, PB - 19-21 October 2016

XIX ENMC e VII ECTM 19 a 21 de Outubro de 2016 Universidade Federal da Para´ıba – João Pessoa - PB

The Caputo fractional derivative of order α > 0 for the function f ∈ C m [0, T ] is defined in the following form  Z t 1  (t − τ )m−α−1 f (m) (τ )dτ, m − 1 < α < m, m ∈ N∗ , Dαt [f (t)] = (3) Γ(m − α) 0  (m) f (t), m = α, where f (m) is the ordinary mth derivative of f , and Γ(.) is the usual Gamma function, with relations Γ(z + 1) = zΓ(z) and Γ(m) = (m − 1)!. In general case however, such an explicit solution is not available and numerical method have to be applied to approximate the solution of (3). If the function is non-differentiable then, this definition isn’t applicable. Similar to integerorder differentiation, Caputo fractional derivative operator satisfies the following properties Dαt [λf (t) + g(t)] = λDαt f (t) + Dαt g(t) Dαt c = 0, ∀α > 0 for all constants c.

(4) (5)

where λ is any scalar and f and g are appropriate functions. For the fractional derivative we have Dαt (tλ ) =

Γ(λ + 1) λ−α t , λ > m − 1, Γ(λ − α + 1)

(6)

this formulation was used extensively by Scott Blair et al. 1947 in rheology. This paper is organized as follows. Section 2 describes fractional cauchy problem. Section 3 is devoted for He’s fractional VIM methods. Section 4 Applications of the fractional VIM Method. Section 5 Illustrative examples: Bernoulli case. Section 6 concludes the paper. 2.

Fractional Ordinary Differential Equation

Let f : [0, T ] × R → R be a continuous function and α > 0 a real. If zα is a solution of the system α Dt y(t) = f (t, y), α ∈ (0, 1) y(0) = y0 , and if the limit lim zα (t) := z(t)

α→1−

exists for all t ∈ [0, T ], then z is solution of the Cauchy problem 0 y (t) = f (t, y) y(0) = y0 . In a recent work by Camargo & Oliveira (2015), it is suggested first consider a fractional differential equation of order α ∈ (0, 1), and as α → 1− , the fractional system converts into an ordinary Cauchy problem.

Proceedings of XVIII ENMC - National Meeting on Computational Modeling and VI ECTM - Meeting on Materials Science and TechnologyJoão Pessoa, PB - 19-21 October 2016


3.

Basic ideas of He’s variational iteration method

Variational iteration method is a powerful device to solve the various kinds of linear and nonlinear functional equations. The VIM was developed by He (1998,1999,2000) for solving linear, nonlinear, and initial value problems. For the purpose of illustration of the methodology to the proposed method, using variational iteration method, we begin by considering a differential equation in the formal form, Ly + N y = g(t)

(7)

where L is a linear operator, N a nonlinear operator and g(t) an inhomogeneous term. According He (1998) to the variational iteration method, we can write down a correction functional as follows Z t λ(t, s)[Lyk + N y˜k (s) − g(s)]ds, (n ≥ 0). (8) yk+1 = yk (t) + 0

Where λ is a general Lagrange multiplier which can be identified optimally via the variational theory, Inokuti et al (1978). The subscript k denotes the kth order approximation, y0 is an initial approximation with possible unknowns, and y˜k is considered as a restricted variation, that is δ y˜k = 0. Taking the variation from both sides of the correct functional with respect to δyk and imposing δyk+1 = 0, the stationary conditions are obtained. For the generalized fractional differential equations, one can have the variational iteration formula Z t yk+1 = yk (t) + λ(t, s)[Dαs yk + N y˜k (s) − g(s)]ds, (α ≥ 0). (9) 0

Guo & Dumitru (2006) proposed the stationary conditions the optimal value of the λ can be identified for any order α as follows λ(t, s) =

(−1)n (s − t)n−1 . (n − 1)!

(10)

Then the Lagrange multiplier can be easily determined for 0 < α ≤ 1 as λ(t, s) = −1. 4.

Applications of the VIM Method Consider the following form of the fractional Bertalanffy differential equation: Dαt y(t) = A(t)y µ − B(t)y q ,

y(0) = 1,

(11)

where A(t) and B(t) are scalar functions, and µ, q ≥ 1. To solve Eq. (11), 0 < α ≤ 1, by means of He’s variational iteration method, we can construct a correct functional as follows: Z t (12) yk+1 (t) = yk (t) − [Dαs yk − A(s)ykµ + B(s)ykq ]ds, 0

and the initial approximation y0 can be freely chosen if it satisfies the initial conditions of the problem. However the success of the method depends on the proper selection of the initial approximation y0 . Finally, the exact solution is obtained at the limit of the resulting successive approximations by y(t) = lim yk (t). k→∞


(13)


5.

Illustrative examples

We derive a numerical method for the solution of fractional Bertalanffy differential equation with A(t) ≡ 1 and B(t) ≡ −1. A fractional Bernoulli ODE of the form Dαt y(t) = y q (t)

with y(0) = 1,

(14)

where α denotes the order of the derivative with 0 < α ≤ 1. Notice that the zero function y = 0 is a solution. So we assume that y is not the zero function. We start with initial approximation y0 (t) = 1, and by using the above iteration formula (12), we can directly obtain the components of the solution. Now, the first three components of the solution y(t) by using (12) are given by t2−α (1 + t)q+1 + 1 y0 (t) = 1, y1 (0) = 1 + t, y3 (t) = 1 + t − − , Γ(3 − α) q+1

(15)

and so on, other components can be obtained in a like manner. When α → 1− , then we have which is the same solution VIM approximated of the equation standard. The analytic solution of the Eq. (14) can be written 1 (1 − q) α 1−q . y(t) = 1 + t Γ(α + 1) 6.

(16)

CONCLUSIONS

Bertalanffy equation is one of the more popular topics in elementary physics. In this paper, we present the application of He’s variational iteration method to obtain the solution of the fractional ordinary differential equation. Some of the advantages of VIM are that, the initial solution can be freely chosen with some unknown parameters. The traditional methods is rescued when α → 1− , the simplicity of the method and the obtained exact results show that the modified variational iteration method is a powerful mathematical tool for solving nonlinear fractional differential equations of derivative type caputo. REFERENCES A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. C. Guo, B. Dumitru, New Application of the Variational Iteration Method form Differential Equations to q-Fractional Difference Equations, Spring Open Journal, 2013, 1-16. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. J. H. He, approximate solution of nonlinear differential equations with convolution product nonlinearities, Computer methods in pplied mechanics and engineering. 167 (1998), 69-73. J. H. He, Variational iteration method- a kind of non-linear analytical technique: some examples, Int. J. Nonlin. Mech., vol.34, pp. 699-708, 1999. J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comp., vol.114, pp. 115-123, 2000. J. H. He: Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B. 20 (10), 1141-1199(2006). K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.



K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974. L. Von. Bertalanffy, A quantitative theory of organic growth, Human Biology, 10(2), (1938), 181-213. M. Inokuti and et al. General use of the Lagrange multiplier in non-linear mathematical Physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, PP. (1978), 156-162. R. F. Camargo e E. Capela de Oliveira, Cálculo Fracionário, Editora Livraria da F´ısica, São Paulo, 2015. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Philadelphia, 1993.
