ON THE NUMERICAL INVESTIGATIONS OF THE TIME-FRACTIONAL MODIFIED BURGERS’ EQUATION WITH CONFORMABLE DERIVATIVE, AND ITS STABILITY ANALYSIS

In this paper, we aim to introduce the cubic non-polynomial spline functions to develop a computational method for solving the fractional modified Burgers’ equation. Using the Von Neumann method, the proposed approach is shown to be conditionally stable. The proposed approach has been implemented on two test problems. The obtained results indicate that the proposed approach is a good option for solving the fractional modified Burgers’ equation. The error norms l2 and l∞ have been determined to validate the accuracy and efficiency of the proposed method. The numerical solution of such kinds of models has been the key interest of researchers due to their wide range of applications in real life, optical fibers, solid-state physics, biology, plasma physics, fluid dynamics, number theory, chemical kinetics, turbulence theory, heat conduction, gas dynamics.


INTRODUCTION
Fractional calculus, as a generalization of integer order calculus, has aided scientists in understanding and modeling a wide range of phenomena in physics and engineering branches [1][2][3]. In the works of literature, there are some common methods used to obtain analytical or approximate solutions of nonlinear-fractional ordinary and partial differential equations. For instance, variational iteration method (VIM) [4] for space-and time-fractional Burgers equations, differential transformation method (DTM) [5] for fractional Fornberg-Whitham equation, combination of DTM and generalized Taylor's formula [6] for nonlinear fractional partial differential equations, Adomian decomposition method (ADM) [7] for the fractional nonlinear Schrödinger equation, radial basis functions (RBFs) [8] for fractional partial differential equations, homotopy analysis method (HAM) [9] for nonlinear fractional differential equations.
The fractional integral is used to define these definitions, which are commonly used in non-integer calculus literature. As a result, fractional derivative operators behave as non-local operators and do not satisfy classical properties of normal (integer) derivatives such as product, chain, and quotient rules, which allow us to obtain analytical solutions in standard calculus. Algebraic operations in non-integer calculus involve many challenges and inconvenience in mathematical handling because these basic rules cannot be used. R. Khalil et al. [10] have presented a new definition, namely "conformable fractional derivative", which obeys basic classical properties and allows us to solve fractional differential equations analytically. This definition is more simple than other fractional definitions because it has received a lot of attention, many phenomena and applications can be modeled based on the conformable sense [10][11][12][13][14][15], and it has a lot of interesting advantages such as generalizes all concepts of standard calculus and can be solved numerous fractional differential equations in all cases.
In recent years, several studies have been made further studies and explanations on the physical applications and physical meaning of the Burgers equation . The modified Burgers   3  NON-POLYNOMIAL SPLINE METHOD   equations, which is a different form of the Burgers' equation, are discussed in this paper and can   be applied to a wide range of scientific fields, including optical fibers, solid-state physics, biology,   plasma physics, fluid dynamics, number theory, chemical kinetics, turbulence theory, heat conduction, gas dynamics, etc. [16][17][18][19]. In this article, the collocation method with cubic nonpolynomial spline functions is used to obtain approximate solutions of the time-fractional modified subject to the conditions and where , p are parameters and, α is the parameter describing the order of the fractional time derivatives. The fractional derivatives are considered in the conformable sense.
The initial condition ( , 0 ) = ( ), for a ≤ ≤ b implies that 0 = ( ), for each = 0, 1, …, . These values can be used in (20) to calculate the value of 1 , for each = 0, 1, …, . If the procedure is reapplied once all the approximations 1 are known, the values of Remark 1. The following steps are used to linearize the nonlinear term in the system (20) I. At = 0 , we approximate 1/2 by 1/2 * computed from 0 and get a first approximation to 1 , then we compute 1/2 from 1 + 0 2 to refine the approximation to 1 .

TRUNCATION ERROR
We can obtain the local truncation error * +1/2 associated with Eq. (10) by rewriting (10) in the form where n ≡ n n and ≡ ( , ).

STABILITY ANALYSIS
The Von Neumann technique will be used to investigate the stability of our system (18). To do this, we must linearize the nonlinear term p of Burgers' equation, Eq. (1), by assuming that the corresponding quantities −1 +1/2 , +1/2 and +1 +1/2 are equal to a local constant * in (18).
Assuming a solution of the form [22,23]: where 2 = −1, ℎ is the element size, ϕ is the mode number, and is the amplification factor at time level . As increases, more time steps are computed. Inserting the latter expression for in the system (18) gives For stability, we must have | | ≤ 1, otherwise in (23) would expand in an unbounded manner. This condition is valid for > 0 and > 0 such that > 2 . Finally, we can say that the proposed method is conditionally stable for > 0 and > 0 such that > 2 .

In this section, we obtain numerical solutions of fractional modified Burgers' equation for two test
problems. The accuracy of our presented numerical method is measured by computing the difference between the numerical and analytical solutions at each mesh point and use these to compute 2 and ∞ error norms. Error norms are defined as

Example 2
Consider the following fractional modified Burgers' equation [17,19]:  Tables 5-7 15 NON-POLYNOMIAL SPLINE METHOD show a very close agreement between the approximate and exact solutions. In Table 8, the error norms have been shown for α = 0.7. The 3D-Graphs of the approximate solutions are shown in Fig. 3, for = 0.5 , = 0.7 and = 0.9 . The exact and approximate solutions of (27) are graphically depicted in Fig. 4, for = 0.5, = 0.7 and = 0.9.

CONCLUSION
In this study, a numerical treatment for the nonlinear fractional modified Burgers' equation is proposed using a collocation method with cubic non-polynomial spline functions. Applying the Von Neumann stability analysis, the proposed method is shown to be conditionally stable. In comparison to exact solutions, the derived approximate solutions have good accuracy. Therefore, we can conclude that the proposed method is a very effective and dependable tool for fractional differential equations that arise in a variety of physics and engineering fields.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.