Residual power series method for fractional Burger types equations

1 Introduction

Most phenomena in real world are described through nonlinear equations and these type of equations have attracted lots of attention among scientists, especially when these equations are fractional differential equations (FDEs) [1, 2]. FDEs which is the generalized version of integer order differential equations has lots of application in the research areas as diverse as dynamical systems, mechanical systems, control, chaos, chaos synchronization, anomalous diffusive, unifcation of diffusion and wave propagation phenomenon and others [3–5].

Therefore it is an important issue for finding the numerical as well as the approximate solution of FDEs. In general, there exists no method that yields an exact solution for a FDEs. Only approximate solution can be obtained by using the different analytical methods. In the open literature, the most common and powerful methods are Adomians decomposition method [6], homotopy perturbation method [7], homotopy analysis method [8, 9], homotopy decomposition method [10, 11], modified reduced differential transform method [12] and homotopy asymptotic method [13].

In this study, an innovative and modified analytical technique namely residual power series method is directly applied to solve time-fractional generalized Burger and B-F equations. RPSM [14–16] is an effective and easy method for constructing power series expansion solutions of non-linear FDEs of different types and orders without linearization, perturbation, or discretization. The advantage of this method over the classical power series method, is it does not require comparing the coefficients of the corresponding terms, and a recursion relation is not necessary.

Consider the following generalized time-fractional B-F equation [17]

with the initial condition:

whose exact solution given as [17]

Here η, β and y are constants. α is the time-fractional derivative lies in 0 < α ≤ 1.

The Burger equation describes the far field of wave propagation in nonlinear dissipative systems. A numerical simulation and explicit solutions of the B-F equation was studied by Kaya and El-Sayed [18]. Wazwaz presented the tanh method [19] to solve generalized form of B-F equation. El-Wakil and Abdou [20] studied non-linear Fisher and B-F equations by using modified extended tan h-function method and Rashidi et. al [21] presented the homotopy analysis method to find an explicit analytical solutions of the generalized Burger and B-F equations.

Rest of the paper systematized as follows: basic idea of residual power series method is presented in section 2. Application of RPSM to generalized B-F equations are presented in Section 3. Results and discussion of B-F equation, Generalized Fisher equation and Fisher equation are presented in Section 4, 5 and 6 respectively. Finally, Section 7 conclude the whole output of the paper.

2 Basic idea of residual power series method

Method is discussed through the following fractional partial differential equation

where, R[x], is the general linear operator in x, N[x], is the general nonlinear operator in x, and g(x, t) are continuous functions. In RPSM method the complete solution equation (2.1) can be expressed as

In fact, u(x, t) satisfy the initial condition given by u(x, 0) = f(x); so from Eq. (2.3), we obtain, ∂lu(x,t0)∂tl=l!f0l(x)=ϕl(x),l=0,1,2,…,m−1. Then f0l(x)=(ϕl(x)l!,l=0,1,2,…,m−1 and the initial guess approximation of u(x, t) can be written as

Using Eq. (2.4)Eq. (2.3) can be written as:

Suppose u_{(a, b)}(x, t) is the (a, b)-truncated series of u(x, t). That is

where a and b running from a = 1,2,3, … and b = 0,1,2,3, m-1.

Next, as per the method for finding the coefficients f_kl(x) in the series expansion of Eq. (2.6), we define the residual function concept for Eq. (2.1) as

and the following (a, b)-truncated residual function:

As described in [14-16], it is clear that Res (x, t) = 0 for each t є [t₀, t₀ + R), x є R, where R is a nonnegative real number. In fact, this shows that Dtk−1αDtlRes (x, t) = 0 for each k = 1, 2, 3, ..., a and l = 0,1,2,3, ..., b, since the fractional derivative of a constant function in the Caputo sense is zero. Again, the fractional derivatives Dtk−1αDtl for each k = 1,2,3,..., a and l = 0,1,2,3,..., b of Res(x, t) and Res(_{x, t}) are matching at (x, t) = (x, t₀). Therefore we have the following equation

This procedure can be repeated till the arbitrary order coefficients of the multiple FPS solution of Eq. (2.7) are obtained. Finally, complete solution of Eq. (2.1) can be found from Eq. (2.3).

3 Application of RPSM to Generalized B-F equation

Taking generalized B-F equation (1.1). According to RPSM technique by taking as ϕ0x=12−12tanhηy21+yx1y, the series solution of Eq. (1.1) can be written as

where u_0,0(x, t) = f(x) is the initial guess approximation. Next, the method consists as, the (a, b)-truncated series of u(x, t) and the (a, b)-truncated residual function of Eq. (1.1) can be defined and thus constructed, respectively, as follows:

For finding f₁₀(x), in the expansion of Eq. (3.2) we substitute the (1, 0)-truncated series u(_{1, 0})(x, t)= f(x) + f₁₀(x)tαΓ1+α into the (1, 0)-truncated residual function

Now from the results of RPSM, in case of (k, l) = (1, 0) and putting t =0, one can obtain

Hence, the (1, 0)-truncated series solution of Eq. (1.1) could be expressed as

In same manner, the rest of the components of f_k0(x) for k ≥ 2 can be completely obtained. Finally, the solutions of Eq. (1.1) are given by equation Eq. (3.1).

3.1 Numerical discussions

In this subsection, we have discussed the numerical computational of the generalized B-F equation through the different graphical representation.

In this regard, Fig. 1 and Fig. 2 show the absolute error curve and the behavior of the approximate analytical solution for different fractional Brownian motions α = 0.7, α = 0.8, α = 0.9 and α = 1 respectively. Whereas, Fig. 3 show the comparison of the approximate solution obtained by RPSM method with respect to the exact solution. All the sub-figures under the Fig. 3 shows that, they are identical in nature.

Fig. 1

Plot of absolute error E₃ = |u(x, t) − u₃(x, t)|.

Fig. 2

Plot of u₃(x, t) vs. time t at x = 1 and different value of α.

Fig. 3

Surface shows the 3rd order approximate solution of the time-fractional generalized B-F equation: (a) u₃(x, t) when α = 1, (b) u₃(x, t) when α = 0.75, (c) u₃(x, t) when α = 0.5, (d) u₃(x, t) when α = 0.25, (e) exact solution u(x, t) when α = 1.

4 B-F equation

When y = 1, Eq. (1.1) is reduced to the B-F equation. Results are verify through the following Figures 4, 5, 6.

Fig. 4

Plot of absolute error E₃ = |u(x, t) − u₃(x, t)|.

Fig. 5

Plot of u₃(x, t) vs. time t at x = 1 and different value of α.

Fig. 6

Surface shows the 3rd order approximate solution of the time-fractional B-F equation: (a) u₃(x, t) when α = 1, (b) u₃(x, t) when α = 0.75, (c) u₃(x, t) when α = 0.5, (d) u₃(x, t) when α = 0.25, (e) exact solution u(x, t) when α = 1.

5 Generalized Fisher equation

When β = 0, Eq. (1.1) is reduced to the generalized Fisher equation. Results are verify through the following Figures 7, 8, 9.

Fig. 7

Plot of absolute error E₃ = |u(x, t) − u₃(x, t)|.

Fig. 8

Plot of u₃(x, t) vs. time t at x = 1 and different value of α.

Fig. 9

Surface shows the 3rd order approximate solution of the time-fractional generalized Fisher equation: (a) u₃(x, t) when α = 1, (b) u₃(x, t) when α = 0.75, (c) u₃(x, t) when α = 0.5, (d) u₃(x, t) when α = 0.25, (e) exact solution u(x, t) when α = 1.

6 Fisher equation

When y = 1 and β = 0, Eq. (1.1) is reduced to the Fisher equation. Results are verify through the following Figures 10, 11, 12.

Fig. 10

Plot of absolute error E₃ = |u(x, t) − u₃(x, t)|.

Fig. 11

Plot of u₃(x, t) vs. time t at x = 1 and different value of α.

Fig. 12

Surface shows the 3rd order approximate solution of the time-fractional Fisher equation: (a) u₃(x, t) when α = 1, (b) u₃(x, t) when α = 0.75, (c) u₃(x, t) when α = 0.5, (d) u₃(x, t) when α = 0.25, (e) Exact solution u(x, t) when α = 1.

7 Conclusion

In this paper, generalized B-F equation, B-F equation, generalized Fisher equation and Fisher equation have been constructed and RPSM was successfully employed to develop the solutions. The reliability of the proposed method is showing with the application of these equations. An important outcome to be noted that only third iterations provide fairly accurate solutions as illustrated by the various Figs. Therefore, from the above discussion we can conclude that RPSM is an one of the most suitable alternative analytical method for handling the FDEs arising in different branches of sciences and technology.