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In the present study, a new method is proposed to obtain solution of nonlinear pantograph delay differential equations (NDDEs) with initial and boundary conditions. The presented method is a combination of differential transform and partial ordinary Bell polynomials. Differential transformation converts the problem into recursive relation and nonlinear terms in the problem is addressed by Bell polynomial. Two problems are discussed to show effectiveness and reliability of the technique. Error analysis are also provided. [ABSTRACT FROM AUTHOR]

This paper presents a new and easy-to-use analytical model for the debonding of steel reinforce grout (SRG)-reinforced concrete specimens subjected to single lap shear tests. The non-linearity is concentrated at the interface between mortar and textile supposed linear elastic, assuming a perfect bond between the support – infinitely rigid and resistant – and mortar. The behavior at the interface, assumed discontinuous and multilinear, is characterized by a tangential stress-slip relationship consisting of four phases: three elastic perfectly brittle and one elastic perfectly ductile. A pseudo-linear hardening phase is observed experimentally in global force-displacement curves when detachment triggers, and consequently, it is essential to assume a non-zero residual tangential resistance at the interface. Admitting such assumptions, the slip at the interface between reinforcement and substrate can be described through a second order differential field equation which states closed form solutions, properly derived. This paper also proposes an implementation in a commercial finite element (FE) software, which discretizes the interface between mortar and steel cords by means of three in parallel cutoff bars, two of which are elastic perfectly brittle and one elastic perfectly ductile, using rigid beams for the connections and modelling SRG with elastic quadrilateral elements. Finally, the reliability of the proposed procedure and model is validated against experimental tests carried out by the authors using five SRG-strengthened specimens tested with direct shear test. [ABSTRACT FROM AUTHOR]

Differential equations are often utilised to express scientific phenomena. In this study, we focus on a system of first order linear differential equations. It was solved in two different algorithms. Even if two different algorithms are applied for the same system, the same solution is expected. However, two algorithms gave the different solutions. From the viewpoint of quantum chemistry and computational chemistry, the fact is discussed. [ABSTRACT FROM AUTHOR]

The paper talks about solution of linear homogeneous fuzzy differential equation of first order, under fuzzy initial condition. Here symmetrical triangular and trapezoidal fuzzy numbers are used as initial conditions of the said differential equation and using concept of α-cut interval arithmetic a general solution to the problem is proposed. [ABSTRACT FROM AUTHOR]

In this article, the theorems that ensure the convergence of the method and the speed of convergence are proved in the numerical modeling of the Orr-Sommerfeld equation describing the problem of hydrodynamic stability using the initial integration method. In the "initial integration" method, the high-order derivative in the hydrodynamic stability problem is sought in the form of a finite series of Chebyshev polynomials of the first type with unknown coefficients. Until the problem is solved, this series is first integrated and lower-order derivatives and the solution are expressed in the form of a finite series. In the process of integration, unknown constants of integration appear corresponding to the number of integrations. They are found through the additional conditions of the given problem and appropriate finite series are set. Then the derivatives of the differential equation and the finite series corresponding to the solution are put into the differential equation and brought to the system of algebraic equations to find the unknown coefficients. Properties of Green's function and convergence theorems of projection methods are used to evaluate the convergence and convergence speed of the method. From the estimation of the speed of convergence, it can be seen that the proposed method approaches the exact solution of the given problem very quickly with the increase in the number of edge series terms. [ABSTRACT FROM AUTHOR]

A numerical algorithm for studying the stress-strain state (SSS) of three-layer rods under spatial loading is discussed in the article. The resulting system of equations for three-layer rod vibrations with the corresponding initial and boundary conditions is solved for six unknowns. The system of differential equations of three-layer rods with the corresponding initial and boundary conditions is presented in vector form. To solve the problem posed, central finite-difference relations of the implicit scheme of the finite-difference method of the second order of accuracy were used, considering the features of the boundary and initial conditions. By setting specific boundary conditions, several practical problems are solved. The methodology and computational algorithm for calculating the statics and dynamics of strain processes in spatially loaded three-layer rods are presented. The procedure for solving the problem is given. The results obtained are presented in the form of graphs. [ABSTRACT FROM AUTHOR]

The article proposes a mathematical model of thermoelasticity, formulated as a dynamic coupled stress problem. A coupled boundary value problem is formulated, consisting of two differential equations for the components of the stress and temperature tensor, as well as the heat influx equation with the corresponding initial and boundary conditions. Explicit and implicit finite-difference relations have been developed that make it possible to calculate numerical solutions to the problem under consideration. As an example, a one-dimensional coupled dynamic problem for isotropic bodies is solved numerically using explicit and implicit finite-difference schemes. The reliability of the obtained numerical results and the developed mathematical model is shown by comparing the results obtained by two methods. The proposed implicit finite-difference equations can be solved by the iterative method. [ABSTRACT FROM AUTHOR]

This paper presents the results of the falling dolomite rocks modelling using the continuum approach and the kinetic theory of granular gas. The two-fluid model was used. In the two-fluid model, the material (rock fragments) is considered as a continuous medium, which is characterized by a continuously distributed volume fraction of rock fragments. The kinetic theory of granular gas is used for describing of the possible fluidization, since this theory takes into account the motion of material both in a dense state and in a rarefied one. The mass and momentum conservation equations and the differential equation for the transport of granular temperature were solved. In this paper we also present initial and boundary conditions for the problem of modelling of the falling dolomite rocks. We used the two-fluid model to determine the collapse zone, which depends on the slope parameters. We compared these results with the results of modeling by discrete element method. When we increased the value of the slope angle, the collapse zone (the distance of the dolomite fragments run) was increased almost linearly in all cases. The results of the two-fluid model describe the experimental data and the results of calculations based on the discrete element method well. Thus, we concluded that the two-fluid model can also be used for modelling of the falling rocks. [ABSTRACT FROM AUTHOR]

A rectilinear mast is considered under combined axial loading with a concentrated force and its own distributed weight. Possible critical states of the mast in terms of strength and loss of stability are determined. A graphonumerical method for solving the eigenvalue problem of the differential equation of longitudinal bending of a rod is proposed. [ABSTRACT FROM AUTHOR]

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 r1

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]

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]