Numerical Method for 2D Quasi-linear Hyperbolic Equation on an Irrational Domain: Application to Telegraphic Equation

We develop a novel three-level compact method (implicit) of second order in time and space directions using unequal grid for the numerical solution of 2D quasi-linear hyperbolic partial differential equations on an irrational domain. The stability analysis of the model problem for unequal mesh is discussed and it is revealed that the developed scheme is unconditionally stable for the Telegraphic equation. For linear difference equations on an irrational domain, the alternating direction implicit method is discussed. The projected technique is scrutinized on several physical problems on an irrational domain to exhibitthe accuracy and effectiveness of the suggested method. Dhaka Univ. J. Sci. 69(2): 116-123, 2021 (July)

Properties of critical and subcritical second order self-adjoint linear equations

We discuss critical and subcritical linear second-order difference equations, and we observe several identities and inequalities which such equations satisfy depending on their coefficients. Later, we investigate the limit behaviour depending on the coefficients of solutions and of the sequences which appear when finding said solutions. We will see that certain identity is preserved in limit under weaker assumptions. Finally, we investigate a class of fourth-order linear difference equations and show that they are always 1-critical.

Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

Instability of second-order nonhomogeneous linear difference equations with real-valued coefficients

In J. Comput. Anal. Appl. (2020), pp. 152--165, the author dealt with Hyers--Ulam stability of the second-order linear difference equation $\Delta_h^2x(t)+\alpha \Delta_hx(t)+\beta x(t) = f(t)$ on $h\mathbb{Z}$, where $\Delta_hx(t) = (x(t+h)-x(t))/h$ and $h\mathbb{Z} = \{hk|\,k\in\mathbb{Z}\}$ for the step size $h>0$; $\alpha$ and $\beta$ are real numbers; $f(t)$ is a real-valued function on $h\mathbb{Z}$. The purpose of this paper is to clarify that the second-order linear difference equation has no Hyers--Ulam stability when the step size $h>0$ and the coefficients $\alpha$ and $\beta$ satisfy suitable conditions. Finally, a necessary and sufficient condition for Hyers--Ulam stability is obtained.