Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation

Abstract

In this paper, we investigate some analytical, numerical and approximate analytical methods by considering time-fractional nonlinear Burger-Fisher equation (FBFE). (1/G )-expansion method, finite difference method (FDM) and Laplace perturbation method (LPM) are considered to solve the FBFE. Firstly, we obtain the analytical solution of the mentioned problem via (1/G )-expansion method. Also, we compare the numerical method solutions and point out which method is more effective and accurate. We study truncation error, convergence, Von Neumann's stability principle and analysis of linear stability of the FDM. Moreover, we investigate the L 2 and L∞ norm errors for the FDM. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace perturbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.

Symmetry Analysis and Conservation Laws of the Boundary Value Problems for Time-Fractional Generalized Burgers' Differential Equation

“…Later Abd-el-Malek and Amin in [16] studied the symmetry analysis of the generalized (1+1)-dimensional Burgers differential equation in the formwith boundary and initial conditions u(0, x) −→ ∞, for x > 0, u(t, 0) = γr(t), for t > 0, γ = 0, and lim x→∞ u(t, x) −→ ∞, for t > 0. Some recent studies of Burgers differential equation the reader can see in [17,18]. In this research, we show the applying of Lie group analysis to study (2+1)-dimensional time-fractional generalized Burgers' differential equation with boundary and initially conditions:…”

“…Also, we find conservation laws for the nonlinear generalized Burgers' differential equation.They obtained the symmetries, according them conservation laws and some analytical solutions for above equation. Later Abd-el-Malek and Amin in [16] studied the symmetry analysis of the generalized (1+1)-dimensional Burgers differential equation in the formwith boundary and initial conditions u(0, x) −→ ∞, for x > 0, u(t, 0) = γr(t), for t > 0, γ = 0, and lim x→∞ u(t, x) −→ ∞, for t > 0.Some recent studies of Burgers differential equation the reader can see in [17,18]. In this research, we show the applying of Lie group analysis to study (2+1)-dimensional time-fractional generalized Burgers' differential equation with boundary and initially conditions:Email addresses and ORCID numbers: gulistan.iskandarova@istanbulticaret.edu.tr, https://orcid.org/0000-0001-7322-1339 (G. Iskenderoglu), dogank@ticaret.edu.tr, https://orcid.org/0000-0002-3420-7718 (D. Kaya)…”

Exact solutions of the Benney–Luke equation via (1/G')-expansion method

“…(1 / ) G -expansion method [6][7][8][9] the Clarkson-Kruskal direct method [10], the auto-Bäcklund transformation method [11], decomposition method [12], homogeneous balance method [13], the first integral method [14], residual power series method [15], collocation method [16], modified Kudryashov method [17], sine-Gordon expansion method [18,19], the improved Bernoulli sub-equation function method, [20] and so on [21][22][23][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Analysis and numerical computations of the fractional regularized long‐wave equation with damping term

“…Avci et al 24 considered a heat conduction equation with respect to the Caputo‐Fabrizio fractional derivative. Many other fractional models have been treated with those fractional‐order derivatives 25–45 …”