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In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green's function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L² ?,q((?0, a); E). [ABSTRACT FROM AUTHOR]

We consider symmetric Dirac operators on bounded time scales. Under general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint, etc.) of these symmetric operators. We construct a self-adjoint dilation of the dissipative operator. Hence, we determine the scattering matrix of dilation. Then we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete. [ABSTRACT FROM AUTHOR]

In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maximal deficiency indices (2, 2) (in limit-circle cases at ±∞1) and acting in the Hilbert space ℓ²Ω(ℤ,ℂ²) (ℤ:= {0,±1,±2, ...}) are considered. We deal with two classes of dissipative operators with separated boundary conditions both at -∞ and ∞. For each of these cases, we establish a self-adjoint dilation of the dissipative operator and construct the incoming and outgoing spectral representations. Then, it becomes possible to determine the scattering function (matrix) of the dilation. Further, a functional model of the dissipative operator and its characteristic function in terms of the Weyl function of a self-adjoint operator are constructed. Finally, we show that the system of root vectors of the dissipative operators are complete in the Hilbert space ℓ²Ω(ℤ,ℂ²). [ABSTRACT FROM AUTHOR]

A space of boundary values is constructed for symmetric discrete Dirac operators in ℓ2A(Z; C2) (Z ≔ {0, ±1, ±2, …}) with defect index (1, 1) (in Weyl's limit‐circle case at ±∞ and limit‐point case at ∓∞). A description of all maximal dissipative (accretive), self‐adjoint and other extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called ‘dissipative at −∞’ and ‘dissipative at ∞’. In each of these cases we construct a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of the self‐adjoint operator. We prove the theorem on completeness of the system of eigenvectors and associated vectors of the dissipative operators. [ABSTRACT FROM PUBLISHER]

In this paper, we attempt to display effective numerical simulations of shock wave propagating represented by the Burgers equations known as a significant mathematical model for turbulence. A high order hybrid approximation based on seventh order weighted essentially non-oscillatory finite difference together with the sixth order finite difference scheme implemented for spatial discretization is presented and applied without any transformation or linearization to the Burgers equation and its modified form. Then, the produced system of first order ordinary differential equations is solved by the MacCormack method. The efficiency, accuracy and applicability of the proposed technique are analyzed by considering three test problems for several values of viscosity that can be caused by the steep shock behavior. The performance of the method is measured by some error norms. The results are in good agreement with the results reported previously, and moreover, the suggested approximation relatively comes to the forefront in terms of its low cost and easy implementation. [ABSTRACT FROM AUTHOR]