Solvability of eight classes of nonlinear systems of difference equations

We have studied recently solvability and semi‐cycles of eight systems of difference equations of the following form: where a ∈ [0, + ∞), the sequences p_n, q_n, r_n, s_n are some of the sequences x_n and y_n, with positive initial values x_?j,y_?j, j = 1,2, in detail. This paper is devoted to the study of the other eight systems of the form. We show that these systems are also solvable in closed form and describe semi‐cycles of their solutions complementing our previous results on such systems of difference equations.     

A problem with normal derivatives in boundary conditions for a system of differential equations

We seek for a solution to a system of differential equations, using linear relations connecting normal derivatives of the desired functions at the domain boundary.     

A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

In this article, a high‐order finite difference scheme for a kind of nonlinear fractional Klein–Gordon equation is derived. The time fractional derivative is described in the Caputo sense. The solvability of the difference system is discussed by the Leray–Schauder fixed point theorem, while the stability and L_∞ convergence of the finite difference scheme are proved by the energy method. Numerical examples are provided to demonstrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 706–722, 2015     

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A high‐order finite difference method for the two‐dimensional complex Ginzburg–Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 876–899, 2015     

Local solvability for a class of evolution equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L² and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg-Treves condition (ψ) which is suitable to our study.     

Shallow water model for lakes with friction and penetration

We deduce a shallow water model, describ‐ ing the motion of the fluid in a lake, assuming inflow–outflow effects across the bottom. This model arises from the asymptotic analysis of the 3D dimensional Navier–Stokes equations. We prove the global in time existence result for this model in a bounded domain taking the nonlinear slip/friction boundary conditions to describe the inflows and outflows of the porous coast and the rivers. The solvability is shown in the class of solutions with L_p‐bounded vorticity for any given p∈(1,∞]. Copyright © 2009 John Wiley & Sons, Ltd.     

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

On a Two-Point Boundary Value Problem for Second Order Singular Equations

The problem on the existence of a positive in the interval ]a, b[ solution of the boundary value problem is considered, where the functions f and satisfy the local Carathéodory conditions. The possibility for the functions f and g to have singularities in the first argument (for t = a and t = b) and in the phase variable (for u = 0) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.