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We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study. [ABSTRACT FROM AUTHOR]

In this paper, we set the mathematical foundations of the Dynamical Low-Rank Approximation (DLRA) method for stochastic differential equations (SDEs). DLRA aims at approximating the solution as a linear combination of a small number of basis vectors with random coefficients (low-rank format) with the peculiarity that both the basis vectors and the random coefficients vary in time. While the formulation and properties of DLRA are now well understood for random/parametric equations, the same cannot be said for SDEs and this work aims to fill this gap. We start by rigorously formulating a Dynamically Orthogonal (DO) approximation (an instance of DLRA successfully used in applications) for SDEs, which we then generalize to define a parametrization independent DLRA for SDEs. We show local well-posedness of the DO equations and their equivalence with the DLRA formulation. We also characterize the explosion time of the DO solution by a loss of linear independence of the random coefficients defining the solution expansion and give sufficient conditions for global existence. [ABSTRACT FROM AUTHOR]

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of Eberlein weakly almost periodic solutions for differential equations of the form u ′ = A u + f (t) and u ′ = A (t) u + f (t) . In the first scenario, when A generates a strongly asymptotically semigroup, we establish the existence of Eberlein-weakly almost periodic solutions, thereby extending and improving a previous result in Zaidman(Ann Univ Ferrara 14(1): 29–34, 1969). In the second case, we consider a more general situation where A(t) is a (possibly nonlinear) operator satisfying a monotony condition. Unlike most existing works in the literature, our approach does not rely on tools of exponential dichotomy and Lipschitz nonlinearity. Lastly, we illustrate the practical relevance of our findings by presenting real-world models, including a hematopoiesis model, that exemplify the key findings. A numerical simulation is also provided. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the existence, uniqueness and stability of solutions for some nonlocal impulsive evolution problems involving conformable fractional derivative. The main results are based on fixed point theorems combined with the semigroup theory. As applications, we give an example to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

A method for studying the secular evolution and stabilization of the shape of rings in small celestial bodies that do not have shepherd satellites is developed. A model of a compound ring consisting of two close, generally non-coplanar elliptical Gaussian rings is constructed. The self-gravitation of the ring is taken into account through the mutual gravitational energy of the boundary rings Wmut. The function Wmut is presented as a series with an accuracy of up to the 4th power of small eccentricities and mutual inclination of the rings. The secular evolution of a compound ring is described by differential equations in special (collective) variables. For rings without a central body (problem 1), a closed system of eight differential equations is obtained using the mutual energy function. The evolution of rings in the azimuthally averaged potential of a rotating triaxial body is also studied (problem 2), for which a second system of eight differential equations is derived. In both problems, besides the general case, two particular ones are considered: (i) the case of coplanar elliptical rings, and (ii) the case of circular rings with a tilt. The theory is applied to study the recently discovered ring of dwarf planet Haumea. It is shown that without taking into account self-gravity, the nodal precession time of the Haumea ring is equal to TΩ = 12.9 ± 0.7d but taking into account the self-gravity of the ring can reduce this period. It is established that self-gravity does indeed contribute to the preservation of the ring shape without invoking the hypothesis of shepherd satellites. Criteria for the preservation of the ring shape are obtained, which made it possible to estimate the interval for the ratio of the ring mass to the mass of Haumea 10–4 < m/M < 10–3. Taking into account the optical thickness of the ring τ ≈ 0.5, it is shown that the Haumea ring with a mass m/M ≈ (1–2) × 10–4 can consist of ice particles with a size of m. [ABSTRACT FROM AUTHOR]

In this article we present strong global solutions for the initial value problem in a reflexive Banach space, when the linear part of the corresponding differential evolution equation is Bohl-Bohr or Stepanov almost periodic function, and the displayed operator is infinitesimal generator of (C0)-semigroup. As an application, we consider a problem from magnetohydrodynamics. [ABSTRACT FROM AUTHOR]

In this paper, we study, in an infinite dimensional Banach space X, the existence and uniqueness of mild solutions to the Cauchy problem for the semilinear differential evolution equations with nonlocal conditions where HDα,β;ψ is the ψ-Hilfer operator, for 0 < α < 1, 0 < β < 1, and A(t): X → X is a continuous operator for each t ∈ [0, T]. The function f: [0, T] × X → X is not necessarily lipschitzian. We first derive a variation of constants formula. Then we use theWessinger and the Krasnoselskii's fixed point theorems to achieve our results. These results generalize a recent work by K. Balachandran et al. We provide an example to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results. [ABSTRACT FROM AUTHOR]

An optimal control problem is investigated for an abstract semilinear differential equation of the first order in time in a Hilbert space with an unbounded operator and control involved linearly in the right-hand side. The cost functional is assumed to be additively separated with respect to state and control, with a rather general dependence on the state. For this problem, the existence of an optimal control is proved and the properties of the set of optimal controls are established. The author's previous results on the total preservation of unique global solvability (totally global solvability) and on solution estimation for such equations are developed in the context of the nonlinearity of the equation under study. The indicated estimate is found important for the present study. A nonlinear heat equation and a nonlinear wave equation are considered as examples. [ABSTRACT FROM AUTHOR]

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy–momentum tensor calculated from a post-Newtonian Maxwell–Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory. [ABSTRACT FROM AUTHOR]

In this article, we study the evolution of the two-qubit system with Ising spin-chain environment by choosing the Lindblad equation to describe this process. First, we expand the Hamiltonian of a two-qubit system with its eigenstates and give the Lindblad equation for the evolution of the two-qubit system with Ising spin-chain as the environment. Compared with directly calculating the differential equation between elements of density matrix, we choose a method to solve the Lindblad equation with a smaller calculation dimension. Finally, by constructing an operator solution with operators, we obtain the dynamics of the two-qubit system with Ising spin-chain environment. [ABSTRACT FROM AUTHOR]

For the Lax-type Hamiltonian flows in the space dual to the Lie algebra of fractional integrodifferential operators, we develop a rational factorization method, which enables one to get new integrable hierarchies of nonlinear fractional-differential dynamical systems on the Lie algebra of ordinary integrodifferential operators and obtain infinite sequences of their conservation laws. By using the Bäcklund transformation, we show that the system of two flows of this kind for a pair of fractional integrodifferential operators related by a gauge transformation is equivalent to a system of two evolutionary equations for fractional differential operators, which determine the corresponding rational factorization, and find the Hamiltonian representation for this system of evolutionary equations. We show that its quasiclassical approximation is a system of two evolutionary equations for polynomials of fractional degree in a certain complex parameter. The proposed method is used to construct a new integrable hierarchy of nonlinear fractionaldifferential systems on the Lie algebra of ordinary integrodifferential operators and an infinite sequence of its conservation laws, as well as a new integrable hierarchy of Benney-type hydrodynamic systems, which is its quasiclassical approximation. [ABSTRACT FROM AUTHOR]

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

The Transformer has emerged as the predominant model in Natural Language Processing due to its exceptional performance in various sequence modeling tasks, particularly in handling long-term dependencies. However, the traditional absolute and relative position encoding methods, which do not learn from data, tend to ignore the inherent structure of natural language sequences due to the position embedding layer at the input end of the Transformer model. This paper introduces a novel learnable neural Ordinary Differential Equation Position Encoding (ODEPE) method that can implicitly capture the natural position relationships within a sequence without requiring additional position embeddings. ODEPE can model continuous sequences and leverage differential equations to simulate the evolution of position information along the sequence, enabling position information to flow seamlessly between sequences. Additionally, a highly effective recurrent attention framework is proposed, which hybridizes attention with the ODEPE method to improve model performance. Compared to the Transformer-based sequence modeling network, our framework demonstrates a performance improvement of 24.0 points on the WikiText-103 dataset, while also achieving a performance improvement of 1.06 points on the Enwik8 dataset. This corresponds to an improvement of 4.9% and 0.17%, respectively. • Novel ODE method for implicit local position encoding in Transformers. • Capture natural sequence position without extra embeddings. • Recurrent attention network captures longer dependencies. • Our method is adaptable, applicable in long-term language modeling. [ABSTRACT FROM AUTHOR]

In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called 'diffusion-wave-type solutions'. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. [ABSTRACT FROM AUTHOR]

Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating a mathematical problem that characterizes the adhesive contact between an electro-elastic-viscoplastic material with damage and a foundation. The contact phenomenon is represented using the Signorini condition, while the progression of damage is delineated through a parabolic-type inclusion. Additionally, the adhesion process is rendered via a bonding field applied to the contact surface. The entire process occurs quasistatically. We establish a weak formulation of the system and subsequently demonstrate the existence of a unique weak solution to the problem. The proof relies on a comprehensive result concerning evolution equations involving maximal monotone operators, parabolic inequalities, differential equations, and fixed points. [ABSTRACT FROM AUTHOR]

In this paper we present a new approach to solve a risk default, under asymmetric information, between Micorfinance Institutions MFIs and borrowers excluded from the conventional financial systems (financial exlusion). To solve this problem we propose the model, as a system of two differential equations to describe the evolution of some safe and risky borrowers. In fact, the population of borrowers is divided into two compartments: Safe and Risky having an interaction, it is a SR model. With numerical simulations of the SR model, we can describe the evolution of some safe and risky borrowers. The result provides to manage the portfolios of all agencies controlled from the MFIs, by predictions of safe and risky borrowers. Then, define or redefine the management strategies of development and investment, and thus strengthen financial inclusion. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to integrate a modified Korteweg-de Vries-Liouville (mKdV-L) equation with an additional term in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator is introduced, and the coefficient of the Dirac operator is a solution for a modified Korteweg-de Vries-Liouville equation with an additional term. A simple algorithm for deriving the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for a Dubrovin infinite system of differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proven. It is proven that there is a global solution of the Cauchy problem for a modified Korteweg-de Vries-Liouville equation with an additional term for sufficiently smooth initial conditions. [ABSTRACT FROM AUTHOR]

The method of molecular dynamics is applied to study the features of the plastic flow of hexagonal magnesium, namely, the evolution of dislocation density of magnesium single crystals of hard and soft orientations during the deformation by compression. The question on the validity of describing the dislocation plasticity at the atomic level by the theoretical dislocation density evolution equations valid at the macroscopic level is considered. The Orowan–Taylor equations, the law of hardening due to interaction of dislocations and their clusters with each other, as well as the differential equation of dislocation dynamics are tested. A relationship between the deformation rate, the defect density, and the Burgers dislocation vectors and active sliding systems is revealed. The simulation results presented in this report are based on the embedded-atom method (EAM) interatomic potential for Mg. Results of calculations of theoretical equations are presented based on the data obtained by molecular dynamics simulation of magnesium deformation by compression. [ABSTRACT FROM AUTHOR]

The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the 'forcing term'. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The 'forcing term' does not express a 'classical' force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology. [ABSTRACT FROM AUTHOR]

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. [ABSTRACT FROM AUTHOR]

This article focuses on the controllability for a class of semilinear evolution equations and applications to some specific differential equations. Without assuming compactness or equicontinuity of the related semigroups, the existence of mild solutions of semilinear equations in Hilbert space is demonstrated via the noncompact measure tool and the fixed point trick. In order to study the controllability of semilinear equations, new concepts are proposed, namely, exact controllability along any A-bounded Lipschitz continuous curve, and approximate controllability along any continuous curve. Furthermore, two new approximation methods, "bisection method"" and "equisection method,"" are introduced. The exact controllability along any A-bounded Lipschitz continuous curve and the approximate controllability along any continuous curve in the sense of the graph norm of semilinear evolution equations are obtained under the asymptotic condition on the norm of the controllability Gramian inverse operator near the zero point. In fact, our conclusions show that this is not only a result control but also a process control. Finally, the results presented in this paper are employed in resistance, inductance, voltage source type electrical circuit systems, one-dimensional nonhomogeneous transport systems, as well as differential equations with delay which have important effects on economic systems. [ABSTRACT FROM AUTHOR]

We study the negative-order modified Korteweg–de Vries equation and show that it can be integrated by the inverse spectral transform method. We determine the evolution of the spectral data for the Dirac operator with periodic potential associated with a solution of the negative-order modified Korteweg–de Vries equation. The obtained results allow applying the inverse spectral transform method for solving the negative-order modified Korteweg–de Vries equation in the class of periodic functions. Important corollaries are obtained concerning the analyticity and the period of a solution in spatial variable. We show that a function constructed using the Dubrovin–Trubowitz system and the first trace formula satisfies the negative-order modified Korteweg–de Vries equation. We prove the solvability of the Cauchy problem for the infinite Dubrovin–Trubowitz system of differential equations in the class of three-times continuously differentiable periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present the concept of S v -asymptotically T-periodic functions taking values in a Banach space and investigate some of their properties. Also, we establish conditions under which semi linear evolution equations in a Banach space have a unique global mild S v -asymptotically T-periodic solution. Then, we apply the results obtained to prove the existence and uniqueness of S v -asymptotically ω -periodic mild solutions to a nonautonomous semi linear differential equations. [ABSTRACT FROM AUTHOR]

This paper designates two important properties of the existence-uniqueness of the mild solution for a fractional controlled fuzzy evolution equation involving the Caputo-derivative by using nonlocal conditions, where β ∈ (1 , 2) , and it is formulated as 0 c D ω β (ω) = (ω) + χ (ω , (h (ω))) + (ω) ℋ (ω) , (0) = 0 + (− 1) g () , ′ (0) = 1 , where is a strongly continuous fuzzy semigroup's generator. Furthermore, many numerical solutions have been presented to ensure that key results are accurate and reliable. Finally, an example about fuzzy fractional evolution equations is presented. [ABSTRACT FROM AUTHOR]

A classical non-stationary three-body problem with two bodies of variable mass moving around the third body on quasi-periodic orbits is considered. In addition to the Newtonian gravitational attraction, the bodies are acted on by the reactive forces arising due to anisotropic variation of the masses. We show that Newtonian's formalism may be generalized to the case of variable masses and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasiconic sections. As equations of motion are not integrable the perturbative method is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations which are assumed to be small. Averaging these equations over the mean longitudes of the bodies in the absence of a mean-motion resonances, we obtain the differential equations describing the evolution of orbital parameters over long period of time. We solve the evolution equations numerically and demonstrate that the mass change modify essentially the system evolution. [ABSTRACT FROM AUTHOR]

We present here a computational numerical operator, and we name it as Time Invariance Operator (TIO). This operator can add obstacles to the domain of the differential equation that describes a physical phenomenon. After the TIO acts, the wave equation recognizes the introduced points as non-interacting zones without affecting the rest of the domain. Computational physics has been consolidated as an important field of study, especially when connected with the fundamentals of physics. In many cases, simulations are conducted considering the ideal case of a wave in an infinite domain and open space without considering obstacles, barriers, or other aspects of the real world. The results presented in this paper allow us to infer that the TIO is the easiest way to apply the physical domain to wave propagation simulations and successfully recreate wave interaction experiments through computer simulations. Our motivation is to perform wave simulations that interact with obstacles, barriers, single slits, and double slits. We aim to investigate the results obtained in images to determine if the methodology we used to introduce realistic physical characteristics was successful in presenting the expected phenomenology. The simplicity of the TIO's action in creating locally time-invariant regions over the domain makes it suitable not only for waves but also for equations with transient terms. Heat transfer, mass transfer, computational fluid dynamics, and other time evolution equations can take some benefit from the operator presented in this paper. The TIO ensures local conservation that mimics interaction regions or ensures free space characteristics if it is the case like a 2D tensor of local conservation. The principal result from this paper is the validation of the TIO to impose realistic conditions with minimal modifications over a running code of wave equation simulation originally in free space. The TIO is innovative because it imposes dynamic conditions that mimic realistic interacting zones. [ABSTRACT FROM AUTHOR]

In statistical theory, the Tsallis entropy is an extended form of the Boltzmann–Gibbs entropy. The dimensionless parameter δ is employed to state the quantitative difference from the standard scenario. The concepts of Tsallis entropy and the future event horizon are employed in formulating the present new Tsallis holographic dark energy (NTHDE) model. The model attempts to explain the properties of dark energy using the foundation of quantum gravity. The differential equation characterizing the evolution of the NTHDE density parameter is obtained. Expressions stating the dynamic behavior such as equation of state (EoS), deceleration and jerk parameters are obtained in terms of the NTHDE density parameter. For δ < 1 , the quintessence nature of scalar field could completely characterize the NTHDE. A reconstruction of the scalar field's dynamics and quintessence potential is attempted. We demonstrate that the diagnosis made by statefinder is adaptive enough to distinguish between quintessence and cosmological constant-based dark energy models. Additionally, observational data obtained from CC + SNIa + 5 8 0 union 2.1 sources are used to evaluate the model's effectiveness. [ABSTRACT FROM AUTHOR]

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. The obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments. [ABSTRACT FROM AUTHOR]

The paper provides a new result concerning the existence of solutions for second-order evolution problems associated with time-dependent subdifferential operators involving both single-valued and mixed semi-continuous set-valued perturbations. Optimal control problems corresponding to such differential inclusions using relaxation theorems with Young measures are investigated. The existence of solutions for a coupled system governed by a second-order differential equation with an evolution problem is also addressed. [ABSTRACT FROM AUTHOR]

The classical porous medium equation is widely used to model different natural phenomena related to diffusion, filtration and heat propagation. In this short communication, we prove that the solution of porous medium equation can be locally approximated by the solution of a class of nonlocal dispersal equation. Our work is a counterpart to the important works (see Berestycki et al. [J. Funct. Anal. 271 (2016), pp. 2701–2751; J. Math. Biol. 72 (2016), pp. 1693–1745]; Dipierro et al. [J. Eur. Math. Soc. (JEMS) 19 (2017), pp. 957–966; J. Geom. Anal. 29 (2019), pp. 1428–1455]; Hansen and Netuka [Potential Anal. 2 (1993), pp. 67–71]; Ignat and Rossi [J. Funct. Anal. 251 (2007), pp. 399–437]; Shen and Xie [J. Differential Equations 259 (2015), pp. 7375–7405]; Sprekels and Valdinoci [SIAM J. Control Optim. 55 (2017), pp. 70–93]). [ABSTRACT FROM AUTHOR]

The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov's hydrodynamic system of equations is considered. [ABSTRACT FROM AUTHOR]

The editorial for the special issue "Advances in Differential and Difference Equations and Their Applications" in Mathematics showcases recent progress in the study of differential and difference equations. The collection of articles covers various topics, including boundary value problems, fractional equations, and applications in scientific and engineering fields. The papers presented in this issue reflect the latest theoretical advancements and applications in these important research areas, providing valuable insights for researchers in the global mathematical community. [Extracted from the article]

We establish the correct solvability of a multipoint nonlocal (in time) problem for the evolutionary equation with operator of fractional differentiation and an initial function, which is an element of the space of generalized functions of the distribution type. The analytic representation of the solution is presented. We also analyze the behavior of the solution in the case of unlimited growth of the time variable (stabilization of the solution). [ABSTRACT FROM AUTHOR]

An existing approximately neutral surface, the ω surface, minimizes the neutrality error and hence also exhibits very small fictitious dianeutral diffusivity Df that arises when lateral diffusion is applied along the surface, in nonneutral directions. However, there is also a spurious dianeutral advection that arises when lateral advection is applied nonneutrally along the surface; equivalently, lateral advection applied along the neutral tangent planes creates a vertical velocity esp through the ω surface. Mathematically, esp = u ⋅ s, where u is the lateral velocity and s is the slope error of the surface. We find that esp produces a leading-order term in the evolution equations of temperature and salinity, being similar in magnitude to the influence of cabbeling and thermobaricity. We introduce a new method to form an approximately neutral surface, called an ωu·s surface, that minimizes esp by adjusting its depth so that the slope error is nearly perpendicular to the lateral velocity. The esp on a surface cannot be reduced to zero when closed streamlines contain nonzero neutral helicity. While esp on the ωu·s surface is over 100 times smaller than that on the ω surface, the fictitious dianeutral diffusivity on the ωu·s surface is larger, nearly equal to the canonical 10−5 m2 s−1 background diffusivity. Thus, we also develop a method to minimize a combination of esp and Df, yielding the ω u · s + s 2 surface, which is recommended for inverse models since it has low Df and it significantly decreases esp through the surface, which otherwise would be a leading term that cannot be ignored in the conservation equations. [ABSTRACT FROM AUTHOR]

A qualitative analysis for a nonlinear system of fractional pantograph evolution differential equations (FPEDEs) has been established. Conditions about the solution of the underlying equations have been established in the direction of the existence and uniqueness of the solution. Using results of fixed‐point theorems of Krasnoselskii iterations in Banach spaces, we develop the required results. This research work is enriched by establishing the results related to Ulam–Hyers (UH) stability of the problem under consideration via the tools of nonlinear analysis. To validate the reliability of the obtained results, some pertinent examples are given in the manuscript. [ABSTRACT FROM AUTHOR]

In the 1970's, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map's dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows. [ABSTRACT FROM AUTHOR]

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems. [ABSTRACT FROM AUTHOR]

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations. 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations. 2016;16(3):539‐556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water. [ABSTRACT FROM AUTHOR]

This paper studies the existence, regularity, and asymptotic properties of solutions for a class of neutral differential evolution equations with nonlocal initial conditions on an infinite interval. The existence and regularity of solutions of the considered equation are, respectively, investigated by the theory of fractional power operators and fixed point theorems under some assumptions for nonlinear functions. Then, under suitable conditions, asymptotic properties, including stability and existence of global attracting sets and quasi-invariant sets of mild solutions, are also discussed in the context. Finally, an example is presented to illustrate the applications of the obtained abstract results. [ABSTRACT FROM AUTHOR]

The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given. [ABSTRACT FROM AUTHOR]

A mathematical model is formulated for the evolution of plane perturbations in a cosmological two-component statistical system of completely degenerate scalarly charged fermions with an asymmetric scalar Higgs interaction. A complete closed system of differential equations are constructed to describe the unperturbed state of a homogeneous and isotropic system and a system of self-consistent evolution equations of small perturbations. [ABSTRACT FROM AUTHOR]

Ride-share platforms are contemporary businesses that match passengers with drivers, unlike taxis that can be hailed from the street. In the literature, the problem of optimizing the operations of such companies is mostly considered in static settings. We use in this paper a dynamic model and propose differential equations to model the evolution of the system. The objective is to maximize the profit during the planning horizon. Using optimal control theory, we determine the optimal rate of change in the ride price rate. An illustrative example along with sensitivity analyses shows the effect of the system parameters on the optimal solution obtained. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence, uniqueness and some regularity results for the solution of differential evolution equations of fractional order involving an unbounded linear maximal monotone operator in a Hilbert space. The proof relies on spectral properties of the operator. An explicit representation formula of the solution is given, allowing the computation of a sequence of approximate solutions. [ABSTRACT FROM AUTHOR]

In this article, we consider a general time-fractional differential equation, (∂ α v / ∂ t α) = F [ x , v , v x , v xx , v xxx ] , where F is a third-order polynomial operator with quadratic nonlinearities. We describe the operator F which admits two-dimensional invariant subspaces which are solutions of second-order ordinary differential equation with constant coefficients. Using the invariant subspace method, we find explicit solution for fractional differential equations for various cases of operator F. We show that explicit solutions to many of the known equations like time-fractional Rosenau–Hyman equation, Korteweg–de Vries equation, Benjamin equation etc. can be derived using this method. [ABSTRACT FROM AUTHOR]