Study of stiff differential equations of mathematical description of izomerization of pentane-hexane cut process

This paper provides a brief overview of the existing definitions of a stiff system of differential equations. Further, on the example of the accepted scheme of chemical transformations of the catalytic isomerization process of the pentane-hexane fraction, the stiffness of the system of differential equations was studied. In the course of the work, a method for studying the direct kinetic problem for stiffness is presented. In the Matlab software, the results of solving a system of differential equations by five methods (solvers) were compared. The given method can be tried for solving other problems of chemical kinetics.

THREE POINTS BLOCK EMBEDDED DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING ODES

Block Embedded Diagonally Implicit Runge-Kutta (BEDIRK4(3)) me- thod derived using Butcher analysis and equi-distribution of error approach is outperformed standard Runge-Kutta (RK) formulae. BEDIRK4(3) method produces approximation to the solution of initial value problem (IVP) at a block of three points simultaneously. The standard one step RK3(2) method is used to approximate the solution at the first point of the block. At the second points the solution is approximated using RK4(2) method which is generated by the previous research. The same approach is used to obtain the solution at the third point. The code for this method was built and the algorithm developed is suitable for solving stiff system. The efficiency of the method is supported by some numerical results.

Shifted Genocchi Polynomials Operational Matrix for Solving Fractional Order Stiff System

In this paper, we solve the fractional order stiff system using shifted Genocchi polynomials operational matrix. Different than the well known Genocchi polynomials, we shift the interval from [0, 1] to [1, 2] and name it as shifted Genocchi polynomials. Using the nice properties of shifted Genocchi polynomials which inherit from classical Genocchi polynomials, the shifted Genocchi polynomials operational matrix of fractional derivative will be derived. Collocation scheme are used together with the operational matrix to solve some fractional order stiff system. From the numerical examples, it is obvious that only few terms of shifted Genocchi polynomials is sufficient to obtain result in high accuracy.

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

Leasing Activities: The Problems and Prospects for Development

The article is aimed at researching the current state of the leasing services market in Ukraine, identifying problems and prospects for its further development. It is defined that leasing is a very promising direction of economic activity, which facilitates the improvement of financial conditions in the country. The article considers the peculiarities of foreign experience in leasing activities and the functioning of the global leasing market. The indicators on the state and development of financial companies and lessors in Ukraine are analyzed, which leads to a conclusion about their positive dynamics. It is determined that today the level of leasing use in Ukraine remains low and is in the process of development, so it requires detailed consideration and analysis. The advantages of financial leasing as an investment instrument, as well as problems and obstacles that slow down the development of leasing activities in Ukraine are presented. Given that leasing activity in Ukraine is under development, there are many urgent problems, which include: imperfection of Ukrainian legislation; the unformedness at the national level of leasing development policy; stiff system of taxation of leasing operations; low demand for domestic goods; low solvency of all categories of consumers; poorly developed subleasing; non-compliance with international standards; low financial stability of lessors; dependence of leasing companies on credit funds from banks; vague regulation on the part of the State, etc. Methods of problem solving are provided that will contribute to improving the functioning of leasing activities and stimulating its development. It is defined that in order to use financial leasing as an innovative instrument for modernizing the country's economy, provision of the State support is necessary.

CHEMICAL CONVERSIONS OF REDUCED CARBON COMPOUNDS AND NITROGEN OXIDES IN THE CRAMPED AERODYNAMIC CONDITIONS OF THE URBAN ATMOSPHERE

The kinetics of chemical transformations of the most significant pollutants in the cramped aerodynamic conditions of the urban atmosphere is considered, i.e. in the gaps between rows of wide buildings with significant height. High concentrations created in those areas lead to sufficiently high values of reaction rates. The process is described by a stiff system of differential equations, the solution of which is performed by Rosenbrock method. The effect of reduced carbon compounds and nitrogen oxides in a wide range of their concentrations on the rate of formation of a highly toxic secondary pollutant is investigated. In the course of numerical experiments, conditions are determined that correspond to its maximum value. In that case, the transfer of substances from the reaction zone does not have time to occur.

Application of Differential Evolution in the Solution of Stiff System of Ordinary Differential Equations

In recent times, the adaptation of evolutionary optimization algorithms for obtaining optimal solutions of many classical problems is gaining popularity. In this paper, optimal approximate solutions of initial--valued stiff system of first--order Ordinary Differential Equation (ODE) are obtained by converting the ODE into constrained optimization problem. The later is then solve via differential evolution algorithm. To illustrate the efficiency of the proposed approach, two numerical examples were considered. This approach showed significant improvement on the accuracy of the results produced compared with existing methods discussed in literature.

Shared Tuned Mass Dampers for Mitigation of Seismic Pounding

This study explores the effectiveness of shared tuned mass damper (STMD) in reducing seismic pounding of adjacent buildings. The dynamics of STMDs is explored through numerical simulations of buildings idealized as single and multiple degree of freedom oscillators. An optimization method proposed in the literature is revisited. It is shown that the optimization results in two different solutions. The first one corresponds to the device being tuned to one of the buildings it is attached to. The second solution corresponds to a very stiff system where the TMD mass hardly moves. This solution, which has been described as an STMD in the literature, is shown to be impractical due to its high stiffness and use of a heavy stationary mass that plays no role in response mitigation but adds unnecessary load to the structure. Furthermore, it is shown that the second solution is equivalent to a viscous coupling of the two buildings. As for the properly tuned solution, i.e., the first solution, sharing the device with an adjacent building was found to provide no added benefits compared to when it is placed on one of the buildings. Based on results from a large set of real earthquake ground motions, it is shown that sharing a TMD mass with an adjacent building, in contrary to what is reported in the literature, is not an effective strategy.