Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Steklov problem of the first class for a fractional order delay differential equation

Методом функции Грина получено решение задачи Стеклова первого класса для линейного уравнения с дробной производной Герасимова-Капуто с запаздывающим аргументом. Доказана теорема существования и единственности задачи. The solution to the Steklov problem with conditions of the first class for a linear delay differential equation with a Gerasimov-Caputo fractional derivative is obtained by Green function method. The existence and uniqueness theorem to the problem is proved.

Ulam Stability of n-th Order Delay Integro-Differential Equations

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

Gravitational Radius in view of Existence and Uniqueness Theorem

Talking about a black hole, one has in mind the process of unlimited self-compression of gravitating matter with a mass greater than critical. With a mass greater than the critical one, the elasticity of neutron matter cannot withstand gravitational compression. However, compression cannot be unlimited, because with increasing pressure, neutrons turn into some other “more elementary” particles. These can be bosons of the Standard Model of elementary particles. The wave function of the condensate of neutral bosons at zero temperature is a scalar field. If instead of the constraint det gik < 0 we use a weaker condition of regularity (all invariants of the metric tensor gik are finite), then there is a regular static spherically symmetric solution to Klein-Gordon and Einstein equations, claiming to describe the state to which the gravitational collapse leads. With no restriction on total mass. In this solution, the metric component grr changes its sign twice: g rr (r) = 0 at r=rg and r=rh > rg . Between these two gravitational radii the signature of the metric tensor gik is (+, +, -, -). Gravitational radius rg inside the gravitating body ensures regularity in the center. Within the framework of the phenomenological model “λψ4 ”, relying on the existence and uniqueness theorem, the main properties of a collapsed black hole are determined. At r = rg a regular solution to Klein-Gordon and Einstein equations exists, but it is not a unique one. Gravitational radius rg is the branch point at which, among all possible continuous solutions, we have to choose a proper one, corresponding to the problem under consideration. We are interested in solutions that correspond to a finite mass of a black hole. It turns out that the density value of bosons is constant at r < rg. It depends only on the elasticity of a condensate, and does not depend on the total mass. The energy-momentum tensor at r ⩽ rg corresponds to the ultra relativistic equation of state p = ɛ/3. In addition to the discrete spectrum of static solutions with a mass less than the critical one (where grr < 0 does not change sign), there is a continuous spectrum of equilibrium states with grr(r) changing sign twice, and with no restriction on mass. Among the states of continuous spectrum, the maximum possible density of bosons depends on the mass of the condensate and on the rest mass of bosons. The rest energy of massive Standard Model bosons is about 100 GeV. In this case, for the black hole in the center of our Milky Way galaxy, the maximum possible density of particles should not exceed 3 × 1081 cm-3.

INVERSE PROBLEM FOR THE TIME-FRACTIONAL EULER-BERNOULLI BEAM EQUATION

In this paper, the classical Euler-Bernoulli beam equation is considered by utilizing fractional calculus. Such an equation is called the time-fractional EulerBernoulli beam equation. The problem of determining the time-dependent coefficient for the fractional Euler-Bernoulli beam equation with homogeneous boundary conditions and an additional measurement is considered, and the existence and uniqueness theorem of the solution is proved by means of the contraction principle on a sufficiently small time interval. Numerical experiments are also provided to verify the theoretical findings.

Complex uncertain differential equations with application to time integral

In this paper, we propose complex uncertain differential equations (CUDEs) based on uncertainty theory. In order to describe the evolution of complex uncertain phenomenon related to belief degrees, we apply the complex Liu process to CUDEs. Firstly, we pose a concept of a linear CUDE and prove that homogeneous linear CUDE and general linear CUDE have solutions. Then, we prove existence and uniqueness theorem of a special CUDE. Further, we design a numerical algorithm to obtain inverse uncertainty distribution of the solution. Finally, as an application, we analyse the inverse uncertainty distributions of time integral of CUDEs and design numerical algorithms to obtain inverse uncertainty distributions of time integral.

On weak solutions of the boundary value problem within linear dilatational strain gradient elasticity for polyhedral Lipschitz domains

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1–16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

An uncertain SEIR rumor model

In this paper, an uncertain SEIR rumor model driven by one uncertain process is formulated to investigate the influence of perturbation in the transmission of rumor. Firstly, the deduced process of the uncertain SEIR rumor model is presented. Then, we proposed the existence and uniqueness theorem for the solution of the model. Moreover, the study of the stability of the uncertain SEIR rumor model was carried out, and then we came to the conclusion that the model stable in mean. In addition, computer algorithm and numerical simulation is used to verify the accuracy of the theoretical results. The simulation results show that the proposed model can explain the trend of rumor propagation correctly and describe the rumor propagation accurately. Finally, we have compared the propagation process of the uncertain rumor model and the deterministic model according to the numerical algorithm, and drew the conclusion that the model with uncertain perturbation fluctuates around the deterministic model.

An uncertain SIR rumor spreading model

In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model.