Economical factorized schemes for third-order pseudoparabolic equations

Изучены экономичные факторизованные схемы для псевдопараболических уравнений третьего порядка. На основе общей теории устойчивости разностных схем доказаны устойчивость и сходимость разностных схем. Economical factorized schemes for pseudo-parabolic equations of the third order are studied. On the basis of the general theory of stability of difference schemes, the stability and convergence of difference schemes are proved.

On Solutions of Fractional Telegraph Model with Mittag-Leffler Kernel

In this paper, we research the fractional telegraph equation with the Atangana-Baleanu-Caputo derivative. We use the Laplace method to find the exact solution of the problems. We construct the difference schemes for the implicit finite method. We prove the stability of difference schemes for the problems by the matrix method. We demonstrate the accuracy of the method by some numerical experiments. The obtained results confirm the accuracy and effectiveness of the proposed method. Additionally, the numerical results demonstrate that the expected physical properties of the model are also observed.

Generalization of the Lax–Ryabenky–Philippov theorem to nonlinear problems

In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h0. In this case, the value of h0 depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.

Compact difference schemes for Klein-Gordon equation

In this paper, we consider compact difference approximation of the fourth-order schemes for linear, semi-linear, and quasilinear Klein-Gordon equations. with respect to a small perturbation of initial conditions, right-hand side, and coefficients of the linear equations the strong stability of difference schemes is proved. The conducted numerical experiment shows how Runge rule is used to determine the orders of convergence of the difference scheme in the case of two independent variables.

On the Stable Difference Schemes for the Schrödinger Equation with Time Delay

In the present paper, the first and second order of accuracy difference schemes for the approximate solutions of the initial value problem for Schrödinger equation with time delay in a Hilbert space are presented. The theorem on stability estimates for the solutions of these difference schemes is established. The application of theorems on stability of difference schemes for the approximate solutions of the initial boundary value problems for Schrödinger partial differential equation is provided. Additionally, some illustrative numerical results are presented.

Using Difference Scheme Method to the Fractional Telegraph Model with Atangana-Baleanu-Caputo Derivative

The fractional telegraph partial differential equation with fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. Laplace method is used to find the exact solution of this equation. Stability inequalities are proved for the exact solution. Difference schemes for the implicit finite method are constructed. The implicit finite method is used to deal with modelling the fractional telegraph differential equation defined by Caputo fractional of Atangana-Baleanu (AB) derivative for different interval. Stability of difference schemes for this problem is proved by the matrix method. Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the proposed method.