In this article, we give a new notion of (p,q)-derivatives for continuous functions on coordinates. We also derive post quantum Ostrowski–type inequalities for coordinated convex functions. Our significant results are considered as the generalizations of other results that appeared in the literature.

In this paper, we study the simplest quadratic matrix equation: $\mathcal{Q}(X)=X^2+BX+C=0$. We transform this equation into an equivalent fixed-point equation and based on it we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor-corrector iterative scheme with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improves the numerical application of the Newton method when approximating a solution of the quadratic matrix equation.

In this paper, we study the symmetry on a class of multiparameter Markov Processes in Khoshnevisan (2002), and obtain some important and interesting results.

In this paper, we investigate the decay properties of the thermoelastic Bresse system in the whole space. We consider many cases depending on the parameters of the model and we establish new decay rates. We need to mention here that, in some cases we don't have the regularity-loss phenomena as in the previous works in the literature. To prove our results, we use the energy method in the Fourier space to build a very delicate Lyapunov functionals that give the desired results.

In this paper, the link between combinatorial structures and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one constructs and draws the (pseudo)digraph associated with a given alternative algebra and the second one tests if a given combinatorial structure is associated with some alternative algebra.

In this paper, we investigate vector Lakshmanan-Porsezian-Daniel (VLPD) model which can be used to describe the ultrashort pulses in the birefringent optical fiber. Based on the unified transformation method, the Riemann-Hilbert problem is introduced and initial-boundary value problems of the VLPD model are studied. By solving the formulated matrix Riemann-Hilbert problem, the potential function solutions of the VLPD model can be reconstructed. Moreover, that the spectral functions are not independent but meet the so-called global relation is shown.

In this work, we study the Watson-type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the L 2 (R) space and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulated the Plancherel-type theorem for the aforementioned operators and prove a sequence of functions that converge to the original function in the defined L 2 (R) norm. Next, we study the boundedness of the operators (T k ). Besides, showing the obtained results, we demonstrate how to use it to solve the class of integro-differential equations of Barbashin type, the differential equations, and the system of differential equations. And there are numerical examples given to illustrate these.

We study the temporal equation of radiating stars by using three powerful methods for the analysis of nonlinear differential equations. Specifically, we investigate the global dynamics for the given master ordinary differential equation to understand the evolution of solutions for various initial conditions as also to investigate the existence of asymptotic solutions. Moreover, with the application of Lie’s theory, we can reduce the order of the master differential equation, while an exact similarity solution is determined. Finally, the master equation possesses the Painlev\’{e} property, which means that the analytic solution can be expressed in terms of a Laurent expansion.

The solution is obtained and validated by an existence and uniqueness theorem for the following nonlinear boundary value problem \[ \frac{d}{dx}(1+\delta y+\gamma y^{2})^{n}\frac{dy}{dx}]+2x\frac{dy}{dx}=0,\,\,\,x>0,\,\,y(0)=0,\,\,\,y(\infty)=1, \] which was proposed in 1974 by [1] to represent a Stefan problem with a nonlinear temperature-dependent thermal conductivity on the semi-infinite line (0;1). The modified error function of two parameters $\varphi_{\delta,\gamma}$ is introduced to represent the solution of the problem above, and some properties of the function are established. This generalizes the results obtained in [3, 4].

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

This paper investigates the spatial behavior of the solutions of the double-diffusive Darcy plane flow in a semi-infinite channel. Using the energy estimate method and the differential inequality technology, a differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variable. In the case of decay, we obtain the upper bound for the total energy. We also give some remarks to generalize the results of this paper.

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|^{-b}|u|^{β}u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L^{p}, p < 2 under some suitable conditions on b,β and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

In the present paper the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Frechet space are investigated . Theorems are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover the T-property of CCFDB and CCFIB are dis- cussed. The obtained results recover some known results when alpha = 1. Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel and Chebyshev polynomials have been studied.

Sparse (fast) representations of deterministic signals have been well studied. Among other types there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the $L^2$ space the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI); and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.

We study the small initial date Cauchy problem for the generalized incompressible Navier-Stokes-Coriolis equations in critical hybrid-Besov space $\dot{\mathscr{B}}_{2,\, p}^{\frac{5}{2}-2\alpha, \frac{3}{p}-2\alpha+1}(\mathbb{R}^3)$ with $1/2<\alpha<2$ and $2\leq p\leq 4$. We prove that hybrid-Besov spaces norm of a class of highly osillating initial velocity can be arbitrarily small. and we prove the estimation of highly frequency $L^p$ smoothing effect for generalized Stokes-Coriolis semigroup with $1\leq p\leq\infty$, At the same time, we prove space-time norm estimation of hybrid-Besov spaces for Stokes-Coriolis semigroup. From this result we deduce bilinear estimation in our work space. Our method relies upon Bony’s high and low frequency decomposition technology and Banach fixed point theorem.

In this paper, we obtain the stationary elliptic- and hyperbolic-function solutions of the nonlocal reverse-time and reverse-space-time nonlinear Schrödinger (NLS) equations based on their connection with the standard Weierstrass elliptic equation. The reverse-time NLS equation possesses the bounded dn-, cn-, sn-, sech-, and tanh-function solutions. Of special interest, the tanh-function solution can display both the dark- and antidark-soliton profiles. The reverse-space-time NLS equation admits the general Jacobian elliptic-function solutions (which are exponentially growing at one infinity or display the periodical oscillation in x), the bounded dn- and cn-function solutions, as well as the K-shifted dn- and sn-function solutions. At the degeneration, the hyperbolic-function solutions may exhibit an exponential growth behavior at one infinity, or show the gray- and bright-soliton profiles.

A class of time-varying delay distributed parameter systems with input saturation is investigated in this paper. The periodic intermittent control method is adopted to make the system stable in finite time, improve the control performance of the system, and save on control cost. A periodic intermittent controller combined saturated input is designed to ensure the stability of the proposed system in finite time. Lyapunov--Krasoviskii stability theory and Matrix inequality techniques are used to analyze the finite-time stability of the system, and sufficient conditions for the system to be stable in finite time are obtained. Finally, the correctness of the theorems is verified by simulation experiments.