Narrow Results

In this paper, we work on the fundamental collocation strategy using the moved Vieta–Lucas polynomials type (SVLPT). A numeral method is used for unwinding the nonlinear Rubella illness Tributes. The quality of the SVLPT is presented. The limited contrast system is used to understand the game plan of conditions. The mathematical model is given to attest the resolute quality and ampleness of the recommended procedure. The oddity and meaning of the outcomes are cleared utilizing a 3D plot. We examine free sickness harmony, security balance point and the presence of a consistently steady arrangement.

In this brief review we summarize a number of recent developments in the study of vortices in Bose–Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.

We study a nematic crystal model that appeared in [Liu et al., 2007], modeling stretching effects depending on the different shapes of the microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is two-fold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model.

In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines $\left|x\right|=1$ (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

In this paper, we consider a simple equation which involves a parameter $k$, and its traveling wave system has a singular line.

Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties:

• (i)In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit.
• (ii)When $k<\frac{1}{2}$, in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave.
• (iii)When $k\ge \frac{1}{2}$, in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation.

Secondly, we perform numerical simulations to visualize the above properties.

Finally, when $k<\frac{1}{8}$ and the constant wave speed equals $\frac{1}{2}(1\pm \sqrt{1-8k})$, we give exact expressions to the above phenomena.

For linear reaction–diffusion equations, a general geometric singular perturbation framework was developed, to study the impact of strong, spatially localized, smooth nonlinear impurities on the existence, stability, and bifurcation of localized structure, in the paper [Doelman et al., 2018]. The multiscale nature enables deriving algebraic conditions determining the existence of pinned single- and multi-pulses. Moreover, linearity enables treating the spectral stability issue for pinned pulses similarly to the problem of existence. In this paper, we move one step further to treat a special type of nonlinear reaction–diffusion equation with the same type of impurity. The additional nonlinear term generates richer and more complex dynamics. We derive algebraic conditions for determining the existence and stability of pinned pulses in terms of Legendre functions.

In this paper, we investigate the existence and uniqueness of solutions for Riemann–Liouville fractional integro-differential equations equipped with fractional nonlocal multi-point and strip boundary conditions in the weighted space. The methods of our study include the well-known tools of the fixed point theory, which are commonly applied to establish the existence theory for the initial and boundary value problems after converting them into the fixed point problems. We also discuss the case when the nonlinearity depends on the Riemann–Liouville fractional integrals of the unknown function. Numerical examples illustrating the main results are presented.

In this paper, we introduce and investigate a new class of coupled fractional $q$-integro-difference equations involving Riemann–Liouville fractional $q$-derivatives and $q$-integrals of different orders, equipped with $q$-integral-coupled boundary conditions. The given problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed-points coincide with solutions of the problem at hand. The existence and uniqueness results for the given problem are, respectively, derived by applying Leray–Schauder nonlinear alternative and Banach contraction mapping principle. Illustrative examples for the obtained results are constructed. This paper concludes with some interesting observations and special cases dealing with uncoupled boundary conditions, and non-integral and integral types nonlinearities.

The primary goal of this paper is to study a nonlinear fuzzy fractional dynamic system (FFDS) involving a time-dependent variational inequality. We use the monotone argument and Knaster–Kuratowski–Mazurkiewicz (KKM) theorem to prove that the variational system of FFDS is solvable and its solutions become a bounded, closed and convex set. Employing this result together with Bohnenblust–Karlin fixed point theorem and Filippov implicit function, we show the existence of a mild solution to FFDS.

In this paper, we study the question of the existence and nonexistence of solutions for some fractional equations with variable exponents. This paper generalizes some analog results in the classical fractional one. As we know, there are no previous results on the nonexistence of solutions for nonlinear equations with fractional $p(\cdot ,\cdot )$-Laplacian.

Public health awareness programs have been a crucial strategy in mitigating the spread of emerging and re-emerging infectious disease outbreaks of public health significance such as COVID-19. This study adopts an Susceptible–Exposed–Infected–Recovered (SEIR) based model to assess the impact of public health awareness programs in mitigating the extent of the COVID-19 pandemic. The proposed model, which incorporates public health awareness programs, uses ABC fractional operator approach to study and analyze the transmission dynamics of SARS-CoV-2. It is possible to completely understand the dynamics of the model’s features because of the memory effect and hereditary qualities that exist in the fractional version. The fixed point theorem has been used to prove the presence of a unique solution, as well as the stability analysis of the model. The nonlinear least-squares method is used to estimate the parameters of the model based on the daily cumulative cases of the COVID-19 pandemic in Nigeria from March 29 to June 12, 2020. Through the use of simulations, the model’s best-suited parameters and the optimal ABC fractional-order parameter $\tau$ may be determined and optimized. The suggested model is proved to understand the virus’s dynamical behavior better than the integer-order version. In addition, numerous numerical simulations are run using an efficient numerical approach to provide further insight into the model’s features.

In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative

In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP)

In this paper, we analyzed the chaotic complexity of a financial mathematical model in terms of a new generalized Caputo fractional derivative. There are three components in this nonlinear financial model: price indexes, interest rates, and investment demand. Our analysis is based on applying the fixed point hypothesis to determine the existence and uniqueness of the solutions. The bifurcation of the proposed financial system has been analyzed at various parameters of the system. Dynamical phase portraits of the proposed financial model are demonstrated at various fractional-order values. We investigated the possibility of finding new complex dynamical behavior with generalized Caputo fractional derivative. This economic model is solved numerically using a predictor–corrector (PC) algorithm with a generalized Caputo derivative. This algorithm can be viewed as a non-integer extension of the classical Adams–Bashforth–Moulton (ABM) algorithm. Additionally, this numerical algorithm has been studied for stability. A number of diverse dynamic behaviors have been observed in numerical simulations of the system, including chaos. Over a broad range of system parameters, bifurcation diagrams indicate that the system behaves chaotically.

In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.

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 $\beta \in (1,2)$, and it is formulated as

Whether quanta exist or not is a complex question that engages with a variety of issues in the realism/anti-realism debate and concerning the interpretation of quantum mechanics, ranging from the kinds of commitments and requirements needed to determine the existence of quantum particles to the sorts of empirical and instrumental control of the relevant phenomena. In this paper, I argue that one need not settle the issue of the existence of quanta to determine their objectivity, that is, to settle whether they are what they are independently of what one takes them to be. The objectivity of quanta is a separate issue from those concerning their existence and metaphysical specification of the kind of thing quanta are. Along the way, I discuss the similarities and differences between mathematical objectivity and the objectivity of quanta, and consider the role that the framework in which quanta are formulated plays in addressing the issue of their objectivity. In the end, the objectivity of quanta and their existence are importantly different. One can hold on to the former while being agnostic about the latter.

The system of reaction-diffusion equations with zero-flux and periodic boundary conditions, which has stable steady-states, will induce the cell reproduction pattern by basic analysis of Turing Principle. For models with constant coeffcients, their cell division patterns do not relate to the time factor. Time-related process of cell division is called aberrance of cell division, whose patterns have significant meaning to cancer pathology, especially, for the mechanism of split of cancer cell. Unlike normal cells, cancer cells do not carry on maturing once they have been made. In fact, the cells in a cancer can become even less mature over time. With all the reproducing, it is not surprising that more of the genetic information in the cell can become lost. So the cells become more and more primitive and tend to reproduce more quickly and even more haphazardly. As a result, to explain cancer cells reproduction, one has to study time-related patterns. Therefore, we extend Predator-Prey model with constant coefficients to the model with coefficients to be positive T-periodic functions (we call such a model EP-P model). The goal of this research is to study the pattern formations of EP-P model and simulate the process of tumor forming.

In this paper, by employing the powerful and effective coincidence degree method, we show the existence of T-periodic solutions of ESP-P model in , where is a strictly positively invariant region. Furthermore, Floquet theory is provided to show that the T-periodic solution x₀(t) of ESP-P model is unique in and locally uniformly asymptotically stable. This establishes a solid foundation for studying the patterns of EP-P model.

This note discusses the recent development on the existence and stability of two-dimensional and three-dimensional gravity-capillary waves on water of finite-depth. The Korteweg-de Vries (KdV) equation is derived formally from the exact governing equations and the solitary-wave solutions are obtained. The recent results on the existence of solutions of the exact equations near the solitary-wave solutions of the KdV equation are presented and various two-and three-dimensional wave solutions are given. Then, the stability of these solutions using exact equations is discussed. The ideas and methods to obtain these results are briefly mentioned.

This contribution is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers [1–5] in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas and are presented from a new viewpoint.