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Starting with the bilinear form of the generalized ( 2 + 1 )-dimensional Kadomtsev–Petviashvili equation, multiple-order rogue waves have been obtained through symbolic computation. Maximum and minimum values as well as their trajectories for the first order rogue wave are systematically discussed. Moreover, the second order and third order rogue waves are established by eliminating the impact of the mixed partial derivative, and the numerical simulations are demonstrated to visualize their temporal evolution.

In this paper, we investigate the numerical solution of general N -coupled nonlinear Schrodinger equations on unbounded domains, that describe the multiple solitary waves in physics. Due to the unboundedness of the physical domain and strong nonlinearity, how to design appropriate numerical approach to obtain the numerical solution and simulate the real physical phenomena is a major challenge. Based on the ideas of artificial boundary method and operator splitting approach, we develop local artificial boundary conditions for the N -coupled nonlinear Schrodinger equations on the introduced artificial boundaries. Then the original initial value problem was reduced into an initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. We rigorously prove the stability of the reduced problem by introducing a series of auxiliary variables, which are vital to overcome the difficulty of the mixed partial derivatives in the local artificial boundary conditions. Numerical simulations are also presented to validate accuracy and the stability of our proposed method.

In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. Our results in special cases yield some of the recent results on Opial’s inequality and provide new estimates on inequalities of this type. As an application, we prove the uniqueness of the solution of the initial value problem involving higher order partial differential equations.

a b s t r a c t In this work a unique solvability of a class of mixed type partial differential equations with some unbounded coefficients is proved in R 2 . The estimates of the weighted norms of the solution u and its partial derivatives ux and uy are derived.

A new linearized oscillation criterion is established for a nonautonomous scalar delay differential equation. The assumptions include a weak recurrence property of a partial derivative of the nonlinearity. The importance of the recurrence assumption is shown by an example.

This paper is concerned with the construction of exact series solution of mixed variable coefficient hyperbolic problems.

The authors use their recently proved integral inequality to obtain bounds for the covariance of two random variables 1. in a general setup and 2. for a class of special joint distributions. The same inequality is also used to estimate the difference of the expectations of two random variables. Finally, the authors study the attainability of a related inequality.

The eigenvalue problem L( p ,λ) x = 0 is considered, where the matrix L depends analytically on the eigenvalue λ and a vector parameter p . Some results on the existence and the numerical computation of partial derivatives of eigenvalues λ and eigenvectors x with respect to the components of p are announced and discussed. Proofs will appear elsewhere.

We formulate an optimal design problem for the selection of best states to observe and optimal sampling times and locations for parameter estimation or inverse problems involving complex nonlinear partial differential systems. An iterative algorithm for implementation of the resulting methodology is proposed.

The incomplete beta function B x ( a , b ) is defined for a , b > 0 and 0 x 1 and its definition was extended to negative integer values of a and b in [E. Ozcag¯, I. Ege, H. Gurcay, An extension of the incomplete beta function for negative integers, J. Math. Anal. Appl., in press (doi:10.1016/j.jmaa.2007.05.075)]. In this work we define the partial derivatives of the incomplete beta function B x ( a , b ) for negative integer values of a and b .