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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]

The Bagley-Torvik equation is an imperative differential equation that considerably arises in various branches of mathematical physics and mechanics. However, very few methods exist for the treatment of the model analytically; in fact, researchers frequently shop for semi-analytical and numerical methods in their studies. Therefore, the main goal of this research is to find the exact analytical solution for the fractional Bagley-Torvik equation fitted with Dirichlet boundary data, as well as a system of fractional Bagley-Torvik equations. Thus, this research aims to show that the modified Adomian decomposition method (MADM) via the proposed two algorithms is a very effective method for treating a class of Bagley-Torvik equations endowed with Dirichlet boundary data. Certainly, MADM is a very powerful approach for solving dissimilar functional equations without the need for either linearization, discretization, perturbation, or even unnecessary restraining postulations. Additionally, the method reveals exact analytical solutions whenever obtainable or closed-form series solutions whenever exact solutions are not feasible. Lastly, some illustrative test problems of the governing model are examined to demonstrate the superiority of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

A comprehensive framework has been developed to apply the monomiality principle from mathematical physics to various mathematical concepts from special functions. This paper presents research on a novel family of multivariate Hermite polynomials associated with Apostol-type Frobenius–Euler polynomials. The study derives the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified. Moreover, the research establishes series representations, summation formulae, and operational and symmetric identities, as well as recurrence relations satisfied by these polynomials. [ABSTRACT FROM AUTHOR]

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the Δ h hybrid special polynomials associated with Hermite polynomials denoted by Δ h H Q m (u , v , w ; h) . Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials. [ABSTRACT FROM AUTHOR]

Based on Fourier analysis, we develop an expression for modeling and simulating nonlinear first order systems. This expression is associated to a nonlinear first order differential equation y = f (x) + g (x) x ′ , where x = x (t) is the dynamical variable, y = y (t) is the driving force, and the f and g functions are the characteristic curves which are associated to dissipative and memory elements, respectively. The model is obtained from the sinusoidal response, specifically by calculating the Fourier analysis of y(t) for x (t) = A 1 sin (ω t) + A 0 , where the model parameters are the Fourier coefficients of the response, and the values of A 0 , A 1 and A 1 ′ = A 1 ω . The same expression is used for two kinds of time-domain simulations: to calculate other driving force y ^ based on a dynamical variable x ^ ; and, to calculate the dynamical variable x ^ based on a driving force y ^ . In both cases, the dynamical variable must remain in the range x ^ ∈ [ A 0 - A 1 , A 0 + A 1 ] . By analyzing this expression, we found an equivalence between the Fourier coefficients and the polynomial regressions of the characteristic curves of f and g. This result allows us to obtain the system modeling and simulation based on the amplitude and phase Fourier spectrum obtained from the Fast Fourier Transform (FFT) of the sampled y n version of y(t). It is shown that this technique has a low computational complexity, and it is expected to be suitable for real-time applications for system modeling, simulation and control, in particular when the explicit expressions of the characteristic curves are unknown. Fourier analysis is a fundamental tool in electronics, mathematics and physics, but to the best of the author's knowledge, no work has found this clear evidence of the connection between the Fourier analysis and a first order differential equation. The aim of this work is to initiate a systematic study on this topic. [ABSTRACT FROM AUTHOR]

A modified form of differential equations is proposed that describes physical processes studied in applied mathematics and mechanics. It is noted that solutions to classical equations at singular points may undergo discontinuities of the first and second kind, which have no physical nature and are not experimentally observed. When new equations describing physical fields and processes are derived, elements with finite dimensions are considered instead of infinitely small elements of the medium. Thus, classical equations include nonlocal functions averaged over the element volume and are supplemented by the Helmholtz equations establishing a relationship between nonlocal and actual physical variables, which are smooth functions having no singular points. Singular problems of the theory of mathematical physics and the theory of elasticity are considered. The obtained solutions are compared with the experimental results. [ABSTRACT FROM AUTHOR]

In mathematical physics, time-dependent Green’s functions (GFs) are the solutions of differential equations of the first and second time derivatives. Habitually, the time-dependent GFs are Fourier transformed into the frequency space. Then, analytical continuation of the frequency is extended to below or above the real axis. After inverse Fourier transformation, retarded and advanced GFs can be obtained, and there may be arbitrariness in such analytical continuation. In the present work, we establish the differential equations from which the retarded and advanced GFs are rigorously solved. The key point is that the derivative of the time step function is the Dirac δ function plus an infinitely small quantity, where the latter is not negligible because it embodies the meaning of time delay or time advance. The retarded and advanced GFs defined in this paper are the same as the one-body GFs defined with the help of the creation and destruction operators in many-body theory. There is no way to define the causal GF in mathematical physics, and the reason is given. This work puts the initial conditions into differential equations, thereby paving a way for solving the problem of why there are motions that are irreversible in time. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find the exact solutions of the Schrödinger equation for the Manning–Rosen plus Graz separable potential through two different approaches to the problem. We express the irregular/Jost and physical solutions in terms of the special functions of mathematical physics. Numerical results of the phase shifts are obtained by utilising the properties of the Jost function and Fredholm determinants for nucleon–nucleon and nucleon–nucleus systems. The results obtained are in good agreement with earlier works. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "On symbology and differential equations of Feynman integrals from Schubert analysis" published in the Journal of High Energy Physics. The authors, Song He, Xuhang Jiang, Jiahao Liu, and Qinglin Yang, discuss the generalization of the "Schubert analysis" method to study symbol letters and differential equations for Feynman integral families in various dimensions. They demonstrate the effectiveness of the method using one-loop integrals and various two-loop planar integral families. The correction notice addresses errors in the affiliations and acknowledgments of the original paper. The article is open access and funded by SCOAP3. [Extracted from the article]

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular. [ABSTRACT FROM AUTHOR]

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines. [ABSTRACT FROM AUTHOR]

Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical wave solutions to the mentioned nonlinear equation in the sense of the nonlinear ordinary differential transform equation are obtained. The techniques are the improved exp − Γ ϖ function method and the improved simple equation method. The nonlinear ordinary transform to concern the generalized Schrodinger equation to convert it for a solvable integer-order differential equation is used. After the successful implementation of the presented methods, the exact solitary wave solutions in the form of trigonometric, rational, and hyperbolic functions are obtained. Hence, the presented methods are relatable and efficient to solve nonlinear problems in mathematical physics. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to prove the existence of extremal solutions for a novel class of -Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results. [ABSTRACT FROM AUTHOR]

Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics. [ABSTRACT FROM AUTHOR]

This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo's time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions' accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations. [ABSTRACT FROM AUTHOR]

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind. [ABSTRACT FROM AUTHOR]

This paper presents the Lucas polynomial solution of second-order nonlinear ordinary differential equations with mixed conditions. Lucas matrix method is based on collocation points together with truncated Lucas series. The main advantage of the method is that it has a simple structure to deal with the nonlinear algebraic system obtained from matrix relations. The method is applied to four problems. In the first two problems, exact solutions are obtained. The last two problems, Bratu and Duffing equations are solved numerically; the results are compared with the exact solutions and some other numerical solutions. It is observed that the application of the method results in either the exact or accurate numerical solutions. [ABSTRACT FROM AUTHOR]

In order to determine the optimal strategy to run a race on a curved track according to the lane number, we introduce a model based on differential equations for the velocity, the propulsive force and the anaerobic energy which takes into account the centrifugal force. This allows us to analyze numerically the different strategies according to the types of track since different designs of tracks lead to straights of different lengths. In particular, we find that the tracks with shorter straights lead to better performances, while the double bend track with the longest straight leads to the worst performances and the biggest difference between lanes. Then for a race with two runners, we introduce a psychological interaction: there is an attraction to follow someone just ahead, but after being overtaken, there is a delay before any benefit from this interaction occurs. We provide numerical simulations in different cases. Overall, the results agree with the IAAF rules for lane draws in competition, where the highest ranked athletes get the center lanes, the next ones the outside lanes, while the lowest ranked athletes get the inside lanes. [ABSTRACT FROM AUTHOR]

Vibrational behaviour of symmetric angle-ply layered circular cylindrical shell filled with quiescent fluid is presented. The equations of motion of cylindrical shell in terms of stress and moment resultants are derived from the first order shear deformation theory. Irrotational of inviscid fluid are expressed as the wave equation. These two equations are coupled. Strain-displacement relations and stress-strain relations are adopted into the equations of motion to obtain the differential equations with displacements and rotational functions. A system of ordinary differential equation is obtained in one variable by assuming the functions in separable form. Spline of order three is applied to approximate the displacement and rotational functions, together with boundary conditions, to get a generalised eigenvalue problem. The eigenvalue problem is solved for eigen frequency parameter and associate eigenvectors of spline coefficients. The study of frequency parameters are analysed using the parameters the thickness ratio, length ratio, angle-ply, properties of material and number of layers under different boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices. [ABSTRACT FROM AUTHOR]

Abstract We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations. Highlights • We perform a comparison study on dynamics between the AC and CH equations. • Linear stability analysis shows that growing and decaying modes are the same. • The growth rates are monotonically decreasing for the AC equation. • The growth rates for the CH equation are non-monotonic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new generalized inverses of matrices, namely, the hi;mi-core inverse and the (j, m)-core inverse. The (i, m)-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler [1] and the core-EP inverse defined by Manjunatha Prasad and Mohana [10]. The (j, m)-core inverse of a complex matrix extends the notions of the core inverse and the DMP-inverse defined by Malik and Thome [9]. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with 2p-order Hamiltonian systems with impulsive effects. We investigate the variational structure associated to this system. In addition, we obtain some results of multiple solutions for asymptotically linear 2p-order Hamiltonian systems via variational methods and critical point theorems. Meanwhile, some examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

Abstract: A brief review of the methods for the study of residual process stresses (RSs) in the elements of structures and specimens is given. The possibilities and the fields of the application of nondestructive testing of RSs are indicated. Destructive experimental and calculation methods for the study of two-dimensional and three-dimensional nonuniform fields of RSs in spatial structures, which are based on the interpretation of experimental data as an inverse problem of elasticity theory, are particularly noted. It is recommended to use optical-digital methods for the recording of displacement fields in order to obtain significant collections of experimental information necessary in this case, which are caused by the formation of the cuts of various configurations in the object under study. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, the complete form of Toupin–Mindlin strain gradient theory of elasticity is utilized to determine the elastic field of a spherical inclusion undergoing a time-harmonic eigenfield in an infinite isotropic body. The theory adopted herein involves three characteristic lengths, two ones of which are present therein due to the incorporation of strain gradients in the strain energy density function of the material under consideration, and the presence of the third one arises from taking into account the role of velocity gradients in its kinetic energy density function, reflecting the effect of micro inertia. After the derivation of an expression for the associated Green's function of the problem, closed-form solutions are obtained for the interior and exterior elastic fields of a spherical inclusion subjected to a time-harmonic eigenfield whose spatial distribution is expressible as a function of the radial distance of points from the center of the inclusion. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where u (t , x) and v (t , x) denote the densities of two competing species, d > 0 is some constant, r i (t) , a i (t) and b i (t) are T -periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]

In this article, we establish new oscillation criteria for the second-order Emden-Fowler neutral delay differential equation ..., where z(t) = x(t) + p(t) x (τ(t)), α > 0 and β > 0. Our results improve some well-known results which were published recently in the literature. Some illustrative examples are also provided to show the significance of our results. [ABSTRACT FROM AUTHOR]

The article concerns the initial boundary value problem for a nonlinear Kirchhoff-Love equation. First, by applying the Faedo-Galerkin, we prove existence and uniqueness of a solution. Next, by constructing Lyapunov functional, we prove a blow-up of the solution with a negative initial energy, and establish a sufficient condition for the exponential decay of weak solutions. [ABSTRACT FROM AUTHOR]

We consider the problem ..., where a > 0, c ≥ 0, α ∈ (0, 1), h:(0, 1] → (0, ∞) is a continuous function which may be singular at t = 0, but belongs to L¹(0, 1) ∩ C¹(0, 1), and g:[0, ∞) → [0, ∞) is a continuous function. We discuss existence, uniqueness, and non existence results for positive solutions for certain values of a, b and c. [ABSTRACT FROM AUTHOR]

This article concerns the existence of non-oscillatory solutions to the equation ..., where ak ≥ 0 and hk(t) ≥ t. We generalize existing results and then give an answer to the open question stated in [4]. Moreover we present a new condition based on the integral of (∑ ak)². [ABSTRACT FROM AUTHOR]

We study the well-posedness of abstract time evolution fractional integro-differential equations of variable order u(t) = u0 + ∂-α(t) Au(t) + f(t). Also we study the asymptotic behavior as t → + ∞, and the regularity of solutions. Moreover, we present the asymptotic behavior of the discrete solution provided by a numerical method based on convolution quadratures, inherited from the behavior of the continuous solution. In this equation A plays the role of a linear operator of sectorial type. Several definitions proposed in the literature for the fractional integral of variable order are discussed, and the differences between the solutions provided for each of them are illustrated numerically. The definition we chose for this work is based on the Laplace transform, and we discuss the reasons for this choice. [ABSTRACT FROM AUTHOR]

Abstract The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Γ × S 1 -spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Γ × S 1 -degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group Γ = D 8 ; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. [ABSTRACT FROM AUTHOR]

Highlights • An inhomogeneous composite structure model with a power volume fraction is built. • A fourth-order ODE with variable coefficients is obtained. • Exact solutions are obtained, including the inhomogeneity and microstructural effect. • Mechanical performance of FG micro-cylinders are investigated. Abstract An exact solution is obtained for a functionally graded (FG) micro-cylinder subjected to internal and external pressures. And in the current model, its material properties are assumed to be isotropic and exponentially-varying elastic modulus in radial direction and a material length scale parameter is incorporated to capture the size effect. To this end, a theoretical formulation including the effect of size dependency is derived in the framework of the first gradient elasticity and through Hamilton's principle. Then, a fourth-order governing ordinary differential equation (ODE) with variable coefficients is developed and the corresponding solution is rather difficult to be determined for inhomogeneous problem. Next, the efficient tensor algorithm operator is utilized to reduce the fourth-order homogeneous ODE to a second-order non-homogeneous one. Finally, by using method of variation of constant, the exact solution for FG micro-cylinder problem containing the material length scale parameter and power index is constructed perfectly, which is qualitatively different from existing Lamé's solution in classical elasticity. When ignoring the inhomogeneity of material, the newly obtained exact solution reduces to the ordinary one. The numerical results reveal that increasing characteristic length parameter leads to the decrease of the maximum radial and tangential stresses, and the power index has also a considerable effect on the stress distribution of FG micro-cylinders. A key physical insight that emerges from our analysis is that the newly obtained solution form can be applied directly to practical engineering structures. Graphical abstract This work studies a functionally graded (FG) micro-cylinder subjected to internal and external pressures in the framework of the first gradient elasticity. Then a theoretical formulation of the fourth-order governing differential equation and associated boundary conditions with variable coefficients for inhomogeneous isotropic materials have been established. And exact solutions for FG micro-cylinders are obtained, including the inhomogeneity and microstructural effect perfectly. The numerical results reveal that the characteristic length parameter and power index have a considerable effect on the stress distribution of FG micro-cylinders. Image, graphical abstract [ABSTRACT FROM AUTHOR]

Abstract We investigate the presence of vortex structures in generalized Maxwell–Higgs and Chern–Simons–Higgs models in the three-dimensional spacetime. Despite the important difference between the Maxwell and Chern–Simons dynamics, we have been able to introduce first order differential equations that solve the equations of motion for static and rotationally symmetric field configurations. In both cases, solutions of the first order equations engender minimum energy, and we have found vortex configurations whose internal structure unveils interesting and unusual ringlike profile. [ABSTRACT FROM AUTHOR]

We investigate an M / M / 1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed by means of a suitable heavy-traffic approximation which leads to a continuous random process. Its distribution satisfies a partial differential equation with randomly alternating infinitesimal moments. For the approximating process we determine the steady-state distribution, the transient distribution and a first-passage-time density. [ABSTRACT FROM AUTHOR]

Offshore well elevation systems often present limit cycles caused by oil-gas slugging. The latest works in control solutions use models based on simple Ordinary Differential Equations to suppress slug flow and increase overall production. However, obtaining parameters for these models is not straightforward due to the occurrence of Hopf bifurcations. Therefore, this work aims to propose a novel methodology to improve parameter estimation. The limit cycle readings of pressure instruments are condensed into a single characteristic cycle with mean value and variance for each sample. Each iteration then uses a modified weighted sum of least squares. This creates a more convex region around the optimum that improves the performance of optimization algorithms. The results were obtained using previously published data for two models. Using the F-test, it was statistically proven that the parameters obtained from the proposed method provided an equivalent or better model fit by employing strictly numeric methods. [ABSTRACT FROM AUTHOR]

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics. [ABSTRACT FROM AUTHOR]

We develop famous Merton’s financial model to more realistic and interesting situation: consumption rate has lower and upper constraints. The aim is to find optimal strategies for consumption and investment. The corresponding HJB equation is a fully nonlinear ordinary differential equation. We use stochastic analysis and differential equation technique to find optimal strategies. The regularity of the value function is obtained as well. The method and result are meaningful and interesting in both of finance and mathematics. [ABSTRACT FROM AUTHOR]

The main goal of controller design in teleoperation systems is to achieve optimal performance, transparency and stability in presence of factors such as time delay in communication channel and modeling uncertainties. The teleoperation systems usually have complex dynamic. Consequently, differential equation solution of optimal control problem is difficult and complex for them. This paper presents a novel method for designing optimal controller based on singular perturbation framework for these systems. Firstly, we use the Taylor expansion to model the time delay, with considering time delay term; we derive a singular perturbation formulation for the teleoperation system. Using singular perturbation model and Chang decoupling transformation, singularly perturbed differential equations of optimal control problem is decomposed into the reduced order slow and fast differential equations. A formula is obtained that produces the solution of original differential equations of optimal control problem in terms of solutions of the slow and fast reduced order matrix differential equations. The reduced-order differential equations decrease the complexity of the optimal control problem for teleoperation systems. The simulations verify the effectiveness of the proposed control method and excellent performances tracking with high speed and small control signal. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

In this paper, a numerical procedure called multistage optimal homotopy asymptotic method (MOHAM) is introduced to solve multipantograph equations with time delay. It was shown that the MOHAM algorithm rapidly provides accurate convergent approximate solutions of the exact solution using only one term. A comparative study between the proposed method, the homotopy perturbation method (HPM) and the Taylor matrix method are presented. The obtained results revealed that the method is of higher accuracy, effective and easy to use. [ABSTRACT FROM AUTHOR]

We consider several sufficient conditions for a function f(z), analytic in |z| < 1, to be strongly starlike or to be a Bazilevič function of order α. [ABSTRACT FROM AUTHOR]

Let Y be a complex algebraic variety, G ↷ Y an action of an algebraic group on Y , U ⊆ Y ( C ) a complex submanifold, Γ < G ( C ) a discrete, Zariski dense subgroup of G ( C ) which preserves U , and π : U → X ( C ) an analytic covering map of the complex algebraic variety X expressing X ( C ) as Γ \ U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative χ ˜ : Y → Z (where Z is some algebraic variety) expressing the quotient of Y by the action of the constant points of G . Under the additional hypothesis that the restriction of π to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil–Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation χ : = χ ˜ ∘ π − 1 is a well-defined, differential constructible function. The function χ nearly inverts π in the sense that for any differential field K of meromorphic functions, if a , b ∈ X ( K ) then χ ( a ) = χ ( b ) if and only if after suitable restriction there is some γ ∈ G ( C ) with π ( γ ⋅ π − 1 ( a ) ) = b . [ABSTRACT FROM AUTHOR]

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices. [ABSTRACT FROM AUTHOR]

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation. [ABSTRACT FROM AUTHOR]

We consider evolution differential equations in sequence spaces and construct a version of the majorant function method to prove existence theorems. We also develop a theory of lower estimates. This theory allows us to study instability and blowups. Applications to stability theory are discussed. [ABSTRACT FROM AUTHOR]

We define normal forms of regular closed polygonal curves in R2, prove that any such curve can be taken to normal form by a regular homotopy, construct two different algorithms (implemented in computer animations) designed to take a given curve to normal form via local moves, present experimental results confirming that this almost always happens, and explain the biological motivation behind the algorithms, as well as their biological interpretation. [ABSTRACT FROM AUTHOR]

We consider a planar double box with four massive external momenta and two massive internal propagators. We derive the system of differential equations for the relevant master integrals, cast it in canonical form, write it as a d log form and solve it in terms of iterated integrals up to depth four. [ABSTRACT FROM AUTHOR]