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Several papers on the dynamics and early identification of COVID-19 by mathematical modelling have been published in the last two years. This review will look at the quantitative analysis and dynamical behaviors of novel coronavirus, with a focus on the Caputo fractional derivative. Currently many mathematical models are used to investigate the Novel coronavirus under Caputo fractional order system of differential equations. In this paper, we have focused on reviewing the Caputo fractional model of Covid 19, local stability and optimal control. The purpose of this work is to provide the research community a complete overview of the Caputo fractional derivative approach used in these investigations. [ABSTRACT FROM AUTHOR]

This paper studies the properties of Generalized Vieta Pell Functions (GVPFs). Then new definition for such functions is considered with some interesting properties. The definite expression for constructing the operation matrix of the derivative of GVPFs is given, which is the direct method to find the higher-order derivative according to the requirement of the total differential equation. The explicit expression that relates the power function with the GVPFs is formulated in this work. Another contribution of the present paper is the generation of words for the product of two Generalized Vieta Pell Function vectors. All the results mentioned above are applicable in a wide range of computational processes and useful for finding more exact results. [ABSTRACT FROM AUTHOR]

In this paper, direct explicit numerical RKM methods for solving classes of higher-order ordinary differential equations (ODEs) have been developed and modified to be appropriate with fuzzy differential equations (FDEs) with more computational efficiency. The aim of this paper is to derive the numerical methods of RKM type to solve some classes of higher-order FDEs. The numerical results of implementation showed that the efficiency and advantage of the proposed integrators comparing with classical RK methods in sense of evaluation computational time. [ABSTRACT FROM AUTHOR]

This paper exposes the controllability problem of the nonlinear smoking model. Mathematically, the form of smoking model is the first order of nonlinear differential equation system. The model consists of six variables and two controls. The most important thing before controlling a smoker is to determine its controllability. If the smoking model is in nonlinear standard control model, then it has drift. Applying the Lie Bracket method, the controllability matrix can be found. After determining the rank of the controllability matrix, the controllability of the model can be determined. The result can be stated that the smoking model is locally accessible. This means the control design can be made to control the smoker. In addition, if the optimal control approach is used in controlling the smoking model, through its cost functional model, we can make minimization of beginners and smokers. Another result of this paper is the presentation of optimal control analysis using the Pontryagin Principle approach. The analysis results can be stated as follows, first, each solution of the equation in the dynamic system model is a positive function, second, the model of the system of differential equations with the given initial conditions, has a solution, and that solution is a single solution. [ABSTRACT FROM AUTHOR]

In the numerical solution of Initial Value Problems (IVP) for ordinary differential equations, computational methods serve to approximate the determination of functions representing the solution of these problems. The problem of finding the most convenient numerical expressions for a function and its connection with methods for improving such approximations plays an important role in practical calculations. It is of great interest to consider the so-called discrete methods, i.e. methods that determine the solution for discrete values of the independent variable. Discrete methods are currently the most widely used. One of the discrete methods is difference formulas for the numerical solution of the IVP. In this paper, in the Sobolev space where all derivatives up to the mth order participate in the norms of functions, we consider the problem of constructing optimal difference formulas. In the optimization of difference formulas, i.e. when constructing optimal difference formulas in function spaces, an important role is played by the extremal function of a given difference formula. In this paper, the extremal function of the difference formula is explicitly found in the Sobolev space. Next, the square of the norm of the error functional of difference formulas is calculated. Minimizing this norm with respect to the coefficients of the difference formulas, a system of equations is obtained. In addition, the existence and uniqueness of a solution to the resulting system are proved. [ABSTRACT FROM AUTHOR]

The paper deals with option pricing under the classical Black-Scholes model and a more sophisticated Merton jump– diffusion model, which allows jumps in the underlying asset price. The Merton model is represented by non–stationary integro– differential equations. Due to the integral term, standard methods are not very efficient because they lead to full discretization matrices. The Black–Scholes model can be considered a particular case of the Merton model, which does not contain the integral term. In the paper, a method is proposed, which is a combination of the Crank–Nicolson scheme and the wavelet–Galerkin method using adaptive quadratic spline wavelet basis selected a priori. This enables to significantly decrease the number of basis functions and size of matrices and vectors involved in computation compared to standard non–adaptive wavelet–Galerkin methods. Furthermore, the proposed wavelet–based method leads to sparse discretization matrices with uniformly bounded condition numbers. [ABSTRACT FROM AUTHOR]

This paper presents a numerical scheme to approximate a set of differential equations establishing the behavior of Newtonian fluid motion. The three primary differential equations are the continuity equation, the momentum equation (aka the Navier-Stokes equation) and the energy equation. The paper uses a numerical approach namely Compact Integrated Radial Basis Function (CIRBF) to approximate space variables whereas Crank-Nicolson and Adams-Bashforth methods are adopted to discretize temporal variables. More specifically, the spatial approximation at a point is obtained by collecting the desired information of two adjacent nodes along with their corresponding derivatives. Our scheme is verified by comparison with the benchmark solution of natural convection flow in a square cavity. The solution is in good agreement with the previously published results, thereby showing the robustness and reliability of CIRBF approach. [ABSTRACT FROM AUTHOR]

Silicon has remarkable mechanical properties, and it is a very promising material for MEMS pressure sensors. The Silicon diaphragm plays an important role in enhancing the sensitivity and linearity of the pressure sensor. In this paper, a silicon diaphragm is modeled by solving the analytical equations for both rigidly clamped and freely supported diaphragms by exploring the solution of differential equation and dependency on the different parameters of the diaphragm. For obtaining maximum deflection of the diaphragm, boundary conditions are applied for different loads. COMSOL Multiphysics is used to simulate the optimum geometry of the diaphragm for enhancing the sensitivity of the pressure sensor. Results reveal that diaphragm thickness of 20 µm shows the high displacement sensitivity of the pressure sensor. [ABSTRACT FROM AUTHOR]

Laplace transform is a principal technique which is discover to find the solution of various problem in a real life. Laplace transform is one of the most efficient tools which is used by the researchers to identify the solutions of various real problems patterned into differential equations or simultaneous differential equations and integral equations which is easily solvable by Laplace method. In this paper, we are proceeding to study the details on Laplace transform, its properties and "study on solving differential equation by Laplace transform". Laplace transform bring to the ordinary differential equations into an algebraic sum. This paper represents how ordinary differential equations is solved through Laplace transform. [ABSTRACT FROM AUTHOR]

An investigation of the use of Banach spaces for finding solutions to non-linear differential equations is conducted in this paper, focusing on weak and strong solutions. We present 12 theorems and several mathematical assumptions to support the findings, which conclude that, under certain conditions, at least one weak solution for the non-linear differential Equation is obtained in Banach spaces. Furthermore, it is suggested in the paper that the same result can be obtained by expanding upon the compactness assumption. The properties of weak and strong solutions of non-linear differential equations using Banach spaces are improved upon by this research, and its potential real-world applications are highlighted. [ABSTRACT FROM AUTHOR]

The main area of application of various spaces of generalized functions lies in the theory of differential equations and in the theory of quadrature and cubature formulas. Therefore, it becomes necessary to study spaces of generalized functions, one way or another related to various domains in Rn. The theory of differential equations in the space of generalized functions differs from the theory of these equations in the space of ordinary functions. Deriving these equations and finding their solutions are important in applications. In the study of various questions arising in the theory of approximate integration and partial differential equations and related departments of analysis, the so-called Functional approach turned out to very fruitful. In this paper we investigate weighted cubature formula in the functional spaces L ¯ P (m) of S.L. Sobolev for the functions defined in the n - dimensional unit cube Kn and obtain an upper estimate for the norm of error functionals of weighted cubature formulas. Based on the Bakhvalov theorem it is proved that considered cubature formulas of the optimal on order of convergence in these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical method for solving nonlinear q-fractional differential equations(q-FDEs). Here, we consider q-fractional derivative in the Caputo sense. The homotopy perturbation method(HPM) is applied to construct the numerical solutions. Using homotopy technique in topology, a homotopy is built with an installing parameter s ∈ (0, 1), and is treated as a small parameter. Numerical provide the applicability and validity of this method. The proposed method is consistently valid for small parameters as well as for large parameters. [ABSTRACT FROM AUTHOR]

Now a day's integral transform play very important role in differential equations. In this paper we apply SEE Transform Method and prove convolution theorem to solve linear homogeneous and non-homogeneous Fractional Differential Equations with constant coefficients. The Caputo sense used to describe fractional derivatives. The validity and applicability of the presented technique is verified by including some examples. [ABSTRACT FROM AUTHOR]

In this paper, the problem of optimization of multi-revolutionary transfers of a spacecraft with a fixed angular range of auxiliary longitude and the free transfer time from a given initial orbit to geostationary orbit in a central Newtonian gravitational field is considered. The purpose of the optimization is to calculate the program of control of the vector of the thrust, which provided optimal multi-revolutionary trajectories to the final orbit with the minimum cost of the propellant. The magnitude of thrust is assumed to be constant. The optimized control is the vector control program of thrust, including its direction and periods of operation. For mathematical modeling of the spacecraft motion, differential equations in equinoctial elements are used, and as an independent variable, an auxiliary longitude is used, introduced in such a way that the differential equation for it coincides with the differential equation for true longitude in unperturbed motion. An approach based on the Pontryagin maximum principle is proposed to solve the trajectory optimization problem. Using the maximum principle, the problem of optimizing the multi-revolutionary transfer of spacecraft with limited thrust is reduced to a two-point boundary value problem and the boundary value problem is solved by the method of continuation on parameter. To get the goal, the problems of optimizing the multi-revolutionary transfer with limited power and limited thrust are considered. [ABSTRACT FROM AUTHOR]

The paper considers a new approach to the identification of dynamic objects in the form of the first-order differential equations system. The peculiarities of the proposed approach are the absence of restrictions on the structure of the differential equations included in the system and the symbolic representation of the obtained solution. The proposed approach includes a self-configuring genetic programming algorithm for selecting the structure of differential equations. The number of algorithms corresponds to the number of equations in the system. The method of differential evolution is used to optimize numeric constants. The authors apply a self-configuring procedure for the indicated evolutionary algorithms. The proposed approach has been tested on a variety of problems. The accuracy of the obtained solution has been investigated depending on the presence of noise in the sample data. [ABSTRACT FROM AUTHOR]

This study presents a procedure to transfer a Lomax distribution into a class of ordinary differential equations. In this proposed designed some functions, which obtained from Lomax distribution, such as Survival function, cumulative distribution function, and hazard function are used. The aim of this research is to convert the Lomax distribution to the set of differential equations, then find the exact solutions for the created equations. Despite the availability of the exact solution and this is sufficient, the approximate solution, both analytical and numerical, remains a question that we discuss in this paper to study the types of solutions to these mentioned equations. an approximate solution is determined by analytically by Variation Iteration method (VIM) and numerically by Runge-Kutta of 4th Order (RK4) method. Finally, a numerical illustration is also realized by utilizing MATLB 2016a, where the new results are shown. [ABSTRACT FROM AUTHOR]

This paper deals with the Markovian queueing system with working vacation in which the server may met with failures. There will be no delay in initiating repair. During this period any new customer is allowed to join the system. Each customer will be provided Erlangian phase service. Whenever the server finds nobody, the server goes for a working vacation during which the server provides service at a slower rate than the normal. Further, we considered two types of customer's impatience. We solved the system of differential equations to find Transient state probabilities and computed various performance indices of the queue. We then executed the sensitivity analysis and observed the impact on different parameters. [ABSTRACT FROM AUTHOR]

In this paper the Bernoulli wavelets technique is applied to determine the numerical solution of singular ordinary differential equations. From a mathematical view, the suggested technique is straightforward as it provides precise data. According to this technique, the Bernoulli operational matrix is obtained from the Bernoulli function. The Bernoulli operational matrix constitutes a powerful means utilized in analyzing systems to identify the suitable numerical solutions to any differential equations. Numerical examples were give to support the effectiveness of the technique. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of behavior of equilibrium trajectories in dynamic bimatrix games. We consider a system of differential equations which describe evolutionary dynamics of behavior of two players on the infinite time horizon. In the first case, we examine an approach based on the idea of guaranteed strategies in the sense of N.N. Krasovskii, for which the construction of the dynamic Nash equilibrium is implemented. In the second case, we consider a dynamic system based on the strategies of players' best replies. In this case, the equilibrium trajectory converges to the point of the static Nash equilibrium. For both cases, equilibrium trajectories are constructed and the comparison is carried out for the values of players' payoff functionals at the points of equilibrium. It is shown that characteristics of trajectories of the dynamic Nash equilibrium are better than properties of trajectories of the best reply dynamics. For demonstration of the behavior of equilibrium trajectories in the dynamic bimatrix games the model is presented in which we consider payoff matrices of two players on the financial markets of stocks and bonds. [ABSTRACT FROM AUTHOR]

The paper considers non-stationary systems of differential equations. The aim of this work is to obtain sufficient conditions for uniform dissipativity using the method of Lyapunov functions. The right side of the original system is presented as a sum of two terms: the first nonlinear approximation (unperturbed part) and its perturbation. For an exponentially stable unperturbed system, conditions are obtained for the order of perturbations under which the perturbed system has the property of uniform dissipativity. [ABSTRACT FROM AUTHOR]

Modern methods of technical diagnostics, such as thermophysical diagnostics, vibrometry, boundary scanning, vector circuit analysis, have serious drawbacks. They are expressed in high inertia, distraction of processor time, or require galvanic contact with the object of study, which is often unacceptable. These shortcomings can be eliminated by passive radio-sensor technical diagnostics (PRTD), which uses its own radiation (signal radio profiles) to assess the technical condition of a radio-electronic device. In modern scientific literature, the issues of extracting the parameters of technical diagnostics of radio-electronic devices that it provides are practically not covered. The aim of the work is to study the extraction of the parameters of the signal radio profile and compare them with the parameters of radio electronic devices, the assessment of which can be provided by PRTD. Numerical modeling methods, methods of correlation analysis of the results were used to obtain signal radio profiles. In order to find the parameters of the signal radio profile, a mathematical method for solving differential equations was used. The paper presents an equation for the signal radio profile emitted by the electronic unit of the device, as well as an expression for its free components. The possibility of using a modified windowed Fourier transform to find the number of elementary components of a complex radio profile is shown, with an assessment of the correctness of the transformation at the initial stages of the analysis. A method for assessing the correctness of restoring the original signal radio profile is highlighted. A table of correspondence between the parameters of the radio-electronic device and the parameters of the signal radio profile is presented. The extraction of parameters opens up new possibilities in the field of studying the technical condition of radio electronic devices. [ABSTRACT FROM AUTHOR]

The paper proposes a simple approximate method for solving differential equations for boundary value problems. A method is proposed for averaging differential equations over a moving volume, which allows obtaining approximate analytical solutions of differential equations. The control volume is the only one in the considered area of the boundary value problem. In this case, the control volume is considered to be moved in the area under consideration. Based on the averaging of boundary value problems over the volume being moved, an algebraic equation is obtained. When averaging over one of the variables (in the case of a two-dimensional problem), ordinary differential equations are obtained. Examples are given. [ABSTRACT FROM AUTHOR]

A fluid queueing system fed by a single server vacation queue in a random environment is considered in this paper with the aim to investigate the buffer content distribution. The system of differential equations is solved using probability generating function technique and the buffer content distribution for the operative state and vacation state are determined. [ABSTRACT FROM AUTHOR]

In this paper, different types of electrical circuits were modelled and their stochastic sources were computed for different kinds of sources using Caputo Fabrizio derivative. Numerical examples were given to support our results. [ABSTRACT FROM AUTHOR]

The object of this paper is to develop quardratic transformation of κ-hypergeometric differential equation and also develop new forms of quardratic transformation. Further we establish some properties of κ-hypergeometric function with κ-Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend an oscillation criterion for the half-linear second order differential equations with delay to a more general form and to discuss relationship between these two variants and the link to the perturbation principle. The proving technique employs the Riccati transformation. [ABSTRACT FROM AUTHOR]

In the field of robot tracing, traditional sensors have the disadvantages of low accuracy, poor robustness, and expensive, which cannot meet the requirements of modern robot tracing. Based on the above analysis, this paper solves the equations of motion of differential wheeled robots, processes the visual sensor data through erosion and expansion operations to further acquire trajectories, and designs a segmented PID control system. Through repeated simulation experiments with ROS2 and gazebo simulation lines, we have come to the following conclusions: visual tracking has a better foresight, and can well solve the shortcomings of traditional sensors with low precision and poor sensitivity. The control error of the segmented PID control strategy is 50% of the traditional PID control strategy, and it can improve the response speed and robustness of the traditional PID. [ABSTRACT FROM AUTHOR]

In this paper, we have solved the Schrodinger equation of hyperbolic function position dependent mass for a Modified Hyperbolic Pöschl-Teller potential under external hyperbolic magnetic and AB forces using Laplace transform and Hypergeometry method. The second order differential equation of Schrodinger equation reduced to first order differential equation by using Laplace transform. Then radial part of the system was solved by using hypergeometric function. The eigen function and eigen energy values are obtained, where the energy spectra are influenced by the quantum number, potential parameters, and external magnetic with the various of B and AB force are affected depending on the diatomic molecules for certain condition. [ABSTRACT FROM AUTHOR]

The paper investigates a second order difference equation of the Emden-Fowler type Δ2u(k) ± kαum(k) = 0, where k is the independent variable taking values k = k0, k0 + 1, ... with k0 a fixed integer, u: {k0, k0 + 1, ...} → ℝ is the dependent variable and Δ2u(k) is its second-order forward difference. New conditions with respect to parameters m ∈ ℝ, m ≠ 1 and α ∈ ℝ are found such that the equation admits a solution asymptotically represented by a power function asymptotically equivalent with the exact solution of second-order differential Emden-Fowler equation y″(x) ± xαym(x) = 0. [ABSTRACT FROM AUTHOR]

This contribution introduces a novel method for formulating differential equations. This method relies on expanding an implicit function that varies with time (denoted as "t") in the space-time domain using Taylor series. This formulation encompasses both ordinary differential equations (ODEs) and partial differential equations (PDEs). In the context of visualizing vector fields, such as fluid flow and electromagnetic fields, the critical points of ODEs play a crucial role in understanding physical phenomena behavior. This paper outlines a general approach for formulating ODEs and PDEs by treating them as time-varying scalar functions using the Taylor expansion. Furthermore, a new condition for identifying critical points is derived and specified specifically for cases where the function is invariant with respect to time (referred to as "t-invariant"). This newly derived formula enhances the detection of critical points, particularly in the context of acquiring and analyzing large 3D fluid flow data. This advancement enables efficient compression of 3D vector data and their representation through radial basis functions (RBFs). [ABSTRACT FROM AUTHOR]

Modeling and Simulation of Cyber-Physical Systems require to deal with new challenges arising from the growing heterogeneity of the involved components and related interactions, which often exhibit both continuous and discrete behaviors. To formalize the entire system's dynamics and take into consideration the interactions between the physical and cyber worlds, Hybrid Systems are widely adopted. Hybrid System models extend the traditional finite state machine by combining differential equations to model the continuous behavior of the system with a finite control graph, which formalizes discrete behaviors. The correct design and implementation of such Hybrid Systems play an essential role in the simulation of real-world Cyber-Physical Systems, especially for the ones involved in critical industrial plants such as nuclear, chemical, and aerospace. Traditional numerical methods engage a lot of computational resources to simulate a Hybrid System, as a consequence, slow down the simulation significantly. To overcome these computational issues, the paper investigates the use of a recently proposed numerical method, based on the Infinity Computer methodology, to design and simulate Hybrid Systems. The proposed method allows to reduce computational resources by generating observations more densely where it is necessary. To show the validity of the proposed method, an important real-life Hybrid System has been studied and the simulation results have been compared with the standard method. [ABSTRACT FROM AUTHOR]

Usually, the complete system of thermoelasticity equations consists of the equation of motion, the Dugamel-Neumann relations, the Cauchy relation, and the heat influx equation. In this case, usually the boundary value problem is reduced to a system of differential equations for displacements and temperature with the corresponding initial and boundary conditions. Coupled thermoelasticity problems can be formulated in both stress and temperature, but this requires the second order of smoothness of the stress tensor. In this regard, there is an additional condition regarding stress. In this paper, the coupled problems of thermoelasticity are proposed to be considered on the basis of the equations of motion, the continuity equation, the Dugamel-Neumann relation, and the heat influx equation. In this case, a system of three interrelated equations for stress, velocity and temperature was obtained. Based on this system, the problem of the stress-strain state of a homogeneous semi-infinite thermoelastic rod under the action of a dynamic thermomechanical load applied to the end of the rod is considered. The exact analytical solution was obtained by the continuation method followed by the application of the Laplace and Fourier integral transformations. The asymptotic behavior of the stress is studied for small and large values of time. The propagation velocity of a thermoelastic wave has also been studied. Numerical calculations are carried out, the qualitative and quantitative nature of the mutual influence of temperature on the distribution of stresses and velocities of the rod sections is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior and oscillation criteria of neutral differential equations of Emden-Fowler type of third order are studied. Where some conditions were formulated to ensure oscillation for all solutions of these equations. The obtained conditions can be generalized to higher order delay differential equations of Emden-Fowler type. The obtained results showed that the Emden-Fowler type in the neutral differential equation plays a major role in the presence or absence of the oscillation property, unlike other types of neutral differential equations. Some examples are presented to illustrate and apply the results obtained. [ABSTRACT FROM AUTHOR]

The paper considers the problem of observability and controllability gramians computation for one-dimensional and two-dimensional heat equations. The heat equation is a parabolic differential equation representing the process of heat flow. In the one-dimensional case, the heat equation models the physical process of a heat flow in a rod. In the two-dimensional case, this equation shows how the heat flow changes in a plate. The solutions of the heat equations are derived by applying the time-space separation principle and by using the Fourier series approximation. The zero input solution participates in computing the observability gramian of the distributed parameter system. The solution of the nonhomogeneous differential equation under zero initial conditions is used for computing the controllability gramian. Both solutions are based on the state space formulation of the infinite dimensional systems into an abstract differential form. The gramians computation procedure uses the Riesz-spectral framework of trajectory representation. The obtained system gramians are derived in explicit form, which allows their simple form of computation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe an epidemic model when two SI-type of diseases are transmitted vertically as well as horizontally through one population. The population contains two subclasses: susceptible and infectious, while the infectious are divided into three subgroups: those infected by AIDS disease, HCV disease, and by both diseases. A nonlinear mathematical model for AIDS and HCV diseases is Suggested and analyzed. Both local and global stability for each feasible equilibrium point are determined theoretically by using the stability theory of differential equations, Routh-Hurwitz and Gershgorin theorem. Moreover, the numerical simulation was carried out on the model parameters in order to determine their impact on the disease dynamics, and the results are displayed graphically and discussed. [ABSTRACT FROM AUTHOR]

In this paper, the experimental description and verification of the airflow in the rotor spinning machine are very difficult, because the process takes place in the chamber. To describe the process, we conducted studies based on a new experimental approach. In the experimental work, the uniformity and stability of the velocity field of each channel for moving fibers in an aerodynamic device were checked. The speed of the airflow was changed from 5 m/s to 30 m/s. Differential equations of motion along the OX and OY axis were created taking into account the air resistance. When determining the movement of fibers in a conical channel, the total velocity was divided into components, constant values were found, and the general equations of motion were derived. Also, the movement of fibers in a conical channel along the OY axis as a function of time on different surfaces, the movement along the OY axis as a function of time at different speeds, the movement of fibers in a conical channel along the OX axis as a function of time on different surfaces, the movement along the OX axis as a function of time at different speeds graphs were obtained. [ABSTRACT FROM AUTHOR]

The paper considers the issues of creating a system of second-order differential equations for controlling a dynamic object described by a system with an elastic property, alignments which are further transformed into two polynomial controls in matrix form. To solve the problem of synthesizing a control system in a control system, two matrix polynomials are introduced into the control system with unknown parameters that were the transfer function of the controller, the order of which is equal to two, which gives the system astatic properties. To solve this problem, matrix polynomial control is converted to matrix linear control with real coefficients. The results of simulation modeling show that the synthesized system meets the requirements of the object, the proposed device allows you to ensure the stability of the system, gives the properties of astaticism, and the duration of the transition process is determined by the location of the poles of the system and the desired quality indicators. [ABSTRACT FROM AUTHOR]

In this paper we give lower and upper estimates for the norm of the Hardy-Volterra operator with Oinarov's kernel in the case 1 < p ≤ q < ∞. The results given are very important in the study of the oscillation and non-oscillation properties of solutions of differential equations, as well as spectral properties. [ABSTRACT FROM AUTHOR]