Large‐Time Asymptotics for Periodic Solutions of the Modified Burgers Equation

Abstract

In this paper, we construct large-time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study.

A fourth-order numerical scheme for solving the modified Burgers equation

“…(i) for t ≥ 100 a convergence of the numerical solution to the analytical one appears, (ii) for t ≥ 250 the MPC method has given results more accurate than the corresponding ones in [45], (iii) from t = 300 instead of t = 450 in [35] no visible differences in the curves of u and U (see Fig. 2) appear.…”

“…The MBE equation, which has the strong nonlinear aspects of the governing equation ( is used in many practical transport problems such as those of nonlinear waves in a medium with low-frequency pumping or absorption, ion reflection at quasi-perpendicular shocks, turbulence transport, wave processes in thermoelastic media, transport and dispersion of pollutants in rivers and sediment transport, etc. Theoretical solutions of the MBE equation are found in, among other references, Harris [33], Sachdev and Rao [34], Sachdev et al [35], Inan and Ugurlu [36], while numerical solutions are found in Ramadan and El-Danaf [37] using the collocation method with quintic splines, Ramadan et al [38] with finite elements with a method based on the collocation of septic B-splines, Dağ et al [39] with finite elements with a method based on the collocation of cubic B-splines, Dağ et al [40] using both quadratic and cubic B-spline Galerkin finite-element methods, Saka and Dağ [41] with the use of a quartic B-spline collocation method, Saka and Dağ [42] with the use of a quintic B-spline collocation method, Irk [43] with the sextic B-spline collocation method, Temsah [44] using the El-Gendi method, Duan et al [45] using a lattice Boltzmann model, etc. The initial condition associated with Eq.…”

“…The MBE3 equation [35] using as initial condition Table 3 MBE2 equation. Comparisons for the proposed method (h = 0.001 and = 10 −5 ).…”

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

“…Example Consider MBE‐III equation (Equation with η = 3) with initial condition using d = π and A = 1 subject to boundary conditions: Sachdev et al constructed a solution to the above problem by means of asymptotic expansion and it can be represented as: where …”

Asymptotics of solutions to the periodic problem for a Burgers type equation

“…Periodic problem for the Hopf equation was studied in paper [15]. The asymptotic behavior for large time of solutions to the periodic problem for the generalized Burgers equations was studied in papers [18,23]. The existence of unique global solutions for the periodic problem to Korteweg-de Vries-Burgers equation was proved, and estimates of the large time decay rates of solutions were found in papers [2,16,22].…”