On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Implicit difference schemes of O(k⁴ + k²h² + h⁴), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.     

Global stability for nonlinear difference equations with variable coefficients

Consider the following nonlinear difference equation with variable coefficients:

     

Comparison theorems for second order nonlinear difference equations

New oscillation results are obtained for the second order nonlinear difference equation

Δ(rnf(Δxn−1))+g(n,xn)=0,     

Some temporal second order difference schemes for fractional wave equations

In this article, motivated by Alikhanov's new work (Alikhanov, J Comput Phys 280 (2015), 424–438), some difference schemes are proposed for both one‐dimensional and two‐dimensional time‐fractional wave equations. The obtained schemes can achieve second‐order numerical accuracy both in time and in space. The unconditional convergence and stability of these schemes in the discrete H¹‐norm are proved by the discrete energy method. The spatial compact difference schemes with the results on the convergence and stability are also presented. In addition, the three‐dimensional problem is briefly mentioned. Numerical examples illustrate the efficiency of the proposed schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 970–1001, 2016     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Homoclinic orbits and subharmonics for nonlinear second order difference equations

We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations.     

Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III

In an earlier work of the author it was proved that the Strichartz estimates for second order hyperbolic operators hold in full if the coefficients are of class . Here we strengthen this and show that the same holds if the coefficients have two derivatives in . Then we use this result to improve the local theory for second order nonlinear hyperbolic equations.

     

Existence of solution for the n-dimension second order semilinear hyperbolic equations

In the paper we establish the local and global existence of solution for the n-dimensional second order semilinear hyperbolic equation with a strongly singular coefficient which appears in the boundary-value problems of fluid dynamics. Based on the analysis about the loss of regularity on the line t=0 for the solution of the corresponding linear equation and the decay at infinity which caused by the singular coefficient, we obtain the existence of a small solution for the semilinear equation by use of fixed point theorem.     

On the oscillatory properties of certain fourth order nonlinear difference equations

Some new criteria for the oscillation of fourth order nonlinear difference equations of the form

     

Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

Consider the first-order neutral nonlinear difference equation of the form

, where τ > 0, σ_i ≥ 0 (i = 1, 2,…, m) are integers, {p_n} and {q_n} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ_n=0^∞ q_n = ∞ or Σ_n=0^∞ nq_n Σ_j=n^∞ q_j = ∞ commonly used in the literature.     

Eigenvalue comparisons for boundary value problems for second order difference equations

We consider the eigenvalue problems for boundary value problems of second order difference equations

(1)

and

(2)

Comparison results for the eigenvalues of the problem (1) and the problem (2) are established.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

具有变系数的偶数阶中立型差分方程的振动性

考虑了一类具有变系数的偶数阶中立型差分方程的振动性,通过建立一个比较定理,获得一些一类具有变系数的偶数阶中立型差分方程的振动性的充分条件.     

Approximate solutions for a class of first order nonlinear difference equations

In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear difference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solution. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given.     

Oscillation of certain second order nonlinear damped difference equations with continuous variable

In this work, a class of second order nonlinear damped difference equations with continuous variable are investigated. Some oscillatory criteria are obtained.     

Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator

In the present paper the first and second orders of accuracy difference schemes for the numerical solution of multidimensional hyperbolic equations with nonlocal boundary and Dirichlet conditions are presented. The stability estimates for the solution of difference schemes are obtained. A method is used for solving these difference schemes in the case of one dimensional hyperbolic equation.     

Positive solutions for boundary value problems of second order difference equations and their computation

We consider the following two classes of second order boundary value problems for difference equation: