Order reduction and μ-conservation law for the non-isospectral KdV type equation a with variable-coefficients

Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

New explicit solutions of the fifth-order KdV equation with variable coefficients

By means of the modified CK’s direct method, we give out the relationship between variable coefficients of the fifth-order KdV equation and the corresponding constant coefficient ones. At the same time, we have studied the generalized fifth-order KdV equation with constants coefficients using the Lie symmetry group methods. By applying the nonclassical symmetry method we found that the analyzed...

Application of the Kudryashov method and the functional variable method for the complex KdV equation

In this present work, the Kudryashov method and the functional variable method are used to construct exact solutions of the complex KdV equation. The Kudryashov method and the functional variable method are powerful methods for obtaining exact solutions of nonlinear evolution equations.

An analytic solution for a non-local initial-boundary value problem including a partial differential equation with variable coefficients

‎This paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficients‎. ‎At first‎, ‎the non-self-adjoint spectral problem is derived‎. ‎Then its adjoint problem is calculated‎. ‎After that‎, ‎for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined‎. ‎Finally the convergence ...

The modified KdV equation with variable coefficients: Exact uni/bi-variable travelling wave-like solutions

In this paper, the modified Korteweg-de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solution...

Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...