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We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated. [ABSTRACT FROM AUTHOR]

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The neural network methods in solving differential equations have significant research importance and promising application prospects. Aimed at the time-fractional Huxley (TFH) equation, we propose a novel fractional physics-informed neural networks (fPINNs) method. By integrating the physical information of the TFH equation into neural networks, the fPINNs are trained as a precise approximation model for solving the TFH equation. The fPINNs method involves calculating the Caputo fractional derivative using the L1 formula and estimating the integer-order derivative through the chain rule. The Adam algorithm is employed to optimize two kinds of loss functions constructed by hard and soft constraints, respectively. Through an analysis of the impact of fPINNs parameters on training results, we identify the optimal parameter configuration for solving both one-dimensional and two-dimensional (2D) TFH equations. Numerical examples validate the efficiency and robustness of the fPINNs method in solving TFH equations. Furthermore, numerical simulation of the 2D TFH problem demonstrates the method's practical applicability to engineering problem. [ABSTRACT FROM AUTHOR]

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results. [ABSTRACT FROM AUTHOR]

We study the inverse problem for a differential equation of 2b$$ 2b $$‐order with the Caputo fractional derivative over time and Schwartz‐type distribution in its right‐hand side. The generalized solution of the Cauchy problem for such an equation, space‐dependent part of a source, and a time‐dependent reaction coefficient in the equation are unknown. We find sufficient conditions for unique local in time solvability of the inverse problem under time‐ and space‐integral overdetermination conditions. [ABSTRACT FROM AUTHOR]

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, including soliton, exponential and periodic solutions, which are useful to analyze the behaviors of the FSH equation. We show that the auxiliary equation method improves the speed of the discovery of exact solutions. [ABSTRACT FROM AUTHOR]

We address asymptotic formulae for the classical Poincar'e-Perron problem of linear differential equations with almost constant coefficients in a half line [t0,+¥) for high order equation n ≥ 5 and some t0 2 R. By using a scalar nonlinear differential equation of Riccati type of order n = 1, we recover Poincar'e's and Perron's results and provide asymptotic formulae with the aid of Bell's polynomials. Furthermore, we obtain some weaker versions of Levinson, Hartman-Wintner and Harris-Lutz type Theorems without the usual diagonalization process. For an arbitrary n ≥ 5, these are corresponding versions to known results for cases n = 2, 3 and 4. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs. [ABSTRACT FROM AUTHOR]

This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small. [ABSTRACT FROM AUTHOR]

The classical lateral–torsional instability of a cantilever beam with continuous elastic lateral restraint and a transverse point-load applied at the free end is discussed here through the lens of asymptotic simplifications. One of our main goals is to provide analytical approximations for the critical buckling load, as well as its dependence on various key non-dimensional groups. The first part of this study is concerned with a scenario in which a doubly symmetric beam is constrained to rotate about a fixed axis situated at an arbitrary height above the shear centroidal axis. The second part examines a beam that features a continuous elastic lateral restraint spanning its entire length. Assuming that the stiffness of the constraint, κ (say), is finite, the buckling equations in the second case are described by a system of two coupled fourth-order differential equations in the lateral displacement and the angle of twist. We show that as κ → ∞ , the problem discussed in the first part provides an outer solution for the aforementioned system; the relevant boundary conditions for the next-to-leading-order outer approximation are also derived using matched asymptotics. Our theoretical findings are confirmed by comparisons with direct numerical simulations of the full buckling problem. [ABSTRACT FROM AUTHOR]

In this work, a class of neutral-type Rayleigh equations involving $$D$$ D operators are proposed, where the damping item is a continuous $$T$$ T -anti-periodic function. Several new stability theorems are obtained to assure the $$T$$ T -anti-periodic exponential stability of the addressed equations by applying differential inequality analyses. Moreover, an illustrative example incorporating its graphical simulations is furnished to illustrate the validity of the derived findings. [ABSTRACT FROM AUTHOR]

Applying a method with vanished phase–lag might potentially eliminate the phase–lag and its first, second, and third derivatives. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost–efficient approach. Equation PF3DPHFITN142SPS demonstrates the unique method. The suggested approach is P–Stable, meaning it is indefinitely periodic. The suggested approach is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost–efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

We study the Kneser-type oscillation criterion for a class of second-order half-linear functional dynamic equations on an arbitrary time scale utilizing the integral averaging approach and the Riccati transformation method. The results show an improvement in Kneser-type when compared to some known results. We provide some illustrative examples to demonstrate the significance of our main results. [ABSTRACT FROM AUTHOR]

There are several common procedures used to numerically integrate second-order ordinary differential equations. The most common one is to reduce the equation's order by duplicating the number of variables. This allows one to take advantage of the family of Runge–Kutta methods or the Adams family of multi-step methods. Another approach is the use of methods that have been developed to directly integrate an ordinary differential equation without increasing the number of variables. An important drawback when using Runge–Kutta methods is that when one tries to apply them to differential algebraic equations, they require a reduction in the index, leading to a need for stabilization methods to remove the drift. In this paper, a new family of methods for the direct integration of second-order ordinary differential equations is presented. These methods can be considered as a generalization of the central differences method. The methods are classified according to the number of derivatives they take into account (degree). They include some parameters that can be chosen to configure the equation's behavior. Some sets of parameters were studied, and some examples belonging to structural dynamics and multibody dynamics are presented. An example of the application of the method to a differential algebraic equation is also included. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to add some new asymptotic and oscillatory results for third-order neutral delay differential equations with noncanonical operators. Without assuming any extra conditions, by using the canonical transform technique, the studied equation is changed to a canonical type equation, and this reduces the number of classes of nonoscillatory solutions into two instead of four. Then, we obtain Myshkis type sufficient conditions for the nonexistence of Kneser type solutions for the studied equation. Finally, employing these newly obtained criteria, we provide conditions for the oscillation of all solutions of the studied equation. Examples are presented to illustrate the importance and the significance of the main results. [ABSTRACT FROM AUTHOR]

The important character of the applied moment with a fixed line of direction as implied in deriving the governing differential equations of LTB for elastic beams, however, is not represented in stating the boundary conditions based on conventional energy approaches. This paper sets out to consider the boundary conditions derived by conventional potentials in order to identify the ignored second-order moments that need to be added for maintaining the initial direction of the applied moment during LTB. By taking into account the non-zero work done by these ignored moments for modifying original potentials, the nonconservative nature of the elastic LTB of beam under external moments is clarified in two aspects — by discovering the path-dependent property of the non-zero work and by demonstrating that only trivial solution of LTB mode is expected from the analytical solution of the governing differential equations. Using the concept of dynamic instability, an analysis procedure combining the numerical solution of an eigen-value problem and the bisection method is proposed to identify the first appearance of imaginary or conjugate complex frequencies and to determine the value of the critical moment. Demonstrative examples are considered for illuminating the elastic LTB behavior using conventional energy methods and revised virtual work equation. It has been found that the critical moments given by conventional potentials are generally not consistent with each other unless the work done by the ignored second-order moments vanishes. However, consistent critical moment, under which the free vibrating amplitude of elastic beam is increased unboundedly with time, can be obtained by the kinetic approach based on any revised virtual work equation, indicating the necessity of considering the nonconservative nature of non-follower applied moment. [ABSTRACT FROM AUTHOR]

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with α -order. Solutions for these equations with constant coefficients are obtained through the method of variation of parameters and the method of undetermined coefficients. The solution space for higher α -order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to α -order linear fractal differential equations are graphically depicted to illustrate their non-differentiability. Additionally, equations of motion governing the behavior of two masses in fractal time are proposed and solved. [ABSTRACT FROM AUTHOR]

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko's attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples. [ABSTRACT FROM AUTHOR]

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method's effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results. [ABSTRACT FROM AUTHOR]

The dynamical wave solutions of the time–space fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation have been obtained in this article using an innovative and efficient technique including the Riccati–Bernoulli sub-ODE method through Bäcklund transformation. Fractional-order derivatives enter into play for their novel contribution to the enhancement of the characterization of dynamic waves while providing better modeling ability compared to integer types of derivatives. The solutions of the above-mentioned time–space fractional Date–Jimbo–Kashiwara–Miwa equation have tremendous importance in numerous scientific scenarios. The regular dynamical wave solutions of the aforementioned equation encompass three fundamental functions: trigonometric, hyperbolic, and rational functions will be among the topics covered. These solutions are graphically classified into three categories: compacton kink solitary wave solutions, kink soliton wave solutions and anti-kink soliton wave solutions. In addition, to explore the impact of the fractional parameter (α) on those solutions, 2 D plots are utilized, while 3 D plots are applied to present the solutions involving the integer-order derivatives. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn–Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in L^1(\mathbb {R}) of the numerical solution. Our results partially answer positively to the open problem posed by J. Cui and J. Hong [J. Differential Equations 269 (2020), pp. 10143–10180] on computing the density of the exact solution numerically. [ABSTRACT FROM AUTHOR]

Applying a phase-fitting method might potentially vanish the phase-lag and its first derivative. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost-efficient approach. Equation PF1DPHFITN142SPS demonstrates the unique method. The suggested approach is P-Stable, meaning it is indefinitely periodic. The proposed method is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost-efficient approach. With an AOR of 14, we can significantly improve our present predicament. [ABSTRACT FROM AUTHOR]

The aim of this text is to elucidate the oscillating patterns (see C. Cheverry [Res. Rep. Math. (2018)]) which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency \varepsilon ^{-1} \gg 1. By exploiting the properties of underlying integrable systems, we can complement the KAM picture (see G. Benettin and P. Sempio [Nonlinearity 7 (1994), pp. 281–303]; M. Braun [SIAM Rev. 23 (1981), pp. 61–93]) and go beyond the classical results about gyrokinetics (see M. Bostan [Multiscale Model. Simul. 8 (2010), pp. 1923–1957]; A. J. Brizard and T. S. Hahm [Rev. Modern Phys. 79 (2007), pp. 421–468]). The purely magnetic situation was addressed by C. Cheverry [Comm. Math. Phys. 338 (2015), pp. 641–703; J. Differential Equations 262 (2017), pp. 2987–3033]. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field. [ABSTRACT FROM AUTHOR]

This paper explores the existence of solutions for non-local coupled semi-linear differential equations involving &#936-Caputo differential derivatives for an arbitrary l 2 (0;1). We use topological degree theory to condense maps and establish the existence of solutions. This theory allows us to relax the criteria of strong compactness, making it applicable to semilinear equations, which is uncommon. Additionally, we provide an example to demonstrate the practical application of our theoretical result. [ABSTRACT FROM AUTHOR]

This work is devoted to studying the distribution of zeros of a first-order neutral differential equation with several delays [y(t)+a(t)y(t-σ)]'+nΣj=1bj(t)y(t-μj)=0 New estimations for the upper bounds of the distance between successive zeros are obtained. The properties of a positive solution of a first-order differential inequality with several delays in a closed interval are studied, and many results are established. We apply these results to a first-order neutral differential equation with several delays and also to a first-order differential equation with several delays. Our results for the differential equation with several delays not only provide new estimations but also improve many previous ones. Also, the results are formulated in a general way such that they can be applied to any functional differential equation for which studying the distance between zeros is equivalent to studying this property for a first-order differential inequality with several delays. Further, new estimations of the upper bounds for certain equations are given. Finally, a comparison with all previous results is shown at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability and Hopf bifurcation of the fractional-order logistic equation with two different delays τ 1 , τ 2 > 0 : D α y (t) = ρ y (t - τ 1) 1 - y (t - τ 2) , t > 0 , ρ > 0 . We describe stability regions by using critical curves. We explore how the fractional order α , ρ , and time delays influence the stability and Hopf bifurcation of the model. Then, by choosing ρ , fractional order α , and time delays as bifurcation parameters, the existence of Hopf bifurcation is studied. An Adams-type predictor–corrector method is extended to solve fractional-order differential equations involving two different delays. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we studied the linear 2-refined neutrosophic differential equations and defined the homogeneous and non-homogeneous 2-refined neutrosophic differential equations. Also to presenting 2-refined neutrosophic differential equation of Bernoulli, which turns into a linear equation and thus facilitates its solution. Apart from talking about how to solve these equations and giving enough examples to support it. [ABSTRACT FROM AUTHOR]

We investigate scalar impulsive differential equations with piecewise constant generalized mixed arguments, abbreviated as the IDEPCAG of mixed type. These equations have general step functions as arguments. We propose criteria for the existence of oscillatory and non-oscillatory solutions, and obtain sufficient conditions for the stability of the zero solution. Our results are novel, and extend and improve upon previous publications. Additionally, we provide several numerical examples and simulations to demonstrate the feasibility of our findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue λ 0 N D + for the general Sturm–Liouville problem y ″ = q (t) y + λ m (t) y , with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation. [ABSTRACT FROM AUTHOR]

In this paper, the application of the HAM-Padé technique for solving a system of differential equations is investigated. This method is a combination of the homotopy analysis method (HAM) and Padé approximation. The equations under consideration are the governing equations of an important physical problem that is also widely used in engineering. This study is a comparative research so that after solving the equations, the solutions obtained from the HAM-Padé method will be compared with the outcomes of the three previous studies. The comparison showed that the HAM-Padé solutions have an interesting agreement with the previous works and have good accuracy. [ABSTRACT FROM AUTHOR]

It is crucial to understand the effects of deadly viruses on humans in advance. One such virus is the infectious hantavirus. Since the effects of viruses vary under different conditions, this study models the virus using distributed order differential equations. Because distributed order differential equations effectively capture variable effects in different conditions through the incorporated density function, this study aims to achieve a solution via the discretization method after presenting the equation system. A nonstandard finite difference scheme (NSFD) is used for the discretization. Then the stability analysis of the discretized system is investigated. [ABSTRACT FROM AUTHOR]

A two-point boundary-value problem for an implicit differential equation with a deviating argument is examined. An existence theorem and an estimate for the solution are obtained, which is similar to the Chaplygin theorem on differential inequalities. We use results on equations with covering and monotonic mappings in partially ordered spaces and conditions for ordered covering of the Nemytsky operator in the space of measurable essentially bounded functions. [ABSTRACT FROM AUTHOR]

The phase-lag and all of its derivatives (first, second, third, fourth, fifth, and sixth) might be eliminated using a phase-fitting technique. The new approach, which is referred to as the economical method, targets maximizing algebraic order (AOR) and reducing function evaluations (FEvs). The one-of-a-kind approach is demonstrated by Equation PF6DPFN142SPS.The proposed method is infinitely periodic i.e. P-Stable. To many periodic and/or oscillatory problems, the suggested strategy can be applied. Using this innovative method, the difficult issue of Schrödinger-type coupled differential equations was tackled in quantum chemistry. Every step of the new approach only requires 5FEvs to execute, making it a economic algorithm. By accomplishing a AOR of 14, this allows us to greatly enhance our current situation. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of a generalized nonlinear differential equation with variable delays via fixed point theory. An asymptotic stability theorem with sufficient conditions is proved, which improves and generalizes some previous results. Two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and $ \mathcal {Z} $ Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are $ \mathcal {O}(\tau ^{p}+h^2) $ O (τ p + h 2) for $ 1 \leq p \leq ~4 $ 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme. [ABSTRACT FROM AUTHOR]

This paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using L 2 and L ∞ norms. [ABSTRACT FROM AUTHOR]

Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r ∗ of the initial data in H 1 (R 3). Motivated by this work, we focus on characterizing the large-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting methods. In particular, for the case − n 2 < r ∗ ⩽ 1 , we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case. [ABSTRACT FROM AUTHOR]

We consider the well‐posedness of a coupled system for generalized Korteweg–de Vries (S‐gKdV) ∂tu+∂x3u+∂xupvp+1=0∂tv+∂x3v+∂xup+1vp=0,$$ \left\{\begin{array}{l}{\partial}_tu&#x0002B;{\partial}_x&#x0005E;3u&#x0002B;{\partial}_x\left({u}&#x0005E;p{v}&#x0005E;{p&#x0002B;1}\right)&#x0003D;0\\ {}{\partial}_tv&#x0002B;{\partial}_x&#x0005E;3v&#x0002B;{\partial}_x\left({u}&#x0005E;{p&#x0002B;1}{v}&#x0005E;p\right)&#x0003D;0,\end{array}\right. $$where p∈ℤ+$$ p\in {\mathrm{\mathbb{Z}}}&#x0005E;{&#x0002B;} $$ and the initial data (u0,v0)$$ \left({u}_0,{v}_0\right) $$ are analytic in a strip and then the solutions of S‐gKdV continue to be analytic in a strip the width of which will decrease as time goes to ∞$$ \infty $$, by using the standard contraction method in analytic Bourgain space. Besides, we obtain algebraic lower bounds on the decreasing rate of the uniform radius of analyticity of the solutions. [ABSTRACT FROM AUTHOR]

This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations. [ABSTRACT FROM AUTHOR]

The current research studies a fractal Duffing oscillator in the presence of periodic force. To find an analytic solution for this oscillator, the aspects explained in the following are considered. First, we obtain an alternative unforced fractal fourthorder equation and then convert it into a continuous space. Therefore, the nonperturbative (NP) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. It is seen that the analytical and numerical solutions agree very well. The computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. The present study gives reliability in the technique of reducing the order of differential equations. Furthermore, the required periodic solution is also obtained by Galerkin’s technique. In contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. The current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. [ABSTRACT FROM AUTHOR]

We consider a differential equation with a discontinuous delayed neutral-type feedback. In the phase space, we describe classes of initial functions that depend on a number of parameters. We show that in a certain time, solutions return to an analogous class, possibly with other parameters. The analysis of the change in the parameters allows describing periodic solutions and their stability. We show that infinitely many of stable periodic solutions exist. [ABSTRACT FROM AUTHOR]

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation. [ABSTRACT FROM AUTHOR]

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter H∈(1/4,1/2]$H\in (1/4, 1/2]$. We prove a moderate deviation principle for this equation as the scale parameter tends to zero. [ABSTRACT FROM AUTHOR]

This study is concerned with investigating the oscillatory properties of a general class of neutral differential equations. Neutral equations are characterized by being rich in both practical and theoretical aspects. We obtain criteria that guarantee the oscillation of solutions to a fourth-order neutral differential equation with multiple delays. Considering the canonical case, we obtain some new relations and inequalities that help in obtaining improved criteria. We use the reduction method to relate the oscillation of the studied equation to a first-order equation. We apply the results to a special case. Through this application, we evaluated the efficiency of the new results in the oscillation test compared to previous results in the literature. [ABSTRACT FROM AUTHOR]

In pursuing quick and precise progressive flooding simulations for decision-making support, the linearised method has emerged and undergone refinement in recent years, becoming a reliable tool, especially for onboard decision support. This study consolidates and enhances the modelling approach based on a system of differential-algebraic equations capable of accommodating compartments filled with floodwater. The system can be linearised to permit analytical solutions, facilitating the utilization of larger time increments compared to conventional solvers for differential equations. Performance enhancements are achieved through the implementation of an adaptive time-step mechanism during the integration process. Furthermore, here, a correction coefficient for opening areas is introduced to enable the accurate modelling of free outflow scenarios, thereby mitigating issues associated with the assumption of deeply submerged openings used in governing equations. Experimental validation is conducted to compare the method's efficacy against recent model-scale tests, specifically emphasising the improvements stemming from the correction for free outflow. [ABSTRACT FROM AUTHOR]

We consider the problem in a cylindrical domain of variables t and x , which is the Cartesian product of a segment containing zero and a unit circle and is split into two subdomains by a hyperplane t = 0. In each subdomain, the solution of the problem satisfies the corresponding differential equations with multipoint conditions in the case of multiple nodes and the conditions of linear conjugation on the interface t = 0. The form of the domain imposes additional conditions on the periodicity of the solution with respect to the space variable x. We study the conditions of correct solvability of the problem in the Sobolev space closely related to the problem of small denominators and their estimation. With the help of the metric approach, it is shown that these conditions are satisfied for almost all (with respect to the Lebesgue measure) vectors formed by the interpolation nodes of the multipoint conditions. [ABSTRACT FROM AUTHOR]