Numerical solutions of fractional Boussinesq-Whitham- Broer-Kaup and diffusive Predator-Prey equations with conformable derivative

In this article, the residual power series method (RPSM) is u sed to obtain numerical solutions of the time-fractional Boussinesq-Whitham-Broer-Kaup (BBWK) and Diffusive Pred ator-Prey equations using the conformable fractional deri vative definition. This definition is simple, effective and reliabl e in the solution procedure of the fractional differential e quations that have complicated solutions with classical fractional derivati e definitions like Caputo and Riemann-Liouville.


Introduction
Fractional analysis has been the subject of numerous applications in many areas of science and engineering branches such as fractional differential equations (FDEs) [7,24], mathematical physics, biology, image and signal processing, electrical network, fluid flow, viscoelasticity and control in recent years [8,9].
There are some common methods that are used to obtain approximate or analytical solutions of nonlinear fractional ordinary and partial differential equations in the literature [28,29,30,31,32,33,34,35,36,37]. These methods include, Laplace analysis method (LAM) [15] for the constant coefficient fractional differential equations, Adomian decomposition method (ADM) [13] for the dynamics of the fractional giving up smoking model of fractional order, homotopy perturbation method (HPM) [25] for the nonlinear fractional Schrödinger equation, homotopy analysis method (HAM) [18] for the conformable fractional Nizhnik-Novikov-Veselov system, differential transformation method (DTM) [19] for the convergence of fractional power series, Elzaki projected differential transform method [22] for system of linear and nonlinear fractional partial differential equations and perturbation-iteration algoritm (PIA) [20] for ordinary fractional differential equations.
In this article, the residual power series method [17,14,4,12,2] is used to obtain new approximate solutions of time-fractional Boussinesq-Whitham-Broer-Kaup equations of the form and diffusive predator-prey equations of the form where u = u(x,t) and v = v(x,t). In the RPSM, the coefficients of the power series are calculated by means of the concept of residual error with the help of one or more variable algebraic equation chains, and finally, a so-called truncated series solution is obtained [17].
The major improvement of the RPSM is that it can be implemented to the problem directly without linearization, perturbation or discretization and without any transformation by selecting appropriate initial conditions [14].
After giving a few preliminary definitions and brief description of the RSPM, we presented the solution procedure of two nonlinear fractional partial differential equations that shows the reliability and efficiency of the method. Also figures and tables are presented in order to compare numerical solutions of considered equations. Finally, we discussed the obtained results in a section as conclusion.

Preliminaries
There are a few definitions of fractional derivative of order α > 0 in the literature. The most widely used are the Riemann-Liouville and Caputo fractional derivatives.

Definition 1.
The Riemann-Liouville fractional derivative operator D α f (x) for α > 0 and q − 1 < α < q defined in [8,9] as: Definition 2. The Caputo fractional derivative of order α > 0 for n ∈ N, n − 1 < α < n, D α * , defined in [6] as: Definition 4. Let f is a function with n variables x 1 , ..., x n , and the conformable partial derivatives of f of order α ∈ (0, 1] in x i is defined as follows [5] Definition 5. The conformable integral of a function f for a 0 is defined in [23] as

Description of the residual power series method
In this section we are going to introduce some important definitions and theorems about residual power series Theorem 2. Suppose that f is an infinitely α−differentiable function at a neigborhood of a point t 0 for some 0 < α ≤ 1, then f has the fractional power series expansion of the form [1]: Here T t 0 α f (k) (t 0 ) represents the application of the fractional derivative k−times.

Definition 6.
A multiple fractional power series about t 0 = 0 is defined by a variable and f n (x) are functions called the coefficients of the series. [3,12].
Theorem 3. Assume that u(x,t) has a multiple fractional power series representation at t 0 = 0 of the form [3] u( If u To clarify the basic concept of RPSM, let's take a nonlinear fractional differential equation of the form: expressed by initial condition is a non-linear operator and c(x,t) are continuous functions.
The RPSM method made up of stating the solution of the equation (12) subject to (13) as a fractional power series expansion around t = 0.
The expansion form of the solution is given by In the next step, the k.truncted series of u(x,t), namely u k (x,t) can be written as: If the (1) RPS approximate solution u 1 (x,t) is then u k (x,t) could be reformulated as x ∈ I and k = 2, 3, 4, ... First we express the residual function as and the k. residual function as It is clear that Res(x,t) = 0 and lim k→∞ Res k (x,t) = Res(x,t) for each x ∈ I and 0 ≤ t. In fact this lead to ∂ (n−1)α ∂t (n−1)α Res k (x,t) for n = 1, 2, 3, ..., k because in the conformable sense, the fractional derivative of a constant is zero [17,14,4]. Solving the equation ∂ (n−1)α ∂t (n−1)α Res k (x, 0) = 0 gives us the desired f n (x) coefficients. Thus the u n (x,t) approximate solutions can be obtained respectively.
We can express the following theorem for the convergence analysis for the method [21].

Theorem 4.
If there exists a fixed constant 0 < K < 1 such that u n+1 (x, y,t) ≤ K u n (x, y,t) for all n ∈ N and 0 < t < R < 1, then the sequence of approximate solution converges to an exact solution.
Proof. For all 0 < t < R < 1, we have

Solution of time-Fractional Boussinesq-Whitham-Broer-Kaup equations
Consider the nonlinear time fractional BWBK equation with the initial conditions obtained from the exact solutions as [27] u( For residual power series and k.truncated series of u(x,t) The k 1 -th residual functions of time fractional BWBK equation is: For determining the coefficients f 1 (x) and g 1 (x) , in u k (x,t), we should subrogate the 1.th truncated series Now we take t = 0 through the equations Resu 1 (x,t) and Resv 1 (x,t) to obtain following equations Thus, for Res 1 (x, 0) = 0, we get Therefore, we obtain the 1st RPS approximate solutions of the time fractional BWBK equation as Again, to determine the second unknown coefficients f 2 (x) and g 2 (x), we subrogate the 2 nd truncated series solutions 2α 2 into the 2 nd truncated residual functions to obtain following equations Now, applying T α on both sides of Resu 2 (x,t) and equating the equations above to zero for t = 0 gives: Therefore the 2nd RPS approximate solutions of time-fractional BWBK equation is obtained as: c 2019 BISKA Bilisim Technology In the same manner, we apply the same procedure for n = 3 to obtain the following results.
In Table 1 x,t)) and the exact solutions with absolute errors for ω = 0.25, k = 2, and t = 0.1.

Solution of time-Fractional Predator-Prey Equations
Now consider the time-fractional diffusive predator-prey equtions of the form with the initial conditions Now repeating the above residual power series proedure we obtained the following results.
In Table 3-4, the third order approximate RPSM solutions of time-fractional Predator-Prey equations are compared numerically with the exact solutions [26] u( for α = 0.75, α = 0.85 and α = 0.95 and absolute errors are presented. The results show that as the absolute value of the x points increase the absolute errors decrease. Besides, as the α values increase, the absolute errors decrease. Also in Figure  3 and 4, the surface plots of the approximate solutions are illustrated for α = 0.65, α = 0.75, α = 0.85 and α = 0.95.

Conclusion
In this paper, approximate solutions of the nonlinear time-fractional Boussinesq-Whitham-Broer-Kaup and Diffusive Predator-Prey equations are obtained by the residual power series method (RPSM) . With conformable fractional derivative definition, we can easily transform fractional differential equations to the known classical differential equations. By the presented method and the conformable fractional derivative definition, it is shown that there is no need to another complex method or definition. Approximate and exact solutions of time-fractional nonlinear differential equations are compered to show the reliability of the method. Absolute errors were given with approximate and exact solutions with the help of figures and tables. Also our work reported here is a first step towards understanding structural and physical behaviour of mathematical models in biology. We hope that our work will be very useful in better understanding the events occurring at mathematical biology.