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In this study, Hahn multiplicative calculus was introduced. Some basic theorems are proved within this calculus. As an application of this subject, the classical Sturm-Liouville problem was examined under this structure. [ABSTRACT FROM AUTHOR]

In this paper, we investigate symmetric q — Dirac operator acting in L²q((0, a), C²). We describe maximal dissipative, maximal accumulative, self-adjoint and the other extensions of such operators via the boundary conditions. We construct a self-adjoint dilation of the dissipative operator and determine the scattering matrix of dilation. Later, we construct a functional model of the diisipative operator and define its characteristic function. Finally, we prove that all root vectors of the dissipative operator are complete in the Hilbert space L²q((0, a), C²). [ABSTRACT FROM AUTHOR]

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]