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1

An accurate and efficient local one-dimensional method for the 3D acoustic wave equation

Wu Mengling, Jiang Yunzhi, Ge Yongbin

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

528--552

EN

We establish an accurate and efficient scheme with four-order accuracy for solving three-dimensional (3D) acoustic wave equation. First, the local one-dimensional method is used to transfer the 3D wave equation into three one-dimensional wave equations. Then, a new scheme is obtained by the Padé formulas for computation of spatial second derivatives and the correction of the truncation error remainder for discretization of temporal second derivative. It is compact and can be solved directly by the Thomas algorithm. Subsequently, the Fourier analysis method and the Lax equivalence theorem are employed to prove the stability and convergence of the present scheme, which shows that it is conditionally stable and convergent, and the stability condition is superior to that of most existing numerical methods of equivalent order of accuracy in the literature. It allows us to reduce computational cost with relatively large time step lengths. Finally, numerical examples have demonstrated high accuracy, stability, and efficiency of our method.

2

Time-optimal control of linear fractional systems with variable coefficients

Matychyn Ivan, Onyshchenko Viktoriia

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 3

375--386

EN

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle are obtained. Theoretical results are supported by examples.

3

Solutions of recurrences with variable coefficients for slide bearing wear determination

Wierzcholski K.

Journal of KONES

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2013

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Vol. 20, No. 3

427--433

EN

The numerous methods of numerical calculations occurring in power-train tribology and transport concerning wear bearing determination problems demand the more and more information referring the slide bearing wear anticipation in succeeding years of machine operations. Therefore this paper presents the methods of solutions of some specific class of ordinary non-homogeneous recurrence equations of second and higher order with variable coefficients occurring in hydrodynamic theory of bearing wear problems. Contrary to linear recurrence equations with constant coefficients, linear recurrence equations with variable coefficients rarely have analytical solutions.. Numerical solutions of such equations are always practicable. In numerous analytical methods of solutions of linear recurrence equations with variable coefficients there are usually three research directions. The first of them depend upon the successive determination of the linear independent particular solutions of the considered recurrence equation. The second direction to be characterized by the reduction of the order of recurrence equation to obtain an always solved, first order recurrence equation. The third direction of solutions of recurrence equations with variable coefficients, contains the methods of analytical solutions by means of a summation factor. The majority of the general methods of analytical solutions of linear recurrence equations with variable coefficients constitute an adaptation of the methods applied in solutions of suitable differential equations In final conclusions the application of presented theory in this paper contains the the examples referring the wear values determination of HDD bearing system in the indicated period of operating time.