Differential Ones Research Articles

A detailed study of the current‐reuse VCO's operation and a simple circuit technique to generate intrinsic‐balanced outputs

SummaryIn the first part of this paper, we closely examine well‐known methods for balancing the output power level of the pseudo‐differential current‐reuse voltage‐controlled oscillator. We scrutinize their effectiveness in terms of canceling output mismatches, insensitivity to process, voltage, and temperature variations, and their capability of not degrading phase noise performance. Through analyses and simulations, it is shown that contrary to common belief, some techniques are not as effective as previously thought and should be employed only under specific circumstances. The best conventional approach is proven to be the use of a common‐mode shunting capacitor at the inductor's center tap node to create a virtual ground. In the second part, it is demonstrated that the current‐reuse voltage‐controlled oscillator can intrinsically produce balanced outputs without using any additional techniques or extra passive or active components. Using single‐ended tank capacitors instead of differential ones, the common‐mode voltage of the outputs is shunted to the real ground. This technique reintroduces a reference voltage to the tank circuit, forcing the oscillator to achieve equilibrium between positive and negative swings. The proposed current‐reuse oscillator is designed and post‐layout simulated in a 65‐nm CMOS process, covering a frequency range of 4.84–6.38 GHz with an average power consumption of 0.65 mW. It produces balanced outputs with an amplitude error of less than ±0.2% and achieves a phase noise of −126.7 dBc/Hz and FoMT of 201.8 dB at a 3‐MHz offset frequency.

Key secondary metabolite markers for Wuchang Daohuaxiang rice discrimination in China

This study aimed to comprehensively elucidate the vital secondary metabolites of Wuchang Daohuaxiang (DHX) rice through widely targeted metabolomics analysis. Among the secondary metabolites detected, a total of 30 differential ones were screened out and categorized into 4 different classes, including 6 alkaloids (20%), 15 flavonoids (50%), 6 phenolic acids (20%), and 3 terpenoids (10%) between DHX and control groups. Of these, compounds as zarzissine, fagomine, arbutin, p-Hydroxypheny-β-D-allopyranoside, pimaric acid, kaurenoic acid, and isopimaric acid were more abundant in DHX than control group, with the possibility in serve as key secondary metabolites of DHX rice. Furthermore, arbutin, trigonelline and 6′-O-Feruloyl-D-sucrose were optimized as potential biomarkers for DHX rice discrimination. This study would supply data support for DHX rice authenticity and quality improvement.

Particle Swarm Optimization or Differential Evolution—A comparison

In the mid 1990s two landmark metaheuristics have been proposed: Particle Swarm Optimization and Differential Evolution. Their initial versions were very simple, but rapidly attracted wide attention. During the last quarter century hundreds of variants of both optimization algorithms have been proposed and applied in almost any field of science or engineering. However, no broader comparison of performance between both families of methods has been presented so far. In the present paper ten Particle Swarm Optimization and ten Differential Evolution variants, from historical ones from the 1990s up to the most recent ones from 2022, are compared on numerous single-objective numerical benchmarks and 22 real-world problems. On average Differential Evolution algorithms clearly outperform Particle Swarm Optimization ones. Such advantage of Differential Evolution over Particle Swarm Optimization is in contradiction with popularity: In the literature Particle Swarm Optimization algorithms are two–three times more frequently used than Differential Evolution ones. Problems for which Particle Swarm Optimization performs better than Differential Evolution do exist but are relatively few. Although this result may be an effect of the choice of specific variants, experimental settings or problems used for comparison, some re-consideration of algorithmic philosophy may be needed for Particle Swarm Optimization variants to make them more competitive.

Comprehensive analysis of radiation impact on the transfer of heat and mass micropolar MHD free convective fluid flow across a stretching permeability sheet with suction/injection

AbstractAs part of our research, we investigate the analysis influence of radiation on heat and mass transfer free convection of micropolar MHD fluids over a stretched porosity sheet involving suction and injection. The governing energy, rotational momentum, and concentration and momentum partial differential equations are transformed into ordinary differential equation ones via a similarity transformation. This system of equations is then solved by using MATLAB's built‐in solver. The Sherwood numbers, Nusselt, friction factor, wall couple shear stress, and dimensionless profiles are all influenced by the various physical parameters of the flow. When the material parameter is increased, velocity rises but decreases when the magnetic parameter and surface condition factor are increased.

Discrete and continuous models of linear elasticity: history and connections

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange’s 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier’s partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange’s 1D lattice. Neglecting shear springs in Born–Kármán’s lattice equals to Navier’s assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid (“rari-constant” theory: equal Lamé parameters, or, equivalently, Poisson’s ratio $$\upsilon =1/4$$ ). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born–Kármán’s lattice in plane stress in the limit tends to a continuum with Poisson’s ratio $$\upsilon = 1/3$$ . Contrary to McHenry–Hrennikoff’s truss, Born–Kármán’s lattice leads to a “free” Poisson’s ratio bounded by its “limit’ bound ( $$\upsilon =1/4$$ for plane strain or 3D elasticity; $$\upsilon =1/3$$ for plane stress elasticity). Unfortunately, Born–Kármán’s lattice model does not comply with rotational invariance principle, for non-central forces. The consistent generalization of Lagrange’s lattice in 3D was achieved only by Gazis et al. considering an elastic energy that depends on changes in both lengths and angles of the lattice. An alternative consistent three-parameter elastic lattice is the Hrennikoff’s, with additional structure in the cell. We also discuss the capability of nonlocal continuous models to bridge the gap between continuum isotropic elasticity at low frequencies and lattice anisotropic elasticity at high frequencies.

The <i>n</i>th-order self-resonance law of single wave train for surface capillary-gravity waves in deep water

Wave-wave resonance mechanism plays a fundamental and prominent role in the process of energy transfer and distribution, whether it is in microscopic or macroscopic matter. For the most extensive and intuitive ocean surface wave motion on earth, it is bound to be even more so. Can we extract the general wave-wave resonance law from it, especially the most special and brief resonance law for single wave train? To this end, according to a set of classical methods proposed by Phillips for initiating modern water wave dynamics with the specific 4-wave resonance conditions, and starting from the basic governing equations of ocean deep-water surface capillary-gravity waves, the first-order differential equation, and the second-, third- and fourth-order integral differential ones, which are becoming more and more complex but tend to be complete, of the Fourier components of free surface displacement are respectively given by the Fourier-Stieltjes transformation and perturbation method. Under a set of symbol system, which is self-created, self-evident and concise, these equations are solved in turn to obtain the first-order free surface displacement of single wave train, the Fourier coefficients of the second-, third- and fourth-order non-resonant and resonant free surface displacements, and the second-, third- and fourth-order resonant conditions, thus leading to the general nth-order self-resonance law of single wave train. This completely reveals the rich connotation of single wave resonance dynamics of ocean surface capillary-gravity waves, effectively expands the application range of the classical single wave resonance solutions given by Phillips for ocean surface gravity waves, lays the foundation for depicting single and multiple resonance interaction mechanisms of double and multi-wave trains of ocean surface waves, and so provides a typical example for the exploration of single-wave resonance law in all wave fields.

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.

Gyrotactic Motile Microorganisms Impact on Pseudoplastic Nanofluid Flow over a Moving Riga Surface with Exponential Heat Flux

Background: The improvement of the thermal conductivity of nanofluids is practical for different processes such as drug delivery, manufacturing of crystals, polymer processing, food and drink, cancer treatment, oil and gas, paper making and for many more. The bioconvection phenomenon has engrossed the attention of numerous researchers for its many applications in biotechnology, mechanical and electrical engineering. Bioconvection nanofluids are more prominent in the fields of biomedicine, pharmacy, nanodrug delivery, biomedical, automotive cooling and the military. Purpose: The major purpose of the current work was to determine the numerical and statistical analysis of a novel thermal radiation and exponential space-based heat source on the bioconvective flow of a pseudoplastic 3D nanofluid past a bidirectional stretched Riga surface. The behavior of the Arrhenius activation energy (AAE) and thermal radiation are also disclosed. Methodology: Suitable similarity transformations were used to transmute the partial differential equations of the flow-modeled phenomena into the structure of ordinary differential ones. The numerical solutions for the renewed set of ODEs were tackled by the bvp4c shooting algorithm built-in MATLAB software. Furthermore, the statistical analysis was computed by applying response surface methodology (RSM). Research implications: The numerical analysis is valid for the incompressible three-dimensional, magnetized flow of a pseudoplastic bioconvection nanofluid through a bidirectional surface with Riga plate aspects in the occurrence of activation energy. Social implications: The flow across three dimensions has quite important implementations in various fields, for example, polymer production, material production technology, the manufacturing of nano-biopolymer computer graphics, industry, powered engineering, aeroplane configurations, etc. The current analysis is more applicable in nanotechnology. Results: The consequences of flow control parameters over flow profiles were studied and explained under the graphic structures. Numerical outcomes were computed and discussed in detail. From the results, it was noted that the velocity field was increased via a larger mixed convection parameter. The temperature distribution was boosted via the thermal Biot number. The concentration of nanoparticles declined via the greater Lewis number. Furthermore, the motile microorganisms field was reduced via the Peclet number. Originality: Until now, no investigation has been recognized to examine the consequences of the bioconvection flow of three-dimensional pseudoplastic nanofluids past a Riga plate containing motile microorganisms utilizing the shooting method called bvp4c. Conclusions: From the results, it was concluded that nanofluids are more helpful for heat transfer increments. Furthermore, from the experimental design observed, the response declined via the thermophoresis parameter, which was significant from the ANOVA observed model.

Оптимизация цифрового регулятора системы автоматического регулирования непрерывного объекта

The article gives the analysis of optimizing the digital controller of the automatic control system (ACS) of a continuous object. The procedure for replacing a continuous controller with a discrete controller has been analyzed. The original continuous optimized ACS has been illustrated. It has been stated that the continuous ACS and its elements (all or particular) can be modeled by discrete models described by the difference equations, and not the differential ones, as is done in the case of continuous models. The specification of the sampling period of the PI controller model equal to 0.001 s has been given, the coefficients of the Z-transfer function of the digital PI controller, which is equivalent to a continuous controller, are considered. Of practical importance in the control theory is the representation of a discrete model of a controller, which helps obtain a digital transformation algorithm carried out by a controller over a control error, organize a discrete-digital control of a continuous object using a digital processor. The model with a continuous and discrete controller is illustrated. A formula for calculating the values of the lattice function of the output signal of a discrete controller is obtained. It has been found that sampling and quantization of the error signal, as well as restoring the control signal, should be carried out with the proper accuracy, which should provide the required control accuracy. There are considered the models of discrete-digital control with the correct selection of the parameters of discrete PI controllers, a model with a finite time of the transient process and a structure of its contour, given a fifth-order discrete function with a five-step transition to the steady state. It has been inferred that reducing the sampling period and quantization step significantly increases the processor speed requirements.

A simple one-dimensional map-based model of spiking neurons with wide ranges of firing rates and complexities

This paper introduces a simple 1-dimensional map-based model of spiking neurons. During the past decades, dynamical models of neurons have been used to investigate the biology of human nervous systems. The models simulate experimental records of neurons’ voltages using difference or differential equations. Difference neuronal models have some advantages besides the differential ones. They are usually simpler, and considering the cost of needed computations, they are more efficient. In this paper, a simple 1-dimensional map-based model of spiking neurons is introduced. Sample entropy is applied to analyze the complexity of the model’s dynamics. The model can generate a wide range of time series with different firing rates and different levels of complexities. Besides, using some tools like bifurcation diagrams and cobwebs, the introduced model is analyzed.

Partial velocity slip effect on working magneto non-Newtonian nanofluids flow in solar collectors subject to change viscosity and thermal conductivity with temperature.

Solar thermal collectors distribute, capture, and transform the solar energy into a solar thermal concentration device. The present paper provides a mathematical model for analyzing the flow characteristics and transport of heat to solar collectors (SCs) from non-Newtonian nanofluids. The non-Newtonian power-law scheme is considered for the nanofluid through partial slip constraints at the boundary of a porous flat surface. The nanofluid is assumed to differ in viscosity and thermal conductivity linearly with temperature changes and the magnetic field is appliqued to the stream in the transverse direction. The method of similarity conversion is used to convert the governing structure of partial differential formulas into the system of ordinary differential ones. Using the Keller box procedure, the outcoming ordinary differential formulas along with partial slip constraints are numerically resolved. A discussion on the flowing and heat transport characteristics of nanofluid influenced by power law index, Joule heating parameter, MHD parameter and slip parameters are included from a physical point of view. Comparison of temperature profiles showed a marked temperature increase in the boundary layer due to Joule heating. The thickness of the motion boundary-layer is minimized and the transport of heat through boundary-layer is improved with the partial slip velocity and magnetic parameters rising. Finally, With an increase in the Eckert number, the distribution of temperature within boundary layer is increased.

Identification of miRNAs-mediated seed and stone-hardening regulatory networks and their signal pathway of GA-induced seedless berries in grapevine (V. vinifera L.)

BackgroundStone-hardening stage is crucial to the development of grape seed and berry quality. A significant body of evidence supports the important roles of MicroRNAs in grape-berry development, but their specific molecular functions during grape stone-hardening stage remain unclear.ResultsHere, a total of 161 conserved and 85 species-specific miRNAs/miRNAs* (precursor) were identified in grape berries at stone-hardening stage using Solexa sequencing. Amongst them, 30 VvmiRNAs were stone-hardening stage-specific, whereas 52 exhibited differential expression profiles during berry development, potentially participating in the modulation of berry development as verified by their expression patterns. GO and KEGG pathway analysis showed that 13 VvmiRNAs might be involved in the regulation of embryo development, another 11 in lignin and cellulose biosynthesis, and also 28 in the modulation of hormone signaling, sugar, and proline metabolism. Furthermore, the target genes for 4 novel VvmiRNAs related to berry development were validated using RNA Ligase-Mediated (RLM)-RACE and Poly(A) Polymerase-Mediated (PPM)-RACE methods, and their cleavage mainly occurred at the 9th–11th sites from the 5′ ends of miRNAs at their binding regions. In view of the regulatory roles of GA in seed embryo development and stone-hardening in grape, we investigated the expression modes of VvmiRNAs and their target genes during GA-induced grape seedless-berry development, and we validated that GA induced the expression of VvmiR31-3p and VvmiR8-5p to negatively regulate the expression levels of CAFFEOYL COENZYME A-3-O-METHYLTRANSFERASE (VvCCoAOMT), and DDB1-CUL4 ASSOCIATED FACTOR1 (VvDCAF1). The series of changes might repress grape stone hardening and embryo development, which might be a potential key molecular mechanism in GA-induced grape seedless-berry development. Finally, a schematic model of miRNA-mediated grape seed and stone-hardening development was proposed.ConclusionThis work identified 30 stone-hardening stage-specific VvmiRNAs and 52 significant differential expression ones, and preliminary interpreted the potential molecular mechanism of GA-induced grape parthenocarpy. GA negatively manipulate the expression of VvCCoAOMT and VvDCAF1 by up-regulation the expression of VvmiR31-3p and VvmiR8-5p, thereby repressing seed stone and embryo development to produce grape seedless berries.

Scrutinization of entropy on MHD Eyring–Powell C71500–Ti6Al4V nanoparticles suspended in a C2H6O2−H2O hybrid base fluid with heat generation

AbstractMicroporous channels are extensively elaborated technologies in a variety of engineering, and scientific applications, for example, microelectromechanical systems that are useful in developing the heat exchangers in miniaturized products such as heat sinks of the microchannel, micromixtures, and fuel cells of miniaturized products. Hybrid nanofluids have a better thermal performance as compared to the nanofluid. Based on this, the present study aims to concentrate on the magnetohydrodynamics Powell–Eyring hybrid nanofluid flow in a vertical porous channel with the aspects of porous medium, magnetism, convective condition, dissipation energy, heat generation, and radiation. A nondimensional technique is applied to alter the governing equations into a nonlinear system of ordinary differential ones that contain the convective boundary condition. The RKF 4–5th order method coupled with the shooting method has been utilized to find the numerical solution. From the perspective of numerical results, it is clear that the entropy generation boosts up for Grashof number, Biot number, and radiation parameter and it is the minimum for magnetism, Darcy number, and Eyring–Powell parameters. The comparison of results has been performed and it provides an excellent agreement.

Numerical study for bioconvection transport of micropolar nanofluid over a thin needle with thermal and exponential space-based heat source

Nanomaterials are the latest technique to upgrade thermal conduction of base liquids. The submerging of these nanomaterials in base liquid is known as nanofluid. In recent time the nanofluids are more useful in mechanical engineering, nanotechnology, bioscience, biotechnology and in many sectors. This analysis deals with the bio-convection transport of micropolar liquid containing nanomaterial's and gyrotactic motile microorganisms past a needle moving in parallel flow in occurrence of thermally radiation and activation-energy. Modeling of needle structure for bioconvection flow of micropolar nanofluid is developed. The behavior of temperature and exponential space-based heat source is considered. The boundary layer flow expressions for developed transport issue are represented by PDEs. Suitable similarity variables are used to change transport expressions in nondimensional nonlinear ordinary differential ones. The bvp4c algorithm has been used to numerically tackle dimensionless nonlinear differential equations (ODEs). All results are described by employing bvp4c tool in MATLAB. Impacts of active numbers on flow distributions are examined with sketches and tables. Temperature field and heat transfer rate are quite opposite for exponential space-based heat source sink parameter and heat sink/source parameter. The concentration of nanoparticles is reduced for larger amount of Brownian motion parameter while uplifts for thermophoresis parameter. Concentration is enriched for the growing magnitudes of thermophoresis number. Higher Peclet parameter decayed the microorganism's field.

A decoupled wavelet approach for multiple physical flow fields of binary nanofluid in double-diffusive convection

The paper aims at applications of a decoupled wavelet method for investigating multiple physical steady flow fields of binary nanofluids in double-diffusive mixed convection. The Buongiorno’s mathematical model of nanofluids is further perfected in the presence of Dufour and Soret effects, incorporating with linear and nonlinear diffusiophoresis effects based on experiments [Chemical Engineering Science 176 (2018): 632–640]. Nonhomogeneous thermal boundaries corresponding to heat flux on vertical walls and convection heat transfer at the bottom along with moving top lid are effectively approximated by interpolated Coiflet-type wavelet. Highly coupled and nonlinear governing equations for the complex fields of temperature, nanoparticles volume fraction and solute concentration have been formulated and decomposed into linear differential ones by homotopy transformation. Numerical wavelet solutions with larger range of physical parameters are finally obtained and validated by solving a set of iterative algebra equations applying Galerkin method, which are difficult to be given by traditional numerical methods. The results reveal that nanoparticles and double-diffusive buoyancy parameters, the thermo-nanofluid and thermo-solutal Lewis numbers, the heat conductivity coefficient, the periodical heat flux with different phase differences, the diffusiophoresis parameters, the nanoparticles and solute Dufour parameters, the solute Soret parameter are of great significance on characteristics of heat and mass transfer in the complex flow.

New insight into admissibility analysis for singular systems with time-varying delays

This paper focuses on the admissibility analysis of singular systems with time-varying delays. The goal is to obtain admissibility criterion with less conservatism and fewer decision variables. After decomposing singular systems into differential subsystems and algebraic ones, an improved augmented Lyapunov–Krasovskii functional is established according to decomposed state vectors. Then, combining the auxiliary function-based inequality and the extended reciprocally convex matrix method, a less conservative admissibility criterion with fewer decision variables is obtained. Finally, a numerical example verifies the reduced conservatism and calculation burden of the presented method.

Interchangeability Conditions for Autonomous Processes Models in Finite-dimensional and Infinite-dimensional Spaces

The paper presents the interchangeability conditions for the finite-dimensional, infinite-dimensional, and discrete models of dynamics of autonomous processes and, using the one-dimensional processes as an example, shows that with the certain ratio of the model parameters and appropriate choice of initial functions these conditions are satisfied. Reveals how, subject to the conditions of interchangeability, the process generated by an ordinary differential equation can be represented in the functional space by the trajectory of a discrete dynamic system.Considers dynamic models in the class of ordinary differential equations and delay differential ones. Since the solution spaces of such equations are, in general, different: a finite-dimensional arithmetic space for solutions of the ordinary differential equation and the infinite-dimensional functional space for solutions of the delay differential equation, the problem to reduce dynamic models to the uniform form is associated with representation of processes in both types of spaces. Based on the interchangeability mechanisms of dynamic models of the process under study, the paper proposes methods to reduce the models of dynamics to a uniform form.The results are verified by direct calculations using the specific examples. Interchangeability is used to compare the oppositely directed processes represented by different types of dynamic models. To compare one-dimensional multidirectional processes, presented by dynamic models of various types and having a different behavior pattern, examples of the error models are given. An accuracy of the comparison model of delay type is illustrated by a numerical simulation example.

Numerical Solution of the Vibration Suppression Problem for a Moving Web

Mechanical processes occurring in paper production are modeled. In a papermaking machine, paper moves in the form of a thin sheet. The characteristic thickness of a sheet varies from 0.1 mm (office paper) to 1 mm (cardboard). Every papermaking machine contains open segments where a paper web passes without mechanical support in its motion from one roller to another. On such segments, the web may lose stability, performing transverse vibrations, and, as a result, might tear. The possibility of reducing these vibrations with the help of various control actuators is explored. The transverse vibrations of a moving web with nonzero bending stiffness are modeled by a fourth-order inhomogeneous partial differential equation. The action of control actuators is modeled a function on the right-hand side of the equation. The vibration amplitude is assumed to be identical across the moving web. The vibration suppression problem is reduced to the minimization of a multivariable function. The solution of the problem splits into two stages: the solution of an initial-boundary value problem with a given control and the minimization of a multivariable function. A numerical method is proposed for solving the initial-boundary value problem. The fourth-order differential equation is reduced to a system of two second-order differential ones. The latter are simplified by changing the sought functions. The resulting equations are approximated by a finite-difference scheme, which is proved to be absolutely stable. This scheme is solved using block Gaussian elimination. The multivariable function is minimized by applying the Hooke–Jeeves method. Examples of computations are given for actuators of three types, namely, point actuators, actuators acting on a web segment, and actuators acting along the entire web.

Dissipativity Analysis for Singular Time-Delay Systems Via State Decomposition Method

This paper studies the dissipativity of singular time-delay systems. The objective is to derive a less conservative criterion with less computational demand. Throughout this paper, the augmented Lyapunov–Krasovskii functional (LKF) and the free-matrix-based integral inequality are used to realize less conservativeness. On the basis of these techniques, a less conservative criterion is first given. Then, by decomposing singular time-delay systems into differential equations and algebraic ones, state vectors of the systems are divided into two parts. To reduce redundant decision variables, each part of the state vectors is considered independently to construct a special augmented LKF with less decision variables. As a result, an improved criterion is deduced, and thus less conservativeness and less computational demand are achieved at the same time. Moreover, the presented state decomposition method provides a new research idea for singular time-delay systems not yet reported in the literatures. Finally, the above statements are demonstrated by a numerical example.

Analysis of Dilithium post-quantum electronic signature resistance to fault attacks

Analysis of a perspective variant of post-quantum electronic signature based on algebraic lattices of Dilithium is carried out. The central task of the analysis is to study the resistance of Dilithium to fault attacks, in particular differential ones. First, information is given about the ES scheme itself and its security, fault attacks, their development to differential fault attacks. Possibilities of carrying out these attacks and criteria of their successful execution are considered. The places of the ES algorithm that need protection against fault attacks were identified, such as hash function (the moment of access to it and operation of polynomials multiplying), the stage of loading the private key, the function of expanding seed. Also, nonce reuse and partial nonce reuse when generating keys poses a significant threat, and by carrying out such an attack, the attacker can fully recover the long-term Dilithium private key. Attacks countermeasures are formed based on the sources analysis, their advantages and negative effects are presented. Methods of protection against such attacks are: re-calculation of the signature; verification of signature after signing, which is three times faster than the previous method; introducing additional randomness to the deterministic noise sampling; checking the value of secret and false components (nonce); calculating the average value and variance of the sample, and checking them for belonging to a given range. The results of this work provide researchers with a guide for the development of secure post-quantum electronic signature schemes.