It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer-order nonlinear PDEs. In this paper, based on the separation method of semi-fixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time-fractional thin-film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time-fractional thin-film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann-Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D-graphs.

In this article, the inverse problems for the wave equation in a medium in which multiple types of cavities and inclusion exist in a mixture are considered. From the point of view of the indicator function of the enclosure method, there are two types of heterogeneous parts:“minus group” and “plus group”. For example, cavities with the Dirichlet boundary condition belong to the minus group, while inclusions with smaller propagation velocity belong to the plus group. The heterogeneous part of the minus group gives a negative sign to the indicator function, and the heterogeneous part of the plus group gives a positive sign. In general, the presence of many types of heterogeneous parts causes cancellation of the sign of the indicator function. Such cases are referred to as “mixed cases”. Here we consider the case that the shortest length obtained from the indicator function is attained only by heterogeneous parts of the same group. This case is called the “mixed but separated case” and it is shown that the method of elliptic estimates developed by Ikehata works well. We also show that the case of a two-layered background medium with a flat layer can be considered in the same way as the case of a homogeneous background medium.

In this paper, we consider the Ricker model with delay and constant or periodic stocking. The impact of delay and stocking on stability is known to reflect opposing effect, which motivates investigating the interaction between these factors and their influence on overall stability. We found that the high stocking density tends to neutralize the delay effect. Conditions are established on the parameters in order to guarantee the global stability of the equilibrium solution in the case of constant stocking, as well as the global stability of the 2-periodic solution in the case of 2-periodic stocking. Our approach extensively relies on the utilization of the embedding technique. Whether constant stocking or periodic stocking, the mode has the potential to undergo a Neimark-Sacker bifurcation in both cases. However, the Neimark-Sacker bifurcation in the 2-periodic case results in the emergence of two invariant curves that collectively function as a single attractor. Finally, we pose open questions in the form of conjectures about global stability for certain choices of the parameters.

This paper investigates an optimal investment-reinsurance problem for an insurer who possesses extra information regarding the future realizations of the claim process and risky asset process. The insurer sells insurance contracts, has access to proportional reinsurance business, and invests in a financial market consisting of three assets: one risk-free asset, one bond and one stock. Here, the nominal interest rate is characterized by the Vasicek model; and the stock price is driven by the Heston’s stochastic volatility model. Applying the enlargement of filtration techniques, we establish the optimal control problem in which an insurer maximizes the expected power utility of the terminal wealth. By using the dynamic programming principle, the problem can be changed to four-dimensional Hamilton-Jacobi-Bellman equation. In addition, we adopt a deep neural network method by which the partial differential equation is converted to two backward stochastic differential equations and solved by a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python and the economic behavior of the approximate optimal strategy and the approximate optimal utility of the insurer are analyzed.

In this paper, we introduce some explicit formulas for the positive integer order q-derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between the commutative algebras and the properties of q-operators. Especially, the third formula is a beautiful one involving the determinant of the functions and their q-derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, the last formula is used for proving of some interesting identities. MSC CLASSIFICATION: 05A30, 26A24

The well-posedness and regularity properties of diffusion-aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence argument to treat the non-linearity as well as a Moser iteration. Moreover, we prove a quantitative estimate in probability with arbitrary algebraic rate between the approximative interacting particle systems and the approximative McKean--Vlasov SDEs, which implies propagation of chaos for the interacting particle systems.

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol-Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler’s identity, we give relations among theis characteristic function, the Apostol-Bernoulli polynomials and numbers, and also trigonometric functions including sin w and cos w . A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol-Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas.

We are considered with the following nonlinear Schrödinger equation −∆ u+( λa( x)+1) u= f( u) ,x∈ V, on a locally finite graph G=( V,E), where V denotes the vertex set, E denotes the edge set, λ>1 is a parameter, f( s) is asymptotically linear with respect to s at infinity, and the potential a: V→[0 ,+∞) has a nonempty well Ω. By using variational methods we prove that the above problem has a ground state solution u λ for any λ>1. Moreover, we show that as λ→∞, the ground state solution u λ converges to a ground state solution of a Dirichlet problem defined on the potential well Ω.

A general approach is developed for discriminating strong and hereditary symmetric operators. The recursion operator of the Blaszak-Marciniak (BM) equation hierarchy is proved to be strong and hereditary symmetric. As an example of discrete soliton equations related to 3×3 matrix spectral problems, the τ-symmetries and Lie algebra structure of the BM equation are built firstly.

This paper deals with the existence of weak solutions of the system that describes the two-phase two-component fluid flow in porous media. Both two-phase and possible one-phase flow regions are taken into account. Our research is based on a global pressure, introduced in3, an artificial variable that allows us to decouple the original equations. We introduce another variable that does not have physical meaning in one-phase region and name it capillary pseudo-pressure. By using these two variables we rewrite system's equations in a fully equivalent form. This allows us to derive an existence result for weak solutions in more tractable way compared to15.

In this paper, we investigate the Lipschitz stability of a perturbed impulsive differential system concerning the unperturbed system. We employ the variation of parameters or the constant of variation for impulsive differential systems with an initial time difference.

Pulse injection of insulin analogues is an important strategy to control blood glucose concentrations and can be combined with α– glucosidase inhibitors to reduce adverse effects to improve glucose control. We propose a novel mathematical models with pulse injection insulin and α– glucosidase inhibitors , eating in the form of pulse blood glucose injection. The existence and uniqueness of the positive periodic solution is confirmed In type 1 diabetes, which is globally asymptotically stable. Further, the permanence of the system is given in type 2 diabetes. The numerical analysis verified the correctness of the theoretical calculation results, and show that the period and the dose of insulin injections and α– glucosidase inhibitors are crucial for insulin therapies. In addition, we systematically evaluated a reasonable strategy to treat diabetes combined with α– glucosidase inhibitors, which can provide more reasonable clinical strategies.

A document by Aykut Toplama. Click on the document to view its contents.

In this article, we generalized Mittag-Leffler-type functions F ̵̄ A ( 3 ) , F ̵̄ B ( 3 ) , F ̵̄ C ( 3 ) and F ̵̄ D ( 3 ) , which correspond, respectively, to the familiar Lauricella hypergeometric functions F A ( 3 ) , F B ( 3 ) , F C ( 3 ) and F D ( 3 ) of three variables. Among the various properties and characteristics of these three-variable Mittag-Leffler-type function F ̵̄ D ( 3 ) , which we investigate in this article, include their relationships with other extensions and generalizations of the classical Mittag-Leffler functions, their three-dimensional convergence regions, their Euler-type integral representations, their Laplace transforms, their connections with the Riemann-Liouville operators of fractional calculus, and the systems of partial differential equations which are associated with them.

This paper investigates a stochastic two predators-one prey system with ratio-dependent functional response under regime switching. The stochastic extinction of species and the existence of ergodic stationary distribution for the system are established, and the transition probability of the solution converging to the stationary distribution also is obtained. To illustrate our theoretical results, the numerical examples and simulations are presented. Our findings also demonstrate that the stationary distribution and extinction of species for the stochastic two predators-one prey system are affected by random perturbations, leading to an imbalance in ecology.

This paper focuses on the existence of normalized solutions for the Chern-Simons-Schrödinger system with mixed dispersion and critical exponential growth. These solutions correspond to critical points of the underlying energy functional under the L 2 -norm constraint, namely, ∫ R 2 u 2 d x = c > 0 . Under certain mild assumptions, we establish the existence of nontrivial solutions by developing new mathematical strategies and analytical techniques for the given system. These results extend and improve the results in the existing literature.

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second-order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti-damping. The hybrid system is first reduced to a first order port-Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central-difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter-Kato Theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

Two types of short-time Fourier transforms involving respectively nonlinear modulation and nonlinear translation are designed in this note. Some important properties such as twin in time-frequency domain and orthogonality relations are investigated. Moreover, Lieb type inequalities and uncertainty principle are established.