Analytical solutions of fractional foam drainage equation by residual power series method

Transcription

1 Math Sci () 83 6 DOI.7/s96--- ORIGINAL RESEARCH Analytical solutions of fractional foam drainage equation by residual power series method Marwan Alquran Received October / Accepted 3 February / Published online February Ó The Author(s). This article is published with open access at Springerlink.com Abstract The current work highlights the following issues a brief survey of the development in the theory of fractional differential equations has been raised. A very recent technique based on the generalized Taylor series called residual power series (RPS) is introduced in detailed manner. The time-fractional foam drainage equation is considered as a target model to test the validity of the RPS method. Analysis of the obtained approimate solution of the fractional foam model reveals that RPS is an alternative method to be added for the fractional theory and computations and considered to be a significant method for eploring nonlinear fractional models. Keywords Generalized Taylor series Residual power series Time-fractional foam drainage equation Mathematics Subject Classification 3C Introduction 6A33 3F The last four decades witness fundamental works and developments on the fractional derivative and fractional differential equations. Oldham and Spanier [], Miller and Ross [], Samko el. [3], Podlubny [], Kilbas el. [] and others [6, 7] are the pioneer in this field; their works M. Alquran (&) Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Bo(33), Irbid, Jordan marwan@just.edu.jo; marwan@squ.edu.om M. Alquran Department of Mathematics and Statistics, Sultan Qaboos University, P.O.Bo(36), PC 3, Al-Khod, Muscat, Oman form an introduction to the theory of fractional differential equations and provide a systematic understanding of the fractional calculus such as the eistence and the uniqueness of solutions. Hernandez el. [8] published a paper on recent developments in the theory of abstract differential equations with fractional derivative. Finally, interested various applications of fractional calculus in the field of interdisciplinary sciences such as image processing and control theory have been studied by Magin el. [9] and Mainardi []. In the literature, there eist no methods that produce an eact solution for nonlinear fractional differential equations. Only approimate solutions can be derived using linearization or successive or perturbation methods. Such methods are variational iteration method and multivariate Pade approimations [], Iterative Laplace transform method [], Adomian decomposition method [3 7], Homotopy analysis method [8, 9] and Sumudu transform method []. The main objective of this paper is to conduc new novel technique called residual power series method to study the solution of time-fractional Foam drainage model described by D a t uð; tþ ¼ uð; tþu ð; tþ u ð; tþu ð; tþþu ð; tþ; ðþ subject to the initial conditions uð; Þ ¼f ðþ ðþ It is a simple model of the flow of liquid through channels (Plateau borders) and nodes (intersection of four channels) between the bubbles, driven by gravity and capillarity [, ]. Approimate solutions of Eqs. (.) (.) have been obtained by different methods; Adomian decomposition 3

2 Math Sci () 83 6 method [3], the Homotopy analysis method [] and the variational iteration method []. It should be noted here that none of the previous studies addressed the accuracy of such methods. The pattern of the current paper is as follows In Sect., some definitions and theorems regarding Caputo s derivative and fractional power series are given. In Sect. 3, we derive a residual power series solution to the time-fractional foam drainage equation. Graphical results regarding the foam drainage model are presented in Sect.. Preliminaries Many definitions and studies of fractional calculus have been proposed in the literature. These definitions include Grunwald Letnikov, Riemann Liouville, Weyl, Riesz and Caputo sense. In the Caputo case, the derivative of a constant is zero and one can define, properly, the initial conditions for the fractional differential equations which can be handled using an analogy with the classical integer case. For these reasons, researchers prefer to use the Caputo fractional derivative [6] which is defined as Definition. For m to be the smallest integer that eceeds a, the Caputo fractional derivatives of order a [ are defined as D a uð; tþ ¼ oa uð; tþ 8 ot Z a t ðt sþ m a o m uð; sþ >< Cðm aþ os m ds; m \a\m ¼ > o m uð; tþ ot m ; a ¼ m N Now, we survey some needed definitions and theorems regarding the fractional power series (RPS) where there is much theory to be found in [7, 8]. Definition. A power series epansion of the form X c m¼ mðt t Þ ma ¼ c þ c ðt t Þ a þ c ðt t Þ a þ n \a n; t t is called fractional power series PS about t ¼ t Theorem. Suppose that f has a fractional PS representation at t ¼ t of the form f ðtþ ¼ X c m¼ mðt t Þ ma ; t t\t þ R If D ma f ðtþ; m ¼ ; ; ;...are continuous on ðt ; t þ RÞ, then c m ¼ Dma f ðt Þ CðþmaÞ Definition.3 A power series epansion of the form X f m¼ mðþðt t Þ ma is called multiple fractional power series PS about t ¼ t Theorem. Suppose that uð; tþ has a multiple fractional PS representation at t ¼ t of the form uð; tþ ¼ X f m¼ mðþðt t Þ ma ; I; t t\t þ R If t uð; tþ; m ¼ ; ; ;... are continuous on I ðt ; t þ RÞ, then f m ðþ ¼ Dma t uð;t Þ D ma CðþmaÞ From the last theorem, it is clear that if n þ -dimensional function has a multiple fractional PS representation at t ¼ t, then it can be derived in the same manner, i.e. Corollary.3 Suppose that uð; y; tþ has a multiple fractional PS representation at t ¼ t of the form uð; y; tþ ¼ X g m¼ mð; yþðt t Þ ma ; ð; yþ I I ; t t\t þ R If D ma t uð; y; tþ; m ¼ ; ; ;... are continuous on I I ðt ; t þ RÞ, then g m ð; yþ ¼ Dma t uð;y;t Þ CðþmaÞ Net, we present in details the derivation of the residual power series solution to the generalized fractional DSW system. Residual power series (RPS) of the foam drainage model The aim of this section is to construct power series solution to the time-fractional Foam drainage model by substituting its power series (PS) epansion among its truncated residual function [9, 3]. From the resulting equation, a recursion formula for the computation of the coefficients is derived, while the coefficients in the fractional PS epansion can be computed recursively by recurrent fractional differentiation of the truncated residual function. The RPS method proposes the solution for Eqs. (..) as a fractional PS about the initial point t ¼ uð; tþ ¼ X n¼ t na f n ðþ ; \a ; I; t\r Cð þ naþ ð3þ Net, we let u k ð; tþ to denote the k-th truncated series of uð; tþ, i.e., u k ð; tþ ¼ Xk n¼ t na f n ðþ ; \a ; I; t\r Cð þ naþ ð3þ 3

3 Math Sci () 83 6 It is clear that by condition (.) the -th RPS approimate solutions of uð; tþ is u ð; tþ ¼f ðþ ¼uð; Þ ¼f ðþ Also, Eq. (3.) can be written as u k ð; tþ ¼f ðþþ Xk f n ðþ Cð þ naþ ; n¼ \a ; I; t\r; k ¼ ; ; 3;... t na ð33þ ð3þ Now, we define the residual functions, Res, for Eq. (.) Res u ð; tþ ¼D a t uð; tþ uð; tþu ð; tþ ð3þ þ u ð; tþu ð; tþ u ð; tþ; and, therefore, the k-th residual function, Res u;k,is Res u;k ð; tþ ¼D a t u kð; tþ u kð; tþ o o u kð; tþ þ u k ð; tþ o o u kð; tþ o o u kð; tþ ð36þ As in [9, 3], Resð; tþ ¼ and lim k! Res k ð; tþ ¼ Resð; tþ for all I and t. Therefore, D ra t Resð; tþ ¼ since the fractional derivative of a constant in the Caputo s sense is. Also, the fractional derivative D ra t of Resð; tþ and Res k ð; tþ is matching at t ¼ for each r ¼ ; ; ;...; k. To clarify the RPS technique, we substitute the k-th truncated series of uð; tþ into Eq. (3.6), find the fractional derivative formula D ðk Þa t of both Res u;k ð; tþ; k ¼ ; ; 3;...; and then solve the obtained algebraic system D ðk Þa t Res u;k ð; Þ ¼; \a ; I; k ¼ ; ; 3... ð37þ to get the required coefficients f n ðþ; n ¼ ; ; 3;...; k in Eq. (3.). Now, we follow the following steps. Step. To determine f ðþ, we consider ðk ¼ Þ in (3.6) Res u; ð; tþ ¼D a t u ð; tþ u ð; tþ o o u ð; tþ þ u ð; tþ o o u ð; tþ o o u ð; tþ But, u ð; tþ ¼f ðþþf ðþ CðþaÞ. Therefore, ð38þ Res u; ð; tþ ¼f ðþ f ðþþf ðþ f ðþþf ðþ þ f ðþþf ðþ f ðþþf ðþ f ðþþf ðþ ð39þ From Eq. (3.7), we deduce that Res u; ð; Þ ¼ and thus, f ðþ ¼ f ðþf ðþ f ðþf ðþþf ðþ ð3þ Therefore, the st RPS approimate solution is u ð; tþ ¼f ðþþ f ðþf ðþ f ðþf ðþþf ðþ ð3þ Step. To obtain f ðþ, we substitute the nd truncated t series u ð; tþ ¼f ðþþf ðþ a CðþaÞ þ f t ðþ a CðþaÞ into the nd residual function Res u; ð; tþ, i.e., Res u; ð; tþ ¼D a t u ð; tþ u ð; tþ o o u ð; tþ þ u ð; tþ o o u ð; tþ o o u ð; tþ Applying D a t ¼ f ðþþf ðþ f ðþþþf ðþ f ðþþþf ðþ þ f ðþþþf ðþ f ðþþþf ðþ f ðþþþf ðþ ð3þ on both sides of Eq. (3.) gives 3

4 6 Math Sci () 83 6 D a t Res u;ð; tþ ¼f ðþ f ðþþf ðþ f ðþþþf ðþ f ðþþþf ðþ f ðþþf ðþ þ f ðþþþf ðþ f ðþþf ðþ f ðþþþf ðþ þ f ðþþþf ðþ f ðþþf ðþ f ðþþþf ðþ f ðþþf ðþ ð33þ By the fact that D a t Res u;ð; Þ ¼ and solving the resulting system in (3.3) for the unknown coefficient function f ðþ, we get f ðþ ¼ f ðþf ðþþ f ðþf ðþ fðþf ðþf ðþ f ðþf ðþþf ðþf ðþ ð3þ Step 3. We derive the formula of the coefficient function f 3 ðþ. Substitute the 3rd truncated series u 3 ð; tþ ¼f ðþþ t f ðþ a CðþaÞ þ f t ðþ a CðþaÞ þ f t 3ðÞ 3a Cðþ3aÞ into the 3rd residual function Res u;3 ð; tþ. i.e., Res u;3 ð; tþ ¼D a t u 3ð; tþ u 3ð; tþ o o u 3ð; tþ þ u 3 ð; tþ o o u 3ð; tþ o o u 3ð; tþ ¼ f ðþþf ðþ þ f 3ðÞ t 3a f ðþþþf 3 ðþ f ðþþþf3 ðþ t 3a Cð þ 3aÞ t 3a þ f ðþþþf 3 ðþ Cð þ 3aÞ f ðþþþf3 ðþ t 3a Cð þ 3aÞ f ðþþþf3 ðþ t 3a Cð þ 3aÞ ð3þ Now, we apply the operator D a t on both sides of Eq. (3.) D a t Res u;3 ð; tþ ¼D a t D a t Res ;3ð; tþ ¼ f 3 ðþ f ðþþþf 3 ðþ f ðþþþf3 ðþ f ðþþf 3 ðþ f ðþþþf3 ðþ t 3a Cð þ 3aÞ t 3a f ðþþþf 3 ðþ Cð þ 3aÞ f ðþþf3 ðþ þ f ðþþþf 3 ðþ f ðþþþf3 ðþ t 3a Cð þ 3aÞ t 3a þ f ðþþþf 3 ðþ Cð þ 3aÞ f ðþþf 3 ðþ f ðþþþf3 ðþ t 3a Cð þ 3aÞ t 3a þ 8 f ðþþþf 3 ðþ Cð þ 3aÞ f ðþþþf 3 ðþ f ðþþþf 3 ðþ t 3a þ f ðþþþf 3 ðþ Cð þ 3aÞ f ðþþf 3 ðþ f ðþþþf 3 ðþ f ðþþþf3 ðþ t 3a Cð þ 3aÞ f ðþþf 3 ðþ ð36þ Thus, solving the equation D a t Res u;3 ð; Þ ¼ results in the following recurrence formula f 3 ðþ ¼f ðþf ðþþ f ðþf ðþþ f ðþf ðþ f ðþf ðþ fðþf ðþf ðþ 8f ðþf ðþf ðþ f ðþf ðþ þ f ðþ þf ðþf ðþ ð37þ 3

5 Math Sci () (b).7 (a)...t.t.9 (c). (d)....t.t. (e) Eact solution when...t. Fig. The th RPS approimate solution of the Foam model when f ðþ ¼ c tanhð cþ a u ð; t; ; a ¼ 7Þ, b u ð; t; a ¼ 8Þ, c u ð; t; a ¼ 9Þ, d u ð; t; a ¼ Þ, e uð; tþ for a ¼, \\; \t\ 3

6 8 Math Sci () 83 6 Table The obtained errors ju eact u approj for the RPS method t Numerical scheme E E E-.8.E E-..7E E E-7...3E-3. 8.E E E- Finally, we follow the same manner as above and solve the equation D 3a t Res u; ð; Þ ¼ to deduce the following result f ðþ ¼ 3 f ðþf ðþþ3 f ðþf ðþþ f 3ðÞf ðþ þ f ðþf3 ðþ f ðþf ðþf ðþ f ðþf ðþ f ðþf 3 ðþf ðþ 8fðÞf ðþf ðþ 8f ðþf ðþf ðþ f ðþf3 ðþþ6f ðþf ðþ þ f ðþf3 ðþ ð38þ (a). (b).7...t.t. (c)...t. Fig. The th RPS approimate solution of the Foam model when f ðþ ¼ a u ð; t; a ¼ Þ, b u ð; t; a ¼ 7Þ, c u ð; t; a ¼ Þ, \\; \t\ 3

7 Math Sci () (a).7 (b)...t.t. (c) t. Fig. 3 The th RPS approimate solution of the Foam model when f ðþ ¼ sinðþ a u ð; t; a ¼ Þ, b u ð; t; a ¼ 7Þ, c u ð; t; a ¼ Þ, \\; \t\ By the above recurrence relations, we are ready to present some graphical results regarding the time-fractional Foam drainage model. Applications and results The purpose of this section is to test the derivation of residual power series solutions of the following timefractional foam drainage equation. Dat uð; tþ ¼ uð; tþu ð; tþ u ð; tþu ð; tþ þ u ð; tþ; ðþ subject to the initial conditions uð; Þ ¼ c tanhð cþ ðþ Where the eact solution of this model when a ¼ is uð; tþ ¼ c tanhð cð ctþþ. Based on the obtained results from the previous section, we consider only the th RPS approimate solution. Figure represents the th RPS approimate solution of the function uð; tþ for different values of the fractional derivative a. To validate the efficiency and accuracy of the analytical scheme, we give eplicit values of and nd compute the absolute error compared with the eact solution when a ¼ (See Table ). Figure represents solutions to the fractional foam drainage when the initial condition is f ðþ ¼ for a ¼ ;. Finally, Fig. 3 represents solutions to the fractional foam drainage when the initial condition is f ðþ ¼ sinðþ for a ¼ ; 7;. In the theory of fractional calculus, it is obvious that when the fractional derivative n \a\n tends to positive integer n, the approimate solution continuously tends to the eact solution of the problem with derivative n ¼. It is also clear from Figs., and 3 that when a is close to, the solutions bifurcate and provide wave-like pattern. But, when a is close to, there is no pattern. 3

8 6 Math Sci () 83 6 Conclusions A new analytical iterative technique based on the residual power series is proposed to obtain an approimate solution to a nonlinear time-fractional foam drainage equation. Three different initial conditions of the Foam model are considered and accordingly different physical behaviors are produced. We observed when a is close to, the solutions bifurcate and provide wave-like pattern. But, when a is close to, there is no pattern (Figs. and 3). This bifurcation phenomenon will help the researchers to open many new avenues to better understanding of time-fractional derivatives and its relation to real-life phenomena. The accuracy of the proposed method has been tested by studying the absolute errors of the obtained approimate of the Foam model (Table ). The RPS method is a promising technique based on its simplicity and accuracy. It is to be considered an additive tool for the field of fractional theory and computations. As future work, we will etend the RPS method to handle ( þ )-dimensional linear and nonlinear space- and time-fractional physical models. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. References. Oldham, K.B., Spanier, J. The Fractional Calculus Theory and Application of Differentiation and Integration to Arbitrary order. Academic Press, California (97). Miller, K., Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (993) 3. Ray, S.S. Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul., 3 9 (9). Podlubny, I. Fractional Differential Equations. Academic Press, California (999). Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. 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