This paper studies the problem of optimal control of the viscous KdV-Burgers’ equation. We develop a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the viscous KdV-Burgers’ equation. While the viscous KdV-Burgers’ equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show t...

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analy...

In this paper we calculate the structure functions of one dimensional Burgers and KPZ equations with and without noise, and we interpret our findings using shocks. Most of our results are based on direct numerical simulation of the KPZ equation. Using the solution of noiseless Burgers equation we show that the exponent cq (< |h(x+r)−h(x)| q >∝ rqcq ) of the structure function of noiseless KPZ i...

A non-isotropic version of phase equations such as the Burgers equation, the K-dV-Burgers equation, the Kuramoto-Sivashinsky equation and the Benney equation in the three-dimensional space is systematically derived from a general reaction-diffusion system by means of the renormalization group method. PACS codes: 47.20.Ky

In this article, a generalized (G′/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the h...

In this work, an efficient numerical method for the complex solutions of nonlinear partial differential equations based on the direct algebraic approach is proposed, and tested in the case of generalized Burgers–Huxley equation ∂u ∂t + αu ∂u ∂x − ∂ u ∂x2 = βu(1− u)(u − γ) Whenα = 0, δ = 1, is reduced to the Huxley equation. The proposed scheme can be used in a wide class of nonlinear reaction–d...