Related papers: Variable separated ODE method--A powerful tool for…
In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be…
Under the traveling wave transformation, Camassa-Holm equation with dispersion is reduced to an integrable ODE whose general solution can be obtained using the trick of one-parameter group. Furthermore combining complete discrimination…
The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g.,…
We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each…
Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then…
We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also…
Ordinary differential equations (ODEs) are foundational in modeling intricate dynamics across a gamut of scientific disciplines. Yet, a possibility to represent a single phenomenon through multiple ODE models, driven by different…
Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using an elliptic function method which is more general than the $\mathrm{tanh}$-method. The method works by…
In this paper, a variable-coefficient Gardner equation is considered. By using the classical symmetry analysis method symmetries for this equation are obtained. Then, the generalized Jacobi elliptic function expansion method is used to…
The modified method of simplest equation is applied to the extended Korteweg - de Vries equation and to generalized Camassa - Holm equation. Exact traveling wave solutions of these two nonlinear partial differential equations are obtained.…
This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical…
Existence and bifurcation results are derived for quasi periodic traveling waves of discrete nonlinear Schrodinger equations with nonlocal interactions and with polynomial type potentials. Variational tools are used. Several concrete…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…
Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition,…
Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…
This paper is concerned with the traveling wave solutions and asymptotic spreading of delayed lattice differential equations without quasimonotonicity. The spreading speed is obtained by constructing auxiliary equations and using the theory…
We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE…
In this paper, we present a complete classification of traveling wave solutions for monostable systems within a unified framework. To this end, we introduce a novel technique, referred to as the slicing method, which is based on the…