Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation

Invariant analysis and explicit solutions of the time fractional nonlinear perturbed Burgers equation

The Lie group analysis method is performed for the nonlinear perturbed Burgers equation and the time fractional nonlinear perturbed Burgers equation. All of the point symmetries of the equations are constructed. In view of the point symmetries, the vector fields of the equations are constructed. Subsequently, the symmetry reductions are investigated. In particular, some novel exact and explicit solutions are obtained

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms