New analytical method for solving nonlinear time-fractional reaction-diffusion-convection problems

Abstract. In this paper, we propose a new analytical method called generalized Taylor fractional series method (GTFSM) for solving nonlinear timefractional reaction-diffusion-convection initial value problems. Our obtained results are given in the form of a new theorem. The advantage of the proposed method compared with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction. The accuracy and efficiency of the method is tested by means of two numerical examples. Obtained results interpret that the proposed method is very effective and simple for solving different types of nonlinear fractional problems.


Introduction
In recent years, there has been a rapid development in the concept of fractional calculus and its applications [3,4,7,9]. The fractional calculus which deals with derivatives and integrals of arbitrary orders [8] plays a vital role in many fields of applied science and engineering. Recently, nonlinear partial differential equations with fractional order derivatives have been successfully applied to many mathematical models in mathematical biology, aerodynamics, rheology, diffusion, electrostatics, electrodynamics, control theory, fluid mechanics, analytical chemistry and so on. In all these scientific fields, it is important to obtain exact or approximate solutions of nonlinear fractional partial differential equations (NFPDEs). But in general, there exists no method that gives an exact solution for NFPDEs and most of the obtained solutions are only approximations.
The main objective of this paper is to conduct a new analytical method called generalized Taylor fractional series method (GTFSM) to study the solution of nonlinear time-fractional reaction-diffusion-convection initial value problems described by where D α t is the Caputo fractional derivative operator of order α, 0 < α ≤ 1 and 0 < t < R < 1. u = u(x, t) is an unknown function, and the arbitrary smooth functions a(u), b(u) and c(u) denote the diffusion term, the convection term and the reaction term respectively. The reaction-diffusion-convection problems are very useful mathematical models in applied sciences such as biology modeling, physics, chemistry, astrophysics, hydrology, medicine and engineering.
The paper is organized as follows. In Section 2, we give some necessary definitions and properties of the fractional calculus theory. In Section 3, we introduce our results to solve the nonlinear time-fractional reaction-diffusionconvection initial value problems (1) using the GTFSM. In Section 4, we present two examples to show the efficiency and effectiveness of this method. In Section 5, we discuss our obtained results represented by figures and tables. These results were verified with Matlab (version R2016a). Section 6, is devoted to the conclusions on the work.

Basic Definitions
In this section, we give some basic definitions and properties of the fractional calculus theory which are used further in this paper. For more details see [8]. (2) For this definition we have the following properties Definition 2.4. The Mittag-Leffler function is defined as follows A further generalization of (4) is given in the form For α = 1, E α (z) reduces to e z .

Analysis of the Method
Method 3.1. Consider the nonlinear time-fractional reaction-diffusion-convection initial value problems in the form (1).
Then, by GTFSM the solution of (1) is given in the form of an infinite series which converges rapidly to the exact solution as follows where c i (x) are the coefficients of the series.
Proof. In order to achieve our goal, we consider the following nonlinear reactiondiffusion-convection initial value problems in the form (1).
Assume that the solution takes the following infinite series form Consequently, the approximate solution to (1), can be written in the form of By applying the operator D α t on Eq. (7), and using the properties (1) and (2), we obtain the formula Next, we substitute both (7) and (8) in (1). Therefore, we have the following recurrence relations .

NONLINEAR TIME-FRACTIONAL REACTION-DIFFUSION-CONVECTION PROBLEMS 5
We follow the same analogue used in obtaining the Taylor series coefficients. In particular, to determine the function c n (x), n = 1, 2, 3, .., we have to solve the following D .
Now, we determine the first terms of the sequence {c n (x)} N 1 . For n = 1 we have .
In general, to obtain the coefficient function c k (x) we solve Finally, the solution of (1), can be expressed by The proof is complete.

Numerical Examples
In this section, we test the validity of the proposed method to solve some nonlinear Caputo time-fractional reaction-diffusion-convection problems.
We define E n to be the absolute error between the exact solution u and the approximate solution u n , as follows By applying the steps involved in GTFSM as presented in Section 3, we have the solution of the problem (9) is in the form and c i (x) = e x , for i = 1, 2, 3, ...
Taking α = 1 in (11), the solution of (9) has the general pattern form which is coinciding with the following exact solution in terms of infinite series So, the exact solution of (9) in a closed form of elementary function will be which is exactly the same solution obtained by HAM [5].

Example 4.2. Consider the following initial value nonlinear problem
By applying the steps involved in GTFSM as presented in Section 3, we have the solution of problem (12) in the form and Therefore, the solution of (12), can be expressed by where E α (2t α ) is the Mittag-Leffler function, defined by Eq. (4).
Taking α = 1 in (14), the solution of (12) has the general pattern form which is coinciding with the following exact solution in terms of infinite series u(x, t) = 2 e x − e −4x 1 + 2t + (2t) So, the exact solution of (12) in a closed form of elementary function will be which is exactly the same solution obtained by HAM [10].

Numerical Results and Discussion
In this section the numerical results for both Examples 4.1 and 4.2 are presented. Figures 1 and 3 represent the surface graph of the exact solution and the approximate solution u 4 (x, t) at α = 0.6, 0.8, 1. Figures 2 and 4 represents the behavior of the exact solution and the approximate solution u 4 (x, t) at α = 0.7, 0.8, 0.9, 1. Tables 1 and 2 show the absolute errors between the exact solution and the approximate solution u 4 (x, t) at α = 1 for different values of x and t. The numerical results afirm that when α approaches 1, our obtained results by the GTFSM approach the exact solution.

Conclusion
In this paper, a new analytical method called generalized Taylor fractional series method (GTFSM) is presented for finding the solution of the nonlinear timefractional reaction-diffusion-convections problems. The method was applied to two numerical examples. The results show that the GTFSM is an efficient and easy to use technique for finding approximate and analytic solutions for these problems. The obtained approximate solutions using the suggested method is in excellent agreement with the analytic solution. This confirms our belief that the efficiency of our technique gives it much wider applicability for general classes of nonlinear fractional problems.