Scalar Differential Equations Research Articles

Exact solvability of certain linear ODEs

AbstractWe study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.

Periodic Solutions in a Simple Delay Differential Equation

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

Scalar BSDEs of iterated-logarithmically sub-linear generators with integrable terminal values

We establish a general existence and uniqueness result of integrable adapted solutions to scalar backward stochastic differential equations with integrable parameters, where the generator g has an iterated-logarithmically uniform continuity in the second unknown variable z. The result improves our previous one in Fan et al. (2023), and can be applied to the (conditional) g-expectation and the star-shaped risk measure.

On the structure of the infinitesimal generators of scalar one-dimensional semigroups with discrete Lyapunov functionals

AbstractDynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $$C^0[0,1]$$ C 0 [ 0 , 1 ] or on $$C^1[0,1]$$ C 1 [ 0 , 1 ] which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $$C^1[0,1]$$ C 1 [ 0 , 1 ] , the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $$C^0[0,1]$$ C 0 [ 0 , 1 ] , the corresponding equations are generalizations of monotone feedback delay differential equations.

On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

An improved result on bounded/periodic solutions for some scalar delay differential equations

An improved result on bounded/periodic solutions for some scalar delay differential equations

On the stability of a nonlinear nonautonomous scalar equation with variable delay

The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

Rigorous higher-order Poincaré optical vortex modes

The state of polarization of a general form of an optical vortex mode is represented by the vector ϵ^ m , which is associated with a vector light mode of order m>0. It is formed as a linear combination of two product terms involving the phase functions e±imϕ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (Θ P ,Φ P ) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A=ϵ^ m Ψ m , from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of Ψ m can be found by rigorously demanding that the product ϵ^ m Ψ m satisfies the vector paraxial equation. For a given order m this leads to a unique Ψ m , which has no azimuthal phase of the kind e i ℓ ϕ , and it is a solution of a scalar partial differential equation with ρ and z as the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m≥2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos⁡Θ P =0.

The modified truncated Euler–Maruyama method for stochastic differential equations with concave diffusion coefficients

Influenced by Gyöngy and Rásonyi (2011), many scholars established the strong convergence of several numerical methods for scalar stochastic differential equations (SDEs) with superlinearly growing drift and Hölder continuous diffusion coefficients. However, their methods depend on the Yamada-Watanabe method and therefore fail to work for multi-dimensional SDEs. In this paper, we study the strong Lp−convergence, for all p⩾2, of the modified truncated Euler–Maruyama method for multi-dimensional SDEs with superlinearly growing drift and concave diffusion coefficients satisfying the Osgood condition. We also discuss an example with computer simulations to illustrate our theoretical results.

Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

AbstractThe global bifurcation diagrams for two different one-parametric perturbations ($$+\lambda x$$ + λ x and $$+\lambda x^2$$ + λ x 2 ) of a dissipative scalar nonautonomous ordinary differential equation $$x'=f(t,x)$$ x ′ = f ( t , x ) are described assuming that 0 is a constant solution, that f is recurrent in t, and that its first derivative with respect to x is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation $$+\lambda x$$ + λ x , a so-called generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work.

A two-stage multi-step numerical scheme for mixed convective Williamson nanofluid flow over flat and oscillatory sheets

The novelty of this contribution is to propose an implicit numerical scheme for solving time-dependent boundary layer problems. The scheme is multi-step and consists of two stages. It is third-order accurate in time and constructed on three-time levels. For spatial discretization, a fourth-order compact scheme is adopted. The stability of the proposed scheme is analyzed for scalar linear partial differential equation (PDE) that shows its conditional stability. The convergence of the scheme is also provided for a system of time-dependent parabolic equations. Moreover, a mathematical model for heat and mass transfer of mixed convective Williamson nanofluid flow over flat and oscillatory sheets is modified with the characteristic of the Darcy–Forchheimer model. The results show that the temperature profile rises by developing thermophoresis and Brownian motion parameter values. Also, the proposed scheme is compared with an existing Crank–Nicolson method. It is found that the proposed scheme converges faster than the existing one for solving scalar linear PDE as well as the system of linear and nonlinear parabolic equations, which are dimensionless forms of governing equations of a flow phenomenon. The findings provided in this study can serve as a helpful guide for future investigations into fluid flow in closed-off industrial settings.

Design of Sturm global attractors 1: Meanders with three noses, and reversibility.

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0<x<1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1⇝v2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d-1)-sphere boundary of the global attractor.

Exponentially decaying solutions for models with delayed and advanced arguments: Nonlinear effects in linear differential equations

We consider a scalar linear mixed differential equation with several terms, both delayed and advanced arguments and a bounded right-hand side. Assuming that the deviations of the argument are bounded, we present sufficient conditions when there exists a unique bounded solution on the positive half-line. Explicit tests for a bounded solution of a homogeneous equation to decay exponentially are obtained. Existence of exponentially decaying solutions for this class of differential equations is studied for the first time, and we illustrate sharpness of the results with examples. We show that the standard approach when convergence of all solutions is stated does not work for mixed equations; in addition to an exponentially decaying, there may be a growing solution. All the coefficients and the mixed arguments are assumed to be Lebesgue measurable functions, not necessarily continuous. Though the equation is linear, some properties, as well as the methods applied, are more typical for nonlinear models, for example, fixed-point theorems are used in the proofs.

Mean square asymptotic stability characterisation of perturbed linear stochastic functional differential equations

In this paper we investigate the mean square asymptotic stability of a perturbed scalar linear stochastic functional differential equation. Specifically, we are able to give necessary and sufficient conditions on the forcing terms for convergence of the mean square, exponential convergence of the mean square, and integrability of the mean square of solutions. It is also essential that the underlying unperturbed SFDE is mean–square asymptotically stable for these results to hold.

Stability and bifurcations in scalar differential equations with a general distributed delay

For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model.

Bounded/periodic solutions for a class of scalar delay differential equations

In this paper we use upper and lower solutions, the monotone iterative method, and the Schauder fixed point theorem to establish a new result on the existence of bounded/periodic solutions over R \mathbb R for a class of scalar delay differential equations of the form d u d t = f ( t , u ( t ) , u ( t − r 1 ) , ⋯ , u ( t − r m ) ) \begin{equation*} \frac {du}{dt}=f(t,u(t), u(t-r_1), \cdots , u(t-r_m)) \end{equation*} with constant delays, where f f satisfies a one-sided Lipschitz condition on the state variables u ( t ) u(t) , u ( t − r 1 ) u(t-r_1) , …, u ( t − r m ) u(t-r_m) . Applications of this result to some population models are given.

Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results.

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations