Constant Step Research Articles

Formation and Chromatographic Detection of Organic Compound Hydrates

The formation of hydrates of inorganic compounds is well known and is widely discussed in the contemporary literature. The reasons for the insufficiently detailed characterization of numerous well-known hydrates of organic compounds are analyzed, the main of which is their instability. The CAS numbers assigned to many of these hydrates confirm interest in these hydrates. We found that the reversible formation of the hydrates of various organic compounds can be a reason for anomalous dependences of their retention parameters (tR) on the concentration of an organic eluent modifier (c) in reversed-phase high-performance liquid chromatography. The recurrent approximation of the retention parameters tR(c + Δc) = atR(c) + b (*), where Δc = const is a constant step of c, is the most effective way to detect these anomalies. Deviations of dependences like (*) from linearity are observed with compounds in which variations in c and, consequently, in the water content of the eluent (1 – c) affect the equilibrium of the reaction $${\text{X}} + n{{{\text{H}}}_{{\text{2}}}}{\text{O }} \rightleftarrows {\text{X}} \cdot n{{{\text{H}}}_{{\text{2}}}}{\text{O}}.$$

Convective core sizes in rotating massive stars

Context. Spectroscopic studies of Galactic O and B stars show that many stars with masses above 8 M⊙ have been observed in the Hertzsprung-Russell (HR) diagram just beyond the main-sequence (MS) band, as predicted by stellar models computed with a moderate overshooting. This may be an indication that the convective core sizes in stars in the upper part of the HR diagram are larger than predicted by these models. Aims. Combining stellar evolution models and spectroscopic parameters derived for a large sample of Galactic O and B stars with the inclusion of brand-new information about their projected rotational velocities, we reexamine the question of the convective core size in MS massive stars. Methods. We computed a grid of 120 different stellar evolutionary tracks with three initial rotations at solar metallicity (Z = 0.014), spanning a mass range from 7 to 25 M⊙, and combining different values for the initial rotation rate and overshooting parameter. For the rotating models, we considered two cases, one with a moderate and one with a strong angular momentum transport, the latter imposing a solid body rotation during most of the MS phase. We confront the results with two observed features: the position of the terminal age main sequence (TAMS) in the HR diagram and the decrease of the surface rotation when the surface gravity decreases at the end of the MS phase. Results. We confirm that for stars more massive than about 8 M⊙, the convective core size at the end of the MS phase increases more rapidly with the mass than in models computed with a constant step overshoot chosen to reproduce the main sequence width in the low mass range (around 2 M⊙). This conclusion is valid for both the cases of non-rotating models and rotating models either with a moderate or a strong angular momentum transport. The increase of the convective core mass with the mass obtained from the TAMS position is, however, larger than the one deduced from the surface velocity drop for masses above about 15 M⊙. Although the observations that are available at present cannot determine the best choice between the core sizes given by the TAMS and the velocity drop, we discuss various methods of escaping this dilemma. At the moment, comparisons with eclipsing binaries seem to favor the solution given by the velocity drop. Conclusions. While we confirm the need for larger convective cores at higher masses, we find tensions among different methods for stars more massive than 15 M⊙. The use of single-aged stellar populations (non-interacting binaries or stellar clusters) would be a great asset in resolving this tension.

Conversion of geared DC motor into stepper motor using switching delay time signals

In solar tracking system, general stepper motor is used to control the stepwise movement and low speed of solar panel in vertical motion. Degree of rotation is directly proportional to stepwise movement (stepping method) of stepper motor. But it is cumbersome process to hold a solar panel at a particular vertical position depending on the sun’s position using low cost stepper motors. So, geared DC motor is implemented into stepper motor for low-speed applications using Stepping Method (GDCSM). Degree per step movement of geared DC motor is identified using Step angle switching delay time signal (A) of microcontroller. Speed of geared DC motor is controlled by passing fixed time interval between the pulses where the pulses have fixed width. Controlling speed is implemented by using Step delay time signal (D) of microcontroller. Combining effect of switching delay time signals A and D represents the Stepping Method. Stepping method resembles the step movement and controlling speed of the stepper motor. Speed of 10RPM & 30RPM geared DC motors is operated at 10V DC power supply. Microcontroller ATmega 328P with switching delay time signals is used to control the geared DC motors. Performance of 10RPM & 30RPM GDCSM is analyzed with stepper motors in terms of Relative slip degree error per revolution (RE_mathrm{{S}}) and acceptable slippage degree tolerance under open loop condition. Change in consuming voltage tolerance (varDelta) during rotation of GDCSM is another parameter which is used to maintain the constant actual total step count per revolutions of GDCSM. At 40 ms of step angle delay time (A), 10RPMGDCSM has 120 steps per revolution. At 25 ms of A, 30RPMGDCSM has 55 steps per revolution. Speed of 10RPM & 30RPM GDCSM is controllable up to 2RPM when its RE_mathrm{{S}} or acceptable slippage degree tolerance value is less than or equal to 1%. If varDelta value of GDCSM is less than or equal to − 0.4%, then it behaves like stepper motor. GDCSM is suitable to hold and control vertical position of solar tracking system with low speed and step movement. Performance of geared DC motor experimentally showed better result than commercial available stepper motors like 28BYJ-48 or STP-43d1027-01.

Prognostics of radiation power degradation lifetime for ultraviolet light-emitting diodes using stochastic data-driven models

With their advantages of high efficiency, long lifetime, compact size and being free of mercury, ultraviolet light-emitting diodes (UV LEDs) are widely applied in disinfection and purification, photolithography, curing and biomedical devices. However, it is challenging to assess the reliability of UV LEDs based on the traditional life test or even the accelerated life test. In this paper, radiation power degradation modeling is proposed to estimate the lifetime of UV LEDs under both constant stress and step stress degradation tests. Stochastic data-driven predictions with both Gamma process and Wiener process methods are implemented, and the degradation mechanisms occurring under different aging conditions are also analyzed. The results show that, compared to least squares regression in the IESNA TM-21 industry standard recommended by the Illuminating Engineering Society of North America (IESNA), the proposed stochastic data-driven methods can predict the lifetime with high accuracy and narrow confidence intervals, which confirms that they provide more reliable information than the IESNA TM-21 standard with greater robustness.

Walking humans trade off different task goals to regulate lateral stepping

People walk in complex environments where they must adapt their steps to maintain balance and satisfy changing task goals. How people do this is not well understood. We recently developed computational models of lateral stepping, based on Goal Equivalent Manifolds that serve as motor regulation templates, to identify how people regulate walking movements from step-to-step. In normal walking, healthy adults strongly maintain step width, but also lateral position on their path. Here, we used this framework to pose empirically-testable hypotheses about how humans might adapt their lateral stepping dynamics when asked to prioritize different stepping goals. Participants walked on a treadmill in a virtual-reality environment under 4 conditions: normal walking and, while given direct feedback at each step, walking while trying to maintain constant step width, constant absolute lateral position, or constant heading (direction). Time series of lateral stepping variables were extracted, and variability and statistical persistence (reflecting step-to-step regulation) quantified. Participants exhibited less variability of the prescribed stepping variable compared to normal walking during each feedback condition. Stepping regulation results supported our models’ predictions: to maintain constant step width or position, people either maintained or increased regulation of the prescribed variable, but also decreased regulation of its complement. Thus, people regulated lateral foot placements in predictable and systematic ways determined by specific task goals. Humans regulate stepping movements to not only “just walk” (step without falling), but also to achieve specific goal-directed tasks within a specific environment. The framework and motor regulation templates presented here capture these important interactions.

Features of the Recurrent Approximation of Retention Parameters of Polyfunctional Compounds in Reversed-Phase High-Performance Liquid Chromatography

Five antitumor drugs are selected as examples of polyfunctional organic compounds to test the possibilities of approximating their retention parameters in reversed-phase high-performance liquid chromatography using recurrent relationships of the form of tR(C + ΔС) = atR(C) + b, where C is the concentration of organic modifier in the composition of the eluent (acetonitrile), ΔC = const is a constant step of its change in concentration, and a and b are coefficients calculated using the least squares method. Examples of both ideally linear dependences in all ranges of acetonitrile concentrations and deviations from linearity in different ranges of its concentrations are revealed. Discussion of the possible reasons for such deviations (both structural factors and physicochemical properties) suggests that acid–base equilibria in the eluent are not the main ones, the positions of which depend on the concentration of the organic component. The reversible formation of hydrated forms typical of compounds whose molecules contain such polar fragments as amide ‒CO–NH– and sulfonamide –SO2–NH groups, seems more likely. Recurrent approximation of the retention parameters can be considered the simplest and most obvious way to reveal the specific interactions of the test compounds with components of the eluent.

Application of the Modified Incremental Step Test for Pulmonary Rehabilitation.

A Modified Incremental Step Test (MIST) performed in the home may facilitate entirely home-based pulmonary rehabilitation programs. The aims of this study were to investigate the reliability and responsiveness, and the utility of the MIST for exercise prescription in people with stable chronic lung disease. The MIST was undertaken at the center and home in random order, before and after pulmonary rehabilitation, with 2 tests at each time point. Reliability was assessed using intraclass correlation coefficient. Responsiveness was evaluated as effect size. The minimal important difference was appraised using distribution and anchor-based methods. In a substudy, physiological responses to MIST were measured by a portable metabolic system, followed by a constant step rate test at 60% of peak oxygen uptake (VO2peak), to evaluate utility for exercise prescription. Forty-six participants were recruited (29% of eligible candidates). There was excellent reliability for number of steps recorded in home- and center-based settings (intraclass correlation coefficient = 0.954, 95% CI = 0.915-0.976). A small-moderate effect size was demonstrated following pulmonary rehabilitation (0.34), and the minimal detectable change was 7 steps. All participants in the substudy achieved 60% of VO2peak and achieved steady state by the fourth minute, with 60% of VO2peak corresponding to a mean 37% (95% CI = 29-44) of the MIST final level. The MIST is reliable and responsive to pulmonary rehabilitation in people with stable chronic respiratory disease. It provides new opportunities to assess exercise capacity, prescribe exercise training, and reassess exercise program outcomes in environments where established field walking tests are not feasible. Pulmonary rehabilitation is a highly effective treatment that is underutilized worldwide. Home-based pulmonary rehabilitation may improve access for patients and deliver equivalent clinical outcomes but is limited by the availability of a robust exercise test that can be used at home to assess exercise capacity and prescribe training intensity. This study tested the clinimetric properties of the MIST and demonstrated a new way to assess exercise capacity, prescribe exercise training of an appropriate intensity, and reassess exercise capacity in environments where established field walking tests are not feasible.

A kMap optimized VMD-SVM model for milling chatter detection with an industrial robot

Industrial robots play an important role in the milling of large complex parts. However, the robot is less rigid and prone to vibration-related problems; chatter, which affects machining quality and efficiency, is more complex and difficult to monitor. In this paper, a variational mode decomposition-support vector machine (VMD-SVM) model based on information entropy (IE) is built to detect chatter in robotic milling. Significantly, the vibration signals are classified into four states for the first time: stable, transition, regular chatter, and irregular chatter. To improve the accuracy of the identification model based on VMD-SVM, a novel hyper-parameter optimization strategy—the kMap method—is proposed in this paper for optimizing three-dimensional hyper-parameters in the VMD-SVM model. The hyper-parameters of VMD-SVM are jointly optimized by the kMap method, with constant step sizes. As an improved grid search (GS), kMap reduces the operation time to the same order of magnitude as the heuristic algorithm (HA) [comprising particle swarm optimization (PSO) and genetic algorithm (GA)]. The VMD-SVM model with the hyper-parameters optimized by kMap exhibits higher accuracy and better stability than the hyper-parameters optimized by PSO and GA. The results of the validation experiments show that the kMap-optimized identification model is effective in industrial robotic milling.

Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning

This paper discusses a stochastic equilibrium problem for which the function is in the form of the expectation of nonmonotone bifunctions and the constraint set is closed and convex. This problem includes various applications such as stochastic variational inequalities, stochastic Nash equilibrium problems, and nonconvex stochastic optimization problems. For solving this stochastic equilibrium problem, we propose an inexact stochastic subgradient projection method. The proposed method sets a random realization of the bifunction and then updates its approximation by using both its stochastic subgradient and the projection onto the constraint set. The main contribution of this paper is to present a convergence analysis showing that, under certain assumptions, any accumulation point of the sequence generated by the proposed method using a constant step size almost surely belongs to the solution set of the stochastic equilibrium problem. A convergence rate analysis of the method is also provided to illustrate the method’s efficiency. Another contribution of this paper is to show that a machine learning algorithm based on the proposed method achieves the expected risk minimization for a class of least absolute selection and shrinkage operator (lasso) problems in statistical learning with sparsity. Numerical comparisons of the proposed machine learning algorithm with existing machine learning algorithms for the expected risk minimization using LIBSVM datasets demonstrate the effectiveness and superior classification accuracy of the proposed algorithm.

Application of the generalized finite-element method to the acoustic wave simulation in exploration seismology

The generalized finite-element method (GFEM) has been applied frequently to solve harmonic wave equations, but its use in the simulation of transient wave propagation is still limited. We have applied GFEM for the simulation of the acoustic wave equation in models relevant to exploration seismology. We also perform an assessment of its accuracy and efficiency. The main advantage of GFEM is that it provides an enhanced solution accuracy compared to the standard finite-element method (FEM). This is attained by adding user-defined enrichment functions to standard FEM approximations. For the acoustic wave equation, we consider plane waves oriented in different directions as the enrichments, whose argument includes the largest wavenumber of the wavefield. We combine GFEM with an unconditionally stable time integration scheme with a constant time step. To assess the accuracy and efficiency of GFEM, we compare the GFEM simulation results against those obtained with the spectral-element method (SEM). We use SEM because it is the method of choice for wave propagation simulation due to its proven accuracy and efficiency. In the numerical examples, we first perform a convergence study in space and time, evaluating the accuracy of both methods against a semianalytical solution. Then, we consider simulations of relevant models in exploration seismology that include low-velocity features, an inclusion with a complex geometric boundary and topography. Results using these models indicate that GFEM presents comparable accuracy and efficiency to those based on SEM. For the given examples, the GFEM efficiency stems from the combined effect of local mesh refinement, nonconforming or unstructured, and the unconditionally stable time integration scheme with a constant time step. Moreover, these features provide great flexibility to the GFEM implementations, proving advantageous when using, for example, unstructured grids that impose severe time step size restrictions in SEM.

Метод формирующего оператора для моделирования циклостационарных случайных процессов

The paper proposes the shaping operator technique for modelling cyclostationary random process (CSRP) which belong to the class of second-order processes being described by the parametric pulse train model. Generating CSRP is expressed as the propagation result of the specially chosen elementary CSRP through the linear time-invariant system with a known impulse response. The paper describes the approach to forming the elementary CSRP based on the train made of Dirac delta-function uniformly following in time domain with constant step and defining the desired structural periodicity of the process being modelled. The analytical expressions for the main characteristics used for describing cyclostationary properties of the elementary process, including two-dimensional, cyclic and spectral corelation functions, were obtained. In addition to the case of the elementary CSRP with correlated weighting coefficients of delta-functions, the case of statistically independent coefficients was considered. It was shown that utilization of the spectral correlation function (SCF) for describing the cyclic property reveals the explicit analytical relation between characteristics of the elementary and modelled CSRPs provided the filter frequency response is known. The paper presents the comparative example which describes the modelling of two CSRPs: one of them was chosen as the train of rectangle pulses which is pulse amplitude modulated by stationary random time series, while the other is considered as a signal formed by the method of direct sequence spread spectrum. The chosen short-length Barker sequence as the code allowed performing visual comparison between absolute value of SCF components taken at the cyclic frequencies multiple of the chip frequency and cyclic frequencies multiple of the symbol frequency. The future development of the methods proposed in the paper opens the road to improving the performance of modern radar and telecommunication systems by means of utilizing cyclic frequencies which are non-random parameters describing the structural properties of signals under processing.

Digital implementation of continuous-time observers for nonlinear networked control systems

We propose a digital implementation of continuous-time observers to networked control systems. We introduce a sufficiently small constant time step to construct the Euler model of continuous-time o...

EXPERIENCE OF FAMILIES FACING CANCER IN PALLIATIVE CARE

ABSTRACT Objective: to understand the experience of families facing cancer in palliative care of one of its members. Method: this is a qualitative research based on the theoretical-methodological framework of Grounded Theory. Six families participated, totaling 15 people. Data were collected from January to December 2019 in the homes of families in a municipality in southern Brazil, through an in-depth interview with the construction of a genogram. Analysis followed the constant comparative method steps: open, selective and theoretical coding. Results: the concepts discovering cancer; suffering with the reality imposed by the illness; preparing to face illness; deciding on treatment; facing the finitude of life, they are part of the central category “Living one day at a time”, which symbolically defines families’ experiences. The substantive theory elaborated shows that the definitions, actions and strategies adopted by families, in family dynamics and in interpersonal interactions, along a trajectory that culminates in the terminality of a relative’s life, is permeated by hope maintenance. Conclusion: throughout the illness experience, families promote changes in their daily lives and in family dynamics, undergoing adaptations and using their internal strengths in an attempt to re-establish the balance prior to the disease. Families live with uncertainty about the future, but it is the hope that drives them to live one day at a time.

Phase Difference Compensation Method for Range Estimation in an MFCW–CW Radar

Multifrequency continuous-wave-continuous-wave (MFCW-CW) radars simultaneously transmit multifrequency continuous-wave (MFCW) signals sequentially at a constant time step and a continuous-wave (CW) signal constantly. As MFCW signals are transmitted sequentially, only one phase difference exists at each time. All phase differences between the MFCW and CW signals should be exploited to resolve ambiguous ranges. In this article, we propose a method to compensate the phase difference between the sequential MFCW signals and CW signal using the Doppler frequency of the CW signal. To verify the phase difference compensation result, we estimated the range of multiple targets. The simulation results showed that the proposed phase difference compensation method performed well regardless of compensation direction and target motion. Moreover, we proposed applying the maximum likelihood estimation (MLE) and reduced-complexity off-grid sparse Bayesian learning (RC-OGSBL) to estimate ranges and analyzed their performance. The range accuracy of the MLE using a 0.1- m grid search showed superior performance and the running time of the MLE using the simple method is the fastest among the simulated methods. The RC-OGSBL using a 5- m grid showed good results in terms of range accuracy and running time.

Finite element analysis for three-dimensional hyper-elastic problems

Hyper-elastic materials are special elastic material that the stress can be derived by the strain energy density function. The most attractive property of these rubber-like materials is their ability to undergo large strains with small loads and recover their initial shape after unloading. Hyper-elastic materials are widely used in engineering such as tire, elastomeric bearing pad, belt drives, rail pad and so on. Due to the large strain state in most of application of this type of material, the behavior of hyper-elastic is often considered in finite deformation analysis. In comparison with linear material, analysis for hyper-elastic material which have nonlinearities is more complicated because both material nonlinearity and geometric nonlinearity should be considered. Along with the development of computer and numerical methods, the finite element method (FEM) is known as a powerful numerical computation tool that can helps to solve engineering problems in many fields such as structural, thermal, electronic, and biomedical analyses. Actually, most of practical engineering problems appear in three-dimensional (3D) state, especially in nonlinear analyses. So, the development for an effective numerical method for 3D nonlinear problems is very necessary. In this study, a finite element approach with 8-node hexahedron element is presented for analyses of large deformation problems of hyper-elastic material. The compressible neo-Hookean is used as the constitutive model for 3D hyper-elastic problems and the total Lagrange formulation is applied for the discretization of large deformation problems. To obtain the nonlinear solutions, the standard Newton-Raphson algorithm with constant load step is chosen as the iteration method. The computing programs are built with Matlab language. To verify the accuracy of the present approach, the obtained results are compared with the reference solutions given by Ansys program.

ВРАЩЕНИЯ ТОРОВ В СТРУКТУРЕ ЖИДКОГО КРИСТАЛЛА

Liquid crystals are a collection of flattened molecules. On the one hand, they have a welldefined structure, on the other hand, this structure is deformable, since its elements can change their position in space. In columnar structures, consisting of disks, plates, and tori, the fluid lines and cords are distinguished. In the case of tori and circular disks, a hexagonal structure of the arrangement of molecules is observed in a combination of the cord. The aim of this work is to obtain the stability of tori positions in an elementary fragment of a liquid crystal and to analyze their rotations in a self-consistent field of the surrounding toroidal molecules. To solve this problem, the method of mathematical modeling was used, based on classical models of molecular dynamics. The calculation is carried out on the basis of the model of cross-atom-atom interactions for molecular tori. The minimal fragment of the cord is selected, which makes it possible to determine the characteristic dynamic state of the central torus in the fragment. To describe the motion of the molecular tori, the equations of motion for their centers of mass and the Euler equations for their angular displacements are used. The minimal fragment of the material contains twenty-nine tori. The equations for displacements of the centers of mass of the tori are initially represented as ordinary differential equations of the second order. However, by introducing fictitious points into the consideration of velocities, they can be reduced to a system of first-order equations with a doubled number of lower-order equations. The resulting system of the first-order differential equations is integrated numerically using a highorder accuracy step-by-step scheme. All calculations are performed with a constant time step. The accuracy of the obtained numerical results is verified in terms of the balance of total energy of the system. Calculations show that the central torus of the presented fragment executes angular oscillations around its main axis with amplitude of more than one revolution. Thus, the performed calculations show that a representative fragment of the liquid crystal structure of molecular tori can be used as a generator of high-frequency mechanical vibrations.

Distributed Stochastic Consensus Optimization With Momentum for Nonconvex Nonsmooth Problems

While many distributed optimization algorithms have been proposed for solving smooth or convex problems over the networks, few of them can handle non-convex and non-smooth problems. Based on a proximal primal-dual approach, this paper presents a new (stochastic) distributed algorithm with Nesterov momentum for accelerated optimization of non-convex and non-smooth problems. Theoretically, we show that the proposed algorithm can achieve an $\epsilon$-stationary solution under a constant step size with $\mathcal{O}(1/\epsilon^2)$ computation complexity and $\mathcal{O}(1/\epsilon)$ communication complexity. When compared to the existing gradient tracking based methods, the proposed algorithm has the same order of computation complexity but lower order of communication complexity. To the best of our knowledge, the presented result is the first stochastic algorithm with the $\mathcal{O}(1/\epsilon)$ communication complexity for non-convex and non-smooth problems. Numerical experiments for a distributed non-convex regression problem and a deep neural network based classification problem are presented to illustrate the effectiveness of the proposed algorithms.

Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass

We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp., weak sharp minima), local/global convergence (resp., linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular extend and/or improve the corresponding results in [B. Afsari, R. Tron, and R. Vidal, SIAM J. Control Optim., 51 (2013), pp. 2230--2260; G. C. Bento et al., J. Optim. Theory Appl., 183 (2019), pp. 977--992].

Optimization of CNC technology in the manufacturing process of an artistic product

The work analyses the optimisation of the manufacturing process of an artistic wooden product (mirror frame) with a 3-axis CNC machine. The 3D model was first made in ArtCAM, from where the STL file was later uploaded to SolidWorks. The parameters for the HSR (high-speed roughing), and HSM (high-speed machining) operations were determined in SolidCAM. The findings show that a combination of operations from HSR and HSM are the best choice, considering the shape of the model and the capabilities of the machine, as the best ratio between the quality of the surface and the time needed for the operation is provided by these. Operations differ with regard to the tool path and parameters that can be adjusted. The operations were compared and the best results are achieved when the HSR operations “Contour Roughing” and then “Rest Roughing” are used and when the HSM operation “3D Constant Step Over” is used, where the machining time amounted 4 hours and 1 minute. Finally the G code was made for our product.

A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.