Vol 28, No 3 (2024)

Recently, the author defined and developed a new integral transform namely the Khalouta transform, which is a generalization of many wellknown integral transforms. The aim of this paper is to extend this new integral transform to include different fractional derivative operators. The fractional derivatives are described in the sense of Riemann–Liouville, Liouville–Caputo, Caputo–Fabrizio, Atangana–Baleanu–Riemann–Liouville, and Atangana–Baleanu–Caputo. Theorems dealing with the properties of the Khalouta transform for solving fractional differential equations using the mentioned fractional derivative operators are proven. Several examples are presented to verify the reliability and effectiveness of the proposed technique. The results show that the Khalouta transform is more efficient and useful in dealing with fractional differential equations.

In present paper the propagation of plane harmonic coupled waves of temperature increment, translational and spinor displacements in a semiisotropic thermoelastic solid is discussed. Characteristic equations for the wave numbers of plane harmonic coupled thermoelastic longitudinal (bicubic equation) and transverse waves (biquartic equation that naturally splits into two quartic algebraic equations) are obtained and analyzed. For a longitudinal wave, the complex amplitudes of the temperature increment, translational and spinor displacements are also coupled, contrary to a transverse wave. Algebraic forms containing multivalued complex square and cubic radicals for the wave numbers of transverse waves are derived by using the Wolfram Mathematica 13 symbolic computing system.

A numerical method is developed to calculate the relaxation of residual stresses in a rotating surface-hardened prismatic sample with a semicircular notch under high-temperature creep conditions. The problem models the stressed-deformed state of a sample fixed on a disk rotating at a constant speed.
The methodology includes the following steps:
– reconstruction of residual stress and plastic deformation fields after preliminary surface plastic deformation;
– calculation of residual stress relaxation during creep in a rotating prismatic rod.
A detailed analysis is performed on a prismatic sample measuring 150 $\times$ 10 $\times$10 mm made of EP742 alloy. One face of this sample is hardened using mechanical ultrasonic treatment. The problem is analyzed for samples with semicircular notches of 0.1 mm and 0.3 mm radii, located 2 mm and 75 mm from the fixed edge.
For the notched regions after preliminary surface plastic deformation, the problems are solved in both elastic and elastoplastic formulations. The obtained initial fields of residual stresses and plastic deformations are used as input data for the creep problem.
The analysis of the influence of notch radius, location, angular velocity, and initial residual stress fields on the relaxation of residual stresses is conducted at a temperature of 650 °C based on phenomenological flow theory established from known experimental data for this alloy.
Results show that to determine the initial stressed-deformed state after preliminary plastic deformation for a notch radius of 0.1 mm, an elastoplastic solution is necessary, while for a radius of 0.3 mm, the differences between elastic and elastoplastic solutions are minimal.
The study of residual stress relaxation is conducted at angular velocities of 1500 and 2000 RPM over a period of 100 hours. Despite significant relaxation of residual stresses for samples with notches of radii 0.1 mm and 0.3 mm, a substantial level of residual compressive stresses remains in the notch regions after complete thermal-mechanical unloading. This indicates the high effectiveness of surface hardening under high-temperature creep conditions.

As a rule, two piezoelectric elements are used in case of implementing an active strategy for controlling the dynamic behavior of structures that include elements made of piezoelectric materials. One of them acts as a sensor and the other one acts as an actuator. In this case, the key problem is in determining the magnitude of the control signal applied to the actuator, and the hardware implementation of the established control law. Due to the need of constructing of complex electrical circuits representing a control unit, preliminary modeling of the mechanical response to a particular control signal becomes attractive. In this paper, the earlier developed approach was extended to the case of using two piezoelectric elements that perform the functions of a sensor and an actuator, and are located accordingly on the surface of the structure.
This approach allows us to obtain expressions for determining the magnitude of the electric potential generated at the moment of resonance on the electroded surface of a piezoelectric element when it is deformed at the vibration mode under consideration in case of forced steady-state vibrations. All the derivations are performed on the basis of solving the problem of natural vibrations of an electro-viscoelastic structure.
Analytical expressions are derived to determine the magnitude of the control signal which is applied to the actuator and provides damping of a given vibration mode. The control signal is generated by converting the signal received from the sensor.
The applicability of the proposed approach is demonstrated at the example of a cantilever plate made of viscoelastic material, the mechanical behavior of which is described by complex dynamic moduli. Piezoelectric elements acting as a sensor and an actuator are placed on both sides of the plate. Numerical implementation of the proposed approach is carried out based on the finite element method using the ANSYS application software package. A good concordance of the results obtained by the derived formulas with the results of the calculation in ANSYS is demonstrated. The proposed approach makes it possible to significantly reduce time and resource costs in case of mathematical modeling of active control of forced steady-state vibrations of electro-viscoelastic bodies, to determine the conditions that the elements of the control unit must satisfy when implementing an active strategy for controlling the dynamic behavior of such smart systems.

Systematic research into the operations of the regional power system aimed at improving the efficiency of energy complex management, taking into account the contribution of utilized resources, is fundamentally impossible without the enhancement of mathematical models and methods for their identification based on statistical data.
This article presents the results of an analysis of a well-known mathematical description of the functioning of the regional power system, highlighting significant shortcomings that negatively impact both the reliability of assessments of key performance indicators of the energy complex and the accuracy of forecasts made based on the constructed model.
The study examines and systematizes various three-factor regression models and covariance-stationary time series models based on linear and nonlinear regression into three main groups. Algorithms for numerical methods of least squares estimation of the parameters of these models based on observational results are described.
Results of mathematical modeling of the dynamics of energy system output based on statistical data published in the annual reports of regional ministries and energy companies are provided. A statistical analysis of the obtained results is conducted. A comparative analysis of the developed mathematical models based on forecast error assessment allowed for the selection of the most effective mathematical model with minimal forecasting error from the considered set of models over a time period ranging from one to five years.