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This study presents a modified SEIR differential equation model where a vaccination program is being implemented to a population compartmentalized based on geographic location. The resulting system of differential equations is solved numerically for different scenarios to determine the impact of the disparities in vaccine efficacy and rollout rates. The outcomes of the different scenarios are assessed based on the overall cumulative infections. Our simulations show that the best scenario happens when multiple cities share resources: vaccines with high efficacy should be equally distributed and vaccine rollout should proceed at the same pace everywhere. [ABSTRACT FROM AUTHOR]

This research is a literature study on predator-prey models with functional Holling II responses and cannibal predators and natural death of prey. The purpose of this study is to determine of the predator-prey model and to analyse the stability at each equilibrium point found. The novelty of this research is that it assumes a cannibalistic predator-prey model with a type II Holling response function, where the model will impact the stability of different predator prey populations. The predator prey model is an autonomous system with two differential equations. There are ten parameters obtained, namely K, r1, r2, a1, a2, b1, b2, c1, c2 and m. The model obtains four equilibrium points, namely E1(0,0) where the condition is that the predator and prey populations are extinct, E2(0, Y2) where the prey population is extinct, E3(X3, 0) where the predator population is extinct and E4(X5, Y5) where both populations exist. The four equilibrium points are local asymptotically stable with several conditions for E1(0,0) to be stable if r11 and r22, E2(0, Y2) to be asymptotically stable if r 1 − c 1 < a 1 b 2 (r 2 − c 2) a 2 − m (r 2 − c 2) and D* < 0, for E3(X3, 0) to be stable if K 2 < X 3 < b 1 (c 2 − r 2) m (r 2 − c 2) + a 1 , and E5 (X5, Y5) to be asymptotically stable under certain conditions. These results have been confirmed through numerical simulations using matlab. [ABSTRACT FROM AUTHOR]

The principal objective of this paper is to investigate a multiple pursuer evasion game in a Hilbert space. The game is demonstrated by the infinite system of binary differential equations and geometric constraints are imposed on control functions of players. In the differential game, pursuit is said to be ended when the state of a controlled system, at some limited period, falls into the origin of the space 푙2. The pursuer's goal is to bring the state of at least one of the systems into the space's origin, whereas the evader's goal is to avert this. Finally, we created a strategy for evader which ensures evasion in the game. [ABSTRACT FROM AUTHOR]

This paper discusses the dynamics and interaction of prey and predator populations in an ecosystem consisting of two zones, namely the reservation zone and the unreserved zone. The prey population in the ecosystem can migrate from the reservation zone to the unreserved zone and vice versa, while the predator population is assumed to be looking for prey in the unreserved zone. The dynamics of predator and prey are expressed in the form of a system of differential equations based on food intake capacity and some other factors. In the interaction mechanism between the prey and the predator, the Holling type III predation response function is considered. The existence of an interior equilibrium point of the model is analyzed and investigated for its stability using the linearization method and the Routh-Hurwitz stability test. There is a certain condition where, with the appropriate parameter values, the interior equilibrium point exists and is stable. When the migration rate from the unreserved zone is gradually increased within a certain range of values, the equilibrium point also changes but remains stable. The prey and predator populations in an unreserved zone decrease while the prey population in the reservation zone increases. Numerical simulations are carried out to verify the analytical results and the behavior of the populations. [ABSTRACT FROM AUTHOR]

In this study, we take advantage of the elegant hermite cosh-Gaussian laser beams ability to focus on themselves in collisionless magnetized plasma. The ponderomotive force is the primary cause of the nonlinearity in the dielectric constant that is being considered here. Using WKB and paraxial approximations in the context of a parabolic wave equation approach, differential equations for the beam width parameters in the two transverse dimensions of the beam have been developed. Fourth order Runge-Kutta is used to numerically solve these equations. On a graphic, the impact of decentred parameters in the beam's two transverse dimensions as well as polarization modes has been shown and described. [ABSTRACT FROM AUTHOR]

These days, researchers are interested in the study of fuzzy differential equations. Modeling can be applied to this fuzzy differential equation to find solutions related with real-world issues. Yet, it is challenging to solve this fuzzy differential equation analytically. Thus, the purpose of this study is to present the fuzzy Runge–Kutta Cash–Karp method to solve first-order fuzzy differential equation under the interpretation of Hukuhara derivative. By applying the fuzzy Runge–Kutta Cash–Karp method, the approximate solution is obtained and compared with the analytical solution. The results indicated that the approximate solutions of the proposed method are comparable to the analytical solutions in terms of accuracy. [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]

Integral transforms are known for the simplification that they bring about, while dealing with differential equations subject to some boundary conditions. Of all the integral transforms, Laplace transform is handy to solve differential equations easily. In this article we have aimed at studying the Bicomplex Laplace transforms of Prefunctions, their inverses and the properties. We also have applied Bicomplex Laplace transform to some integral equations, Integro differential equations, Matrix integral equations and system of differential equations and obtained the closed form solutions in terms of Pretrigonometric, Prehyperbolic and Extended Prefunctions. It is interesting to see that the closed form solutions are effective when compared to numerical solutions. [ABSTRACT FROM AUTHOR]

Telangana is India's 12th most populous country. An individual circle of relatives' houses is called a maximally typical house or house. Condominiums consisting of several units are called two-family homes or family condominiums. The production area is of particular importance today due to India's young population and accompanying surge in demand for housing. Due to the increasing demand for housing, land prices are soaring. As a result, the cost of the fabric also increases. Limited residential and residential areas should also be considered. Inflation adds significant commissions each day, impacting additional production costs. Optimization strategies are currently achieved with the help of the definition of complex mathematical urgency equations and differential equations. Structural weight-based optimization has many practical advantages in all areas of technology. In the civil engineering sector, weight-optimized additives can be transported more cost-effectively and cleanly to the manufacturing site. In this study, gadget research optimization was advanced to optimize rebar loading considering factors such as shape, size, and topology. In this study, we introduce machine learning (ML) to reduce the cost of designing and constructing a specific type of home. Machine learning (ML) uses operations that mimic plant evolutionary manipulations consisting of reproduction, crossover, and mutation to incrementally improve subsequent population responses to achieve advanced offspring growth. Global search method. In this work, PC software was developed to improve the layout of reinforced concrete houses at minimal cost. Planning consists of the appearance of the (ML) layout and the preparation of the objective function. Then the modified target features arrive with several constraints applied. This will calculate the change price for the required fabric. Concrete and rebar and parametric studies are managed. [ABSTRACT FROM AUTHOR]

The Finite Element Method (FEM) is a powerful numerical technique for solving complex physical problems involving differential equations and partial derivatives. This study presents the mechanical and thermal analysis of a rocker arm shaft support using FEM. Employing the ADINA (Automatic Dynamic Incremental Nonlinear Analysis) software, the behavior of the shaft support has been modelled, leveraging ADINA's extensive material libraries, diverse loading conditions, and customizable behavioral properties. The analysis focuses on determining deformations, stresses, and displacements, particularly in the median plane section at the point of maximum stress. Both treated and untreated material conditions are evaluated to assess stress transmission in the transition zone between surface-hardened and base materials. A 2D analysis is conducted, given the geometry of the rocker arm spindle support, to ensure fidelity to real structural behavior while optimizing computational efficiency. The validated model provides insights into design constraints for new materials, supporting industrial applications. [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 study, we will understand the concept and procedure of Homotopy Analysis Method(HAM), which is a strong semi analytic method to solve highly nonlinear integral, differential equations. This method provides a lot of freedom to control the convergence of the solution so obtained by this method. Here we will see one such aspect, the freedom to choose the Base Function, in approximating the actual solution and how the choice affects the rate of convergence of the approximate series solution. [ABSTRACT FROM AUTHOR]

Currently, despite the abundance of publications on subharmonic oscillations in three-phase circuits, the processes in them are illuminated based on purely experimental data, and theoretical analysis bas been carried out for the single-phase analogue of a three-phase circuit, distorting the quantitative and qualitative aspects of the process. This is due to the solution of nonlinear systems of inhomogeneous differential equations, the right-band side of which bas fixed phase shifts. These equations describe nonlinear three-phase systems. On the other hand, in all published works, subharmonic oscillations have been studied in three-phase circuits without ferromagnetic coupling between phases. The issues of exciting such oscillations in three-phase circuits with a three-phase ferromagnetic element with magnetic coupling between phases have not been sufficiently studied to date. This article discusses the excitation mode of third-order subharmonic oscillations in a three-phase circuit consisting of active, capacitive, and inductive elements with common magnetic linkage, which is an analogue of a transmission line with an unloaded transformer. The equation of motion is derived. Using the averaging method with corresponding phases, shortened equations are obtained. From the condition of the existence of a periodic solution, phase relationships are determined, different from phase relationships for three-phase circuits with group ferromagnetic elements. In the steady-state mode, conditions for excitation, existence regions, dependencies of output parameters on circuit parameters and applied influence are determined. [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 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]

In this document, we present a novel approach for approximating analytical solutions to differential equations, referred to as the Polynomial Approximation on Subintervals (PAS) method. We demonstrate the method's effectiveness when applied to optimal control problems. To begin, the original optimal control problem is transformed into a variational problem. Subsequently, we solve the corresponding Euler-Lagrange Equation and obtain an approximate analytical solution utilizing the innovative PAS method. To evaluate its performance, we provide a numerical example that includes a comparative analysis between the outcomes obtained with PAS and those achieved by other researchers who employed various approximation techniques. [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]

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]

Dynamic analysis of functionally graded carbon nanotubes in axial direction under the follower load effect have been carried out in the present study. Variation of material properties of nanotube is considered in power-law form. Nonlocal Euler-Bernoulli beam model has been used in modeling. Solution of the differential equation of motion with variable coefficients have been obtained with numerical Haar Wavelet method. Performance of the mentioned numerical method have been discussed in terms of approximate error and convergence rate. Present results could be used at design of axially graded nano structures. [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]

A Singular Perturbation Problem (SPP) often will be developed when the perturbation parameter is introduced into a differential equation in such a way that it multiplies the equation's greatest derivative. In the present study, finite difference approach is considered to find the solution of the SPP. ε y " + a (x) y ' (x) + b (x) y (x) = r (x) , 0 ≤ x ≤ 1 ; y (0) = α , y (1) = β (1) In order to support the theory, several linear/nonlinear SPPs and their solutions have been provided. It has been found that the current approach very closely estimates the exact one so closely. [ABSTRACT FROM AUTHOR]

In the article, the vibration behavior of a shaft rotating at a high angular speed, fixed at one end, is studied. Lagrangian differential equations of motion of the 2nd type are formulated for a rotating shaft. These differential equations are integrated and the law of motion is determined (Fig. 1). [ABSTRACT FROM AUTHOR]

In this study, new mathematical models are developed for nonlinear functionallу-grаded pоrous (PFG) nano/micro/macro closed cylindrical Kirchhoff-Love shells. The thеorу by Yang is used as a mathematical framework to model the size-dependent behaviour of composite shells. Geometric nonlinearity is incorporated using the T. von Karman model. The material composition of the shells is heterogeneous, consisting of ceramic and metal phases. The differential equations governing the system, along with the boundary and initial conditions, are derived using Hamilton's principle. To simplify the equations, the Faedo-Galerkin method is employed in high approximations, which reduces the governing PDEs to a Cauchy problem. The mеthоd's cоnvеrgеnce is invеstigаted to ensure accurate results. To solve the resulting Cauchy problem, several Runge-Kutta methods of different orders of accuracy are utilized. This ensures reliable and accurate solutions for the system. Various types of porosity are considered in the study. It is shown that the value of the small scale parameter has an effect on the character of the vibrations in shells. In fact, the phenomenon of hyper-chaos is revealed, indicating complex and unpredictable behaviour in the system. The study contributes to the understanding of the mechanical behaviour of porous nonlinear functionally graded cylindrical shells and sheds light on the impact of small scale parameters and porosity on their oscillations and stability. [ABSTRACT FROM AUTHOR]

The statement of the problem of vibrations of a beam with a moving spring–loaded support carrying an attached mass is obtained. When the support is not absolutely rigid, energy exchange occurs through the moving boundary. In this regard, there is a difficulty in writing the boundary conditions. To formulate the problem, we used the variational principle of Hamilton. In this case, the viscoelastic properties of the beam material are taken into account. The problem posed includes the differential equation of vibrations, initial conditions for the bent axis of the beam and for the added mass, boundary conditions. The conditions on the moving boundary are written as ratios between the values of the function and its derivatives to the left and right of the boundary. [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 work examines the solution of problems of a biological population taking into account reaction-diffusion. This research employs a technique to evaluate the qualitative properties of solutions to a differential equation system with the aim of addressing the challenge of selecting an optimal initial approximation for achieving efficient iterative convergence towards solving the Cauchy problem. The effectiveness of this approach is contingent on the numerical parameters and initial data of the system. Researchers utilize the asymptotic representation of the solution as an initial estimate, enabling them to conduct a numerical experiment and visualize the system's evolution as a function of parameter values. The study's findings indicate that the proposed nonlinear mathematical model aptly captures the dynamics of competitive biological populations characterized by double nonlinearity. Analysis of the derived solution estimates corroborates the preservation of confinement within a finite range and the system's source size. Thus, this work provides a complete, comprehensive picture of the process and can be a valuable contribution to the study of two-component competing systems in biological populations. [ABSTRACT FROM AUTHOR]

In this work, Lie symmetry method has been employed to analyze the symmetries of the Fujimoto Watanabe equation with variable coefficients. The obtained symmetries have been utilized to reduce the governed equation into ordinary differential equations. The optimal system of considerd equation has been by utilizing adjoint representation. The governing equation has been reduced with respect to each member of optimal system. The conservation laws are also obtained for the considered equation with respect to each infinitesimal generator using the new conservation theorem and Noether operator. [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 solution of the diffusion equation for an aggressive hydrogen medium is considered, and the obtained solution is compared with the results of the classical theory, which does not take into account the aggressive effect of a corrosive medium. To solve the system of resolving differential equations, the method of finite differences was applied. [ABSTRACT FROM AUTHOR]

A new solitary solution for type of non-linear (3+1) D- differential equations is presented in this article. Suggested approach has been successfully implemented to demonstrate novelish solitary wave solution for the non-linear 3rd order (3+1) D-mKdV equation, nonlinear 4th order (3+1) D–KP equation and nonlinear 4th order (3+1) D- gKP equation. The accuracy of solution has been discussed and illustrated by three problems. The technique has established a precision, without recourse any computer programming so it is fast approach to solve this type of equations. [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]

Fuzzy differential equations are used to model problems in science and engineering. For example, it is well known that knowledge of dynamical systems modeled by ordinary differential equations is often incomplete or vague. Whereas, fuzzy differential equations are an appropriate method for modeling dynamical systems under uncertainty and uncertainty. In this article, Seikkala derivative-based numerical techniques for solving fuzzy ordinary differential equations are discussed. A thorough error analysis follows a detailed discussion of a numerical approach based on the Runge-Kutta Method RK6. By resolving a few linear and nonlinear fuzzy Cauchy problems, and applied in MATLAB computer. [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]

This article discusses the issue of the possibility of calculating the approximation error. When replacing differential equations with discrete ones, one of the key issues is the closeness of the discrete solution to the exact solution. For the difference solution of the problem, a grid area is formed. The discrete solution is determined at the nodal points. Traditionally, in questions of replacing a differential equation with a descriptive one, one usually indicates the degree of approximation of the O(hp) type. Here h is the grid step. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. On the other hand, at each node one can construct a differential analog of the difference equation. Using simple examples, the calculation of approximation errors is demonstrated and schemes of the collocation type are constructed. [ABSTRACT FROM AUTHOR]

Many real-world problems, when formulated mathematically, lead to differential equations. We can encounter a number of these equations in the study of such phenomena as the motion of an object moving in a straight line, the decay of radioactive material, population growth, and the cooling of a heated object placed in an environment with a lower temperature. Numerical methods for the approximate solution of Initial Value Problems for ordinary differential equations serve to determine and approximate the functions representing the solution of these problems. In these days 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. Currently, discrete methods are most commonly used. One of the discrete method is the difference formula for numerically solving the Initial Value Problems. The difference method is a universally applicable numerical method for solving differential equations. The present work is devoted finding optimal coefficients of an implicit difference formula in the Hilbert space. [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]

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]

In this paper we will discuss the theoretical analysis for Bickley's method and numerical procedure for getting the approximate solution. If (sm, ck) denote the class of spline functions consisting of piecewise polynomial of degree m, joined at "knots" with at-least k continuous derivatives (k < m) then spline functions in (Sm, cm-1) are used as approximating function for two point boundary value problem in the second order differential equations. The result shows that the presented cubic spline function which interpolates the lacunary data was efficient and effective for solving such problems. [ABSTRACT FROM AUTHOR]

An algebraic relationship between functions and their derivatives is all that a differential equation (DE) is. These mathematical constructs give scientists the ability to comprehend a variety of intricate occurrences. In this essay, we provide an overview of a few techniques for solving differential equations. The Turing Machine, Genetic Algorithm, and Quantum Computing are the techniques used. [ABSTRACT FROM AUTHOR]

We design and investigate an infinite first order 3-system of differential equations in Hilbert space l2. We prove the existence and continuity of a unique solution of the system by showing the solution belongs to the space of continuous functions in l2 on some finite time interval. [ABSTRACT FROM AUTHOR]

Differential equations are used in many sectors of science and engineering to describe real-world issues. Unfortunately, sometimes it is difficult to solve using analytical methods and researchers have developed numerous approaches to solve these differential equations. The purpose of this paper is to emphasise the study of Neural Ordinary Differential Equations (Neural ODE) in solving differential equations linked to ethanol production via fermentation. Two analyses were conducted in order to compare the most effective approaches for solving these mathematical equations. The first comparison is between the conventional numerical method and Neural ODE. The data from experimental data were utilised to determine which approach had the least error in this analysis. Neural ODE performed well when compared to Runge-Kutta, the fourth and eighth orders. Following that, Neural ODE was compared to a recently created technique, Hypersolver, with the goal of enhancing Neural ODE performance. A study on the number of epochs and batches were performed in order to determine which hyperparameters gave the best and fastest results. The analysis of the number of epochs reveals that as the number of epochs rises, the error decreases, but the wall clock increases. For the analysis of batches in Hypersolver, increasing the batch size reduces the wall-clock during training, but no fixed error trend between microbe, ethanol and substrate can be identified, and these errors remain minimal when compared to Neural ODE. From these studies, it can be concluded that Neural ODE is capable of solving the presented model in the same way as the conventional numerical technique does, and the addition of training in Hypersolver does not show significant impact because the difference between Hypersolver and Neural ODE is very small. [ABSTRACT FROM AUTHOR]

This paper mainly presents the Taylor method, Runge-Kutta fourth-order (RK4) method and Runge-Kutta Fehlberg (RKF) method for solving initial value problem (IVP) for ordinary differential equations (ODE). These problems can be effectively addressed using any of the three proposed methods, which have demonstrated high efficiency and practical suitability. Two differential equations model which describe the physical situation are chosen; Newton's cooling law and the spring mass damper system. Numerical comparisons between the Taylor method, RK4 and RKF have been presented. For Newton's cooling law problem, the performance and the computational effort of these methods have been compared. In order to verify the accuracy, we compare numerical solutions with the exact solution in the spring mass damper system problem. The step size needs to be decreased to achieve higher accuracy in the solution. The resulting value indicates that RKF and RK4 are the most efficient for solving the ODE in terms of convergence and accuracy, respectively. Meanwhile, Taylor Methods is still compatible but needs more iterations to converge. In the spring mass damper system problem, the Taylor Method diverges. [ABSTRACT FROM AUTHOR]

This paper studies forced vibrations of a rigid rotor mounted on a balancing machine. The dynamic model is with two degrees of freedom and the rotor is unbalanced. The d'Alembert's principle for a mechanical system (dynamic equilibrium) is used to produce the differential equations of motion. Small displacements for the rotor axis are studied. Vector equations are projected along certain coordinate axes to obtain equations without the involvement of reaction forces. Differential equations are non-linear. Three cases of unbalance are examined and the differential equations of motion are solved using MATLAB (Simulink) software. The values of the parameters were determined on the basis of an existing stand for balancing. [ABSTRACT FROM AUTHOR]

Mathematics plays an important role, directly as well as indirectly in our day-to-day life. Generally it has been noticed that in mathematics solution of differential equation is solved tedious. The laplace transformation is a mathematical method that makes solving differential equations much easier. In laplace transformation technique we convert the differential equation of simultaneous differential equation into integral equation and then solve the integral. [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 low decision-making ability of students is also experienced by most of the prospective teachers. The purpose of this study is to reveal the decision-making process carried out by prospective teachers in solving differential equations. This research is a qualitative descriptive study with 2 subjects taken from 21 prospective mathematics teacher students. Each subject was identified to solve differential equation problems correctly and incorrectly. Data collection was carried out through differential equation tests and test-based interviews. The results of the research show that the decision making carried out by students by solving the correct differential equation is: (1) Building ideas, the subject had two solutions and three solutions, right and wrong ideas. (2) Clarifying ideas, the subject gives reasons for each idea correctly and has the wrong reasons because they do not understand the concept. (3) Assessing the reasonableness of the idea, providing logical reasons with one correct choice and logical reasons with one but less precise choice. The results of this study provide suggestions for prospective teacher educators to continue to innovate in order to provide provisions for prospective mathematics teacher students so that they can make good decisions, especially in solving math problems. [ABSTRACT FROM AUTHOR]

Air temperature is one of the climates that needs to be observed for various purposes so that there is a need for forecasting to facilitate use. Air temperature data is in the form of fluctuations that are suitable for use with several forecasting methods, one of which is Grey Forecasting. In addition, to forecast air temperature can also use Fourier Analysis by finding the value of the period and the results of air temperature prediction. So in this study the forecasting used is Grey Forecasting model and Fourier Analysis. This study aims to compare the accuracy of the two methods to determine which is more accurate between the two in air temperature prediction. The Grey Forecasting model used is GM(3,1) using a differential equation 3-order (third) and 1 (one) variable. Meanwhile, Fourier Analysis uses Fast Fourier Transform to determine the period and approximation of the Fourier Series for air temperature forecasting. Based on the calculation using Fast Fourier Transform, the air temperature period at the Karakates Observation Post, Malang Regency from 2015 to 2021 of 10.5≈11 months with an approximation of Fourier Series is 1-order. Furthermore, by using MAPE to determine the error of GM(3,1), a MAPE value of 3.3161 ×105 was obtained in the seventh-eleven data of air temperature and of 1.3690 × 1022 for the entire air temperature data. Because the error of the GM(3,1) > 50% can be categorized as weak and inaccurate forecasting. Meanwhile, the error in the Fourier Analysis was 2.6331 in the seventh-11 (eleven) data and 4.9756 for all air temperature data. This means that the error of Fourier Analysis is less than 10% then it can be categorized as a forecast with a high degree of accuracy. Thus, the best accuracy of the two the method is The Fourier Analysis. [ABSTRACT FROM AUTHOR]