In this paper, using modified FF-expansion method, we present some explicit formulas of exact traveling wave solutions for the foam drainage equation. A modified FF-expansion method is proposed by taking full advantages of FF-expansion method and Riccati equation in seeking exact solutions of non-linear partial differential equations.

Abstract: In this paper, the nonlinear Schrödinger equation with power law nonlinearity is studied. The first integral method, the Riccati sub-ODE method are efficient methods to construct the exact solutions of nonlinear partial differential equations.By means of these methods, the periodic and solitary wave solutions of the nonlinear Schrödinger equation with power law nonlinearity are obtained.

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained.This approach can be suit...

In this paper, optimal control for stochastic linear singular Takagi-Sugeno (T-S) fuzzy system with quadratic performance is obtained using ant colony programming (ACP). To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is computed by solving differential algebraic equation (DAE) using ACP approach. The solution of this novel method is compared with the ...

We consider a nonlinear time fractional partial differential equation with forced term subject to the Neumann boundary condition. Several sufficient conditions are established for oscillation of solutions of such equation by using the integral averaging method and a generalized Riccati technique. The main results are illustrated by examples. 2014 Elsevier Inc. All rights reserved.

Current gain-scheduling approaches only assure stability when the underlying parameter varies sufficiently slowly, and hence stable closed-loop is not guaranteed for more general (i.e. faster) parameter variations. Shamma and Athans (1992) provides a solution to overcome this by computing Riccati Differential Equation (RDE) online for the current parameter value with the offline Algebraic Ricca...

Sufficient conditions are derived for solutions of a general autonomous system of quad-ratic ordinary vector differential equations to exhibit finite escape times. These results are extended to a matrix Riccati-type differential equation using Kronecker matrix products and a "stacking operator". Finally, these results are applied to a class of models used extensively in population dynamics.