Necmettin Aggez

A Note on the Difference Schemes of the Nonlocal Boundary Value Problems for Hyperbolic Equations

Abstract The nonlocal boundary-value problem for hyperbolic equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. Applying the operator approach, we establish the stability estimates for solution of this nonlocal boundary-value problem. In applications, the stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained. The first and second order of accuracy difference schemes generated by the integer power of A for approximately solving this abstract nonlocal boundary-value problem are presented. The stability estimates for the solution of these difference schemes are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments.

Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

Numerical Solution of Stochastic Hyperbolic Equations

A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

On the Solution of NBVP for Multidimensional Hyperbolic Equations

We are interested in studying multidimensional hyperbolic equations with nonlocal integral and Neumann or nonclassical conditions. For the approximate solution of this problem first and second order of accuracy difference schemes are presented. Stability estimates for the solution of these difference schemes are established. Some numerical examples illustrating applicability of these methods to hyperbolic problems are given.

Boundary value problem for a third order partial differential equation

Boundary value problems for third order partial differential equations in a Hilbert space are investigated. The stability estimates for the solution of the boundary value problem is established. To validate the main result, some stability estimates for solutions of the boundary value problems for third order equations are given.

NBVP for semilinear hyperbolic equations

A nonlocal boundary value problem for the semilinear hyperbolic equation in a Hilbert space is considered. The uniqueness of the solution of this problem is established. The first order and the second order of accuracy difference schemes for the approximate solution of this problem are presented. The convergence estimates for the solution of these difference schemes are obtained. To validate the main results, these difference schemes are applied to the one dimensional semilinear hyperbolic problem.

NBVP for hyperbolic equations involving multi-point and integral conditions

Nonlocal boundary value problems involving multi-point and integral conditions for a hyperbolic equation in a Hilbert space are investigated. The stability estimates for the solution of these multi-point NBVP are established. In applications, the stability estimates for the solution of these problems are obtained.

Numerical Solution of NBVP for Hyperbolic Equations

The stable difference schemes for approximate solution of the multidimensional hyperbolic equations with nonlocal integral and nonclasic conditions are presented. The stability estimates for the solution of this difference schemes are established. The theoretical statements supported by numerical examples.