Vol 28, No 2 (2024)

In this study, we propose a new hybrid numerical method called the Khalouta differential transform method to solve the nonlinear fractional Liénard equation involving the Caputo fractional derivative. The convergence theorem of the proposed method is proved under suitable conditions.
The Khalouta differential transform method is a semi-analytical technique that combines two powerful methods: the Khalouta transform method and the differential transform method. The main advantage of this approach is that it provides very fast solutions without requiring linearization, perturbation, or any other assumptions. The proposed method is described and illustrated with two numerical examples. The illustrative examples show that the numerical results obtained are in very good agreement with the exact solutions. This confirms the accuracy and effectiveness of the proposed
method.

A new algorithm for finding approximate symmetries for fractional differential equations with the Riemann–Liouville and Gerasimov–Caputo fractional derivatives, the order of which is close to an integer, is proposed. The algorithm is based on the expansion of the fractional derivative into a series with respect to a small parameter isolated from the order of fractional differentiation. In the first-order, such an expansion contains a nonlocal integrodifferential operator with a logarithmic kernel.
As a result, the original fractional differential equation is approximated by an integro-differential equation with a small parameter for which approximate symmetries can be found. A theorem is proved about the form of prolongation of one-parameter point transformations group to a new variable represented by a nonlocal operator included in the expansion of the fractional derivative. Knowing such a prolongation allows us to apply an approximate invariance criterion to the equation under consideration.
The proposed algorithm is illustrated by the problem of finding approximate symmetries for a nonlinear fractional filtration equation of subdiffusion type. It is shown that the dimension of approximate symmetries algebra for such an equation is significantly larger than the dimension of the algebra of exact symmetries. This fact opens the possibility of constructing a large number of approximately invariant solutions. Also, it is shown that the algorithm makes it possible to find nonlocal approximate symmetries of a certain type. This possibility is illustrated on a linear fractional differential subdiffusion equation.

The article presents a numerical method for calculating the residual stress fields in a surface-hardened prismatic specimen with a non-through V-shaped crack, based on an elastic-plastic solution to the problem. A detailed analysis of the distributions of residual stresses near the defect will be conducted based on the obtained results across several contours. It is determined that at a crack depth of 0.3 mm, almost all studied components of compressive residual stresses have greater (in absolute value) values than at a depth of
0.1 mm or are equal.

The study is dedicated to further research and development of constructive methods for sequential parametric optimization of unknown characteristics of nonstationary processes in technological heat physics on a compact set of continuous and continuously differentiable functions. The proposed methodology extends the algorithmically accurate method developed for solving inverse problems in technological heat physics to the multidimensional case of the inverse heat conduction problem, allowing the identification of a physically justified characteristic on sequentially converging compact sets.
The research focuses on a two-dimensional axisymmetric body of canonical shape. The problem is formulated in a uniform metric for assessing the temperature deviation of the calculated state from the experimental one. The mathematical model of the studied object is based on its modal description, which led to the reduction of the original inverse heat conduction problem, formulated in an extremal setting, to an optimal control problem.
The use of preliminary parameterization of the sought-after characteristic of the process results in its representation in the form of piecewise-parabolic functions defined by a parameter vector. The number of considered parameters determines the specific type of approximating function, and their values are found by solving the obtained parametric optimization problem. To solve the mathematical programming problem for optimal parameter vector values, alternating properties of the sought extremals are used, similar to the one-dimensional case, leading to the formulation of a closed system of relationships.
The obtained results demonstrate the effectiveness of extending the constructive method of sequential parametric optimization, tested on one-dimensional inverse heat conduction problems, to solving two-dimensional problems using their modal representation. Increasing the number of parameters of solutions forming the piecewise-parabolic form of the sought dependence leads to a reduction in the reconstruction error of both the sought concentrated function and the spatial-temporal temperature field throughout the domain of spatial variables.

Mass transfer in electrodialysis systems during intense current modes is accompanied by the emergence of additional transfer mechanisms that significantly affect their operational efficiency. According to modern concepts, for dilute electrolyte solutions, mechanisms such as electroconvection and the dissociation/recombination reactions of water molecules are particularly important. These processes have opposing effects on the effectiveness of electrodialysis technologies.
Mathematical models that take these mechanisms into account are actively used in membrane system research; however, they typically describe only the potentiostatic regime, in which a potential jump is established in the system. The interpretation of a vast database of experimental data for the galvanodynamic regime (at fixed current density) also requires theoretical analysis tools.
The aim of this work is to develop a mathematical model of mass transfer in the electrolyte solution layer at an ion-exchange membrane, considering electroconvection and water dissociation in the galvanodynamic regime. The model is based on a system of coupled Nernst–Planck–Poisson–Navier–Stokes equations, supplemented by a new galvanodynamic boundary condition for the potential.
Using the developed model, chronopotentiograms of the membrane system were calculated for the first time, taking into account the influence of both electroconvection and the dissociation/recombination reactions of water molecules. The results showed that the ratio of the concentration of water dissociation products to the concentration of salt ions determines the balance of the effects of electroconvection and dissociation.
The following options for balancing the effects of electroconvection and dissociation of water molecules are considered:

1. electroconvection significantly influences mass transfer, while the influence of water dissociation is minimal;
2. electroconvection and dissociation substantially affect transport processes:
   the formation of additional charge carriers from the dissociation of water molecules reduces the potential jump in the electrolyte layer, which decreases the intensity of electroconvection, while the development of electroconvection, in turn, slows down the dissociation process;
3. the products of intense water dissociation slow down the development of electroconvection.

The study examines the Fox function with four parameters, which arises in the theory of degenerate differential equations with partial derivatives of fractional order. In terms of this function, explicit solutions to the first and second boundary value problems in a half-space were previously derived for the equation with the Bessel operator acting on the spatial variable and a fractional derivative with respect to time.
For the function under consideration, when two of the four parameters are dependent, a Laplace transform formula has been obtained, expressed in terms of the special MacDonald function. Additionally, integral transformation formulas have been derived, expressed through the generalized Wright function and the more general

The stress-strain state is considered and the time to fracture of a composite tensile rod during creep under the influence of an active environment is determined. The rod consists of three parts arranged symmetrically in thickness. An additional condition is accepted: all parts of the composite rod are rigidly interconnected without slipping. The creep of each part of the rod is described by a power model with different parameters. To determine the time to fracture, a kinetic equation is used that describes the accumulation of damage during the creep process. For each part of the rod, the same form of the kinetic equation is adopted, but the accumulation of damage occurs under the action of stresses that are different for each part of the rod. The influence of the active medium is determined by the diffusion penetration of its elements into the rod material. An approximate method for solving the diffusion equation based on the introduction of a diffusion front is used. The distribution of stresses in time is analyzed under the condition of penetration of the active medium into different parts of the rod with different diffusion coefficients. As a result of the study, it was shown that the ratio of the constants in the constitutive ratios of the creep of the parts of the rod affects the nature of the accumulation of damage and the distribution of stresses, and therefore, there is an influence on the sequence of destruction of the parts of the composite rod. With an increase in the exponents in the constitutive and kinetic relations, the time until the destruction of the composite rod increases. The dependence of the time to fracture on the ratio of the diffusion coefficients of the active medium in the part of the rod is determined.