Abstract
A nonlinear transformation from the solution of a linear equation to the solution of the Boussinesq-Burgers equations is derived by using the simplified homogeneous balance method. Based on the nonlinear transformation and various given solutions of the linear equation, various exact solutions, including solitary wave solutions, rational solutions, the solutions containing hyperbolic functions and the solutions containing trigonometric functions, of the Boussinesq-Burgers equations are obtained.
Highlights
IntroductionEquations ((1) & (2)) emerge in the investigation of fluid flow, and describe the proliferation of shallow water waves
In the present paper, we will investigate the well-known Boussinesq-Burgers equations in the form ut + 2uux − αvx = 0, (1)vt + 2(uv)x − αuxxx = 0, (2)where α = constant
) x and u xxx in Equation (2): 2m +1 = n +1, m + n +1 = m + 3 ⇒ m = 1, n = 2, according to the simplified homogeneous balance method [7] [8] [9], we can suppose that the solution of Equations ((1) & (2)) is of the form u where the constants A and B, as well as the function φ = φ ( x,t ) are to be determined later
Summary
Equations ((1) & (2)) emerge in the investigation of fluid flow, and describe the proliferation of shallow water waves. Equations ((1) & (2)) have been investigated by many authors with different methods (see [1]-[6] and references therein). We will propose a somewhat different method to find various exact solutions of Equations ((1) & (2)).
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