A Crank-Nicolson Approximation for the time Fractional Burgers Equation

Fractional order integral and derivative are the generalizations of classical integral and derivative concepts which are examined in detail by Leibniz and Newton. The concepts of fractional integral and derivative are as old as integer order integral and derivative concepts, and the fractional derivative expression is first mentioned by Leibniz's letter to L’Hospital in 1695 [1]. In the letter, Leibniz's question was ’Can the integer order derivatives be generalized to fractional order derivative’. This is known as the first emergence of the concept of fractional differential. In addition to Leibniz, many scientists such as Liouville, Riemann, Weyl, Lagrange, Laplace, Fourier, Euler and Abel have worked on the same subject [2]. Many definitions are given in the literature for fractional derivative. Some of these are Riemann-Liouville, Caputo, Grünwald-Letnikov, Wely, Riesy fractional derivatives [3]. Some studies have shown that these definitions are equivalent under certain conditions. There is more than one derivative definition in the fractional analysis, making it possible to use the most suitable one according to the problem and thus to get the best solution for this problem.

However, if the derivative is described as how the derivative of the fractional order is defined, the expression that when the order is selected being equal to the integer is the same as the integer order of the derivative.

The description of Caputo fractional derivative was first introduced by the Italian mathematician M.Caputo in the 1960s to eliminate the problem of the calculation of Riemann-Liouville definition of initial values in Laplace transform applications. The fundamental advantage of the Caputo approach is the fact that the appropriate initial conditions defined for Caputo fractional differential equations are identical. Therefore, in recent studies in the literature, in the exact and approximate solutions of fractional differential equations, instead of Riemann-Liouville fractional derivative operator, Caputo fractional derivative operator has been more preferred. Recently, studies on the solution of fractional differential equations have increased. There are several studies about fractional problems and their computational accuracy in the literature [4, 5, 6]. Since the exact solution for many of the fractional differential equations are not found, various methods have been developed to find approximate or numerical solutions.

In this manuscript, we will deal with the nonlinear time fractional Burgers equation with the given initial and boundary conditions as a test problem presented as (1)Dtγ u+uux−νuxx=f(x,t)u(x,0)=g(x), 0≤x≤1.u(0,t)=h1(t),u(1,t)=h2(t), t≥0\matrix{{D_t^\gamma \;u + u{u_x} - \nu {u_{xx}} = f(x,t)} \hfill \cr {u\left({x,0} \right) = g\left(x \right),\;\;\;\;\;\;\;\;\;0 \le x \le 1.} \hfill \cr {u(0,t) = {h_1}(t),u\left({1,t} \right) = {h_2}\left(t \right),\;t \ge 0} \hfill \cr} in which ν is the viscosity parameter, u(x, t) represents the speed of fluid media at time-space position (x, t) and Dtγu(x,t)=1Γ(1−γ)∫at(t−τ)−γ∂u(x,τ)∂τdτ, 0<γ≤1D_t^\gamma u\left({x,t} \right) = {1 \over {\Gamma (1 - \gamma)}}\int_a^t {(t - \tau)^{- \gamma}}{{\partial u(x,\tau)} \over {\partial \tau}}d\tau,\quad 0 < \gamma \le 1 is the fractional derivative given in the Caputo's form [1, 7]. The Burgers’ equation is the one of the most important differential equations arising in applied sciences and mathematical physics and has interesting applications in physics and astrophysics. The Burgers’ equation examines the modeling of fluid mechanics, diffusive waves in fluid dynamics and also it has many application in the theory of shock waves, mathematical modeling of turbulent fluid, sound waves in a viscous medium and so on. In Mathematical modelling, fractional derivatives provide more accurate and applicable models for real life problems. For the same reasons, recent researches have concentrated on investigating and proposing models on Fractional Burgers Equations. During the years, this attractive modelling ability of the equation have taken attention of most of scientific people, many important and useful papers have brighten science world. In this part of our manuscript, we want to remind some of them; Esen and Tasbozan [8] have investigated numerical solutions of the equation via quadratic B-spline Galerkin method which is an useful and efficient type of finite element approach, then they have obtained more accurate numerical solutions with using collocation points and raising the degree of basis in [9]. Yaseen and Abbas, in [12], have used collocation method with the help of cubic trigonometric B-splines basis. Qui et al [10] have proposed an implicit difference scheme with L1 algorithm which is a specific time discretization of the Caputo fractional derivative. A numerical method focused on a finite difference scheme in terms of time and the Chebyshev spectral collocation method in terms of space is used to get approximate solutions of the Burgers’ equation in [11]. Saad and Al-Sharif [16] have applied variational iteration method for solving the equation considering several initial conditions. Asgari and Hosseini [13] have focused on generalized time fractional Burger type equation, they put forward two semi implicit Fourier pseudospectral approximations for seeking solutions of the equation. As a different view to the mentioned equation, Khan et al [14] are used the generalized version of the differential transform method and homotopy perturbation method. Lombard and Matignon [15] present an article for better understanding the competition between nonlinear effects and nonlocal relaxation.

Throughout this manuscript, in order to contribute to literature, we will consider numerical solutions of the time fractional Burgers equations using finite difference approach. For the presented numerical solutions, to get a Crank Nicolson finite difference scheme to solve the time fractional Burgers equation as utilized in explicit difference method in Ref. [17], we will also discretize the derivative respect to time using the widely-known L1 formula [18] (2)∂γf∂tγ|tn≅(Δt)−γΓ(2−γ)∑k=0m−1bkγ[f(tn−k)−f(tn−k−1)], 0<γ≤1{\left. {{{{\partial ^\gamma}f} \over {\partial {t^\gamma}}}} \right|_{{t_n}}} \cong {{{{\left({\Delta t} \right)}^{- \gamma}}} \over {\Gamma \left({2 - \gamma} \right)}}\sum\limits_{k = 0}^{m - 1} b_k^\gamma \left[ {f\left({{t_{n - k}}} \right) - f\left({{t_{n - k - 1}}} \right)} \right],\quad 0 < \gamma \le 1 where bkγ=(k+1)1−γ−k1−γ.b_k^\gamma = {\left({k + 1} \right)^{1 - \gamma}} - {k^{1 - \gamma}}.

Numerical results obtained for the Eq. (1) are got using the Crank Nicolson finite difference methods. The efficiency of the present methods are tested using the error norm L₂L2=‖uexact−(UN)j‖2=Δx∑j=0M(ujexact−(UN)j)2{L_2} = {\left\| {{u^{exact}} - {{\left({{U_N}} \right)}_j}} \right\|_2} = \sqrt {\Delta x\sum\limits_{j = 0}^M {{\left({u_j^{exact} - {{\left({{U_N}} \right)}_j}} \right)}^2}} and also the maximum error norm L_∞L∞=‖uexact−(UN)j‖∞=maxj|ujexact−(UN)j|.{L_\infty} = {\left\| {{u^{exact}} - {{\left({{U_N}} \right)}_j}} \right\|_\infty} = \mathop {max}\limits_j \left| {u_j^{exact} - {{\left({{U_N}} \right)}_j}} \right|.

Example 1.Firstly, we are going to take time fractional Burgers equation (1) given with the following appropriate boundary conditionsU(0,t)=0,U(1,t)=0,t≥0U(0,t) = 0,U(1,t) = 0,t \ge 0and the initial condition asU(x,0)=0,0≤x≤1.U(x,0) = 0,0 \le x \le 1.

The term f (x, t) is of the form [9]f(x,t)=2t2−γsin(2πx)Γ(3−γ)+2πt4sin(2πx)cos(2πx)+4νπ2t2sin(2πx).f(x,t) = {{2{t^{2 - \gamma}}\sin \left({2\pi x} \right)} \over {\Gamma \left({3 - \gamma} \right)}} + 2\pi {t^4}\sin \left({2\pi x} \right)\cos \left({2\pi x} \right) + 4\nu {\pi ^2}{t^2}\sin \left({2\pi x} \right).

The analytic solution for the present problem is described asU(x,t)=t2sin(2πx).U\left({x,t} \right) = {t^2}\sin \left({2\pi x} \right).

A comparison for the exact and approximate solutions got by the Crank-Nicolson methods for fractional Burgers equation for the values of γ = 0.5, Δt = 0.00025 and t_f = 1 for various values of M is presented in Table 1. As one could clearly see from this table, the decreasing values of the error norms L₂and L_∞validate this comment. In Table 2, the approximate solutions obtained for values of M have been compared with those in Ref. [9]. Those error norms for each values of M got by this method are smaller than those presented in Ref. [9].

The solutions obtained at different times in Fig. 1 were given graphically with exact solutions. The results obtained at different times were found to be too close to each other on the graphics.

Fig. 1

In Table 3, solutions for different viscosity values were given for values of γ = 0.5, Δt = 0.00025, t_f =1.0, N = 40. It was observed that the errors increased gradually at small values of viscosity.

Table 4 presents the error norms obtained at various selected values of γ.

Example 2.Secondly, we will consider the model problem with the appropriate boundary conditionsU(0,t)=t2,U(1,t)=−t2,t≥0U(0,t) = {t^2},U(1,t) = - {t^2},t \ge 0and the initial condition asU(x,0)=0,0≤x≤1.U(x,0) = 0,0 \le x \le 1.

The termf(x,t)=2t2−γcos(πx)Γ(3−γ)−πt4cos(πx)sin(πx)+νπ2t2cos(πx)f(x,t) = {{2{t^{2 - \gamma}}\cos \left({\pi x} \right)} \over {\Gamma \left({3 - \gamma} \right)}} - \pi {t^4}\cos \left({\pi x} \right)\sin \left({\pi x} \right) + \nu {\pi ^2}{t^2}\cos \left({\pi x} \right)and the analytic solution for the problemU(x,t)=t2cos(πx).U(x,t) = {t^2}\cos (\pi x).

In Table 5, for various values of N, the error norms obtained for Example 2 have been illustrated. In the table, one can obviously see that the improvement in numerical solutions has been observed as expected for the increasing values of N.

In Table 6, the error norms obtained for different values of Δt have been given for values of ν = 1.0, Δx = 0.025 and γ = 0.5. As the Δt values decreased, it was observed that the errors also decreased accordingly. In Figure 2, the solutions of Problem 2 obtained at different times for values of ν = 1.0, Δx = 0.025, γ = 0.5 and Δt = 0.0025 have been presented together with their analytical values.

Fig. 2

Example 3.Lastly, we consider the model problem with the boundary conditionsU(0,t)=t2,U(1,t)=et2,t≥0U(0,t) = {t^2},U(1,t) = e{t^2},t \ge 0and the initial conditions asU(x,0)=0,0≤x≤1.U(x,0) = 0,0 \le x \le 1.

The function f (x, t) is of the following formf(x,t)=2t2−γexΓ(3−γ)+t4e2x−νt2ex.f\left({x,t} \right) = {{2{t^{2 - \gamma}}{e^x}} \over {\Gamma \left({3 - \gamma} \right)}} + {t^4}{e^{2x}} - \nu {t^2}{e^x}.

The analytical solution of the problem is given byU(x,t)=t2ex.U(x,t) = {t^2}{e^x}.

In Table 7, approximate solutions obtained for values of t_f = 1.0 and, Δx = 0.025 have been given at different Δt values. It is obvious that the errors get smaller and smaller as Δt values get smaller and smaller. Figure 3 shows the numerical and exact solutions obtained at different times for Problem 3. It is seen that the results are indiscriminately close to each other on the graphics.

Fig. 3

As a conclusion, in the present study the numerical solutions of the time fractional Burgers equation have been obtained by using finite difference method based on Crank-Nicolson discretization. The obtained results are compared with analytic and some of the numerical results available in the literature. This comparison has shown that the presented method is efficient and effective and can also be used for a wide range of physical and scientific applications. Moreover to illustrate the accuracy of the present method the error norms L₂ and L_∞ are computed and given in tables. Two test problems have been used to show the accuracy of the present scheme for various values of parameters in the problem. Tables and figures show the results of these various tests and also comparisons with some available results together with the error norms L₂ and L_∞. Finally the present method has been shown to be applicable for more widely used fractional differential equations.