Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space

Investigation of solitons in magneto-optic waveguides with Kudryashov’s law nonlinear refractive index for coupled system of generalized nonlinear Schrödinger’s equations using modified extended mapping method

In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements

Construction of the beta distributions using the random permutation divisors

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order

Finite-time projective synchronization of fractional-order delayed quaternion-valued fuzzy memristive neural networks

In this paper, the finite-time projective synchronization (FTPS) problem of fractionalorder quaternion-valued fuzzy memristor neural networks (FOQVFMNNs) is studied. Through establishing a feedback controller with signed functions and an adaptive controller, sufficient conditions for FTPS for FOQVFMNNs are obtained. Furthermore, the synchronization establishment time is calculated. Finally, the practicability of the conclusions is verified by numerical simulations

Global dynamics and optimal control of a nonlinear fractional-order cholera model

In this article, a fractional-order epidemic model for cholera is proposed and analyzed. Two transmission routes for cholera are considered to develop the compartmental epidemic model. The basic biological properties of the solutions of the fractional-order model are investigated. The global asymptotic stability of the equilibrium points have been established using appropriate Lyapunov functional. Moreover, a fractional-order control problem is presented, and its analytical solution is derived using Pontryagin’s maximum principle. Also, some graphical visualizations of the theoretical results are provided. It is found that the factional-order derivative only affect the time to reach the stationary states. Sensitivity analysis reveals that by reducing the rates of new recruitment and both the disease transmission rates, it may be possible to reduce the value of the basic reproduction number

Spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling

In this paper, we investigate the spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling. We first study the stability of positive equilibrium and Turing instability induced by diffusion. We then investigate the effect of delay, and it turns out that the value of the rate of recycling k plays an important role in the Hopf bifurcation induced by delay. The delay will and will not induce Hopf bifurcation with low and high level of k, respectively. To reveal the spatiotemporal dynamics, Turing–Hopf bifurcation is carried out, and normal form is derived. Many spatiotemporal dynamics are found, including the coexistence of two stable spatially inhomogeneous periodic solutions or two stable spatially inhomogeneous steadystate solutions

An immunity-structured SEIRS epidemic model with variable infectivity and spatial heterogeneity

A mathematical model is proposed for the spread of an epidemic disease of agedependent infectivity through an asexual population with spatial heterogeneity, assuming that some individuals recover from the disease with temporary immunity, another part recover with permanent immunity, and the last part recover with no immunity. The demographic changes such as births and deaths due to natural causes and the chronological age of individuals are not taken into account. The model is based on a system of partial integro-differential equations including a differential equation to describe the evolution of individuals who have recovered with temporary immunity. The existence and uniqueness of the globally defined solution is proved, and its long-time behaviour is studied