In this paper, being investigated an initial - boundary value problem for a one - dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow - up of weak solutions in a finite time interval.

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.

In this paper, we establish the existence and uniqueness of a time periodic solution to the full compressible quantum Euler equations. First, we prove the existence of time periodic solutions under some smallness assumptions imposed on the external force in a periodic domain by a regularized approximation scheme and the Leray-Schauder degree theory. Then the result is generalized to $\mathbb{R}^{3}$ by adapting a limiting method and a diagonal argument. The uniqueness of the time periodic solutions is also given. Compared to classical Euler equations, the third-order quantum spatial derivatives are considered which need some elaborated treatments thereof in obtaining the highest-order energy estimates.

For the following wave equations with structural damping and nonlinear memory source terms \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t =\int_{0}^{t}(t-s)^{\gamma-1} \vert u (s)\vert^{p}\,\text{d}s, \] and \[ u_{tt}+(-\Delta)^{\frac{\alpha}{2}}u +(-\Delta)^{\frac{\beta}{2}}u_t = \int_{0}^{t}(t-s)^{\gamma-1} \vert u_s (s)\vert^{p}\,\text{d}s, \] posed in $(x,t) \in \mathbb{R}^N \times [0,\infty) $, where $u=u(x,t)$ is real-value unknown function, $p>1$, $\alpha,\beta\in (0, 2]$, $\gamma\in (0,1)$, we prove the nonexistence of global solutions. Moreover, we give an upper bound estimate of the life span of solutions.

We shall investigate the asymptotic behavior of solutions to the Cauchy problem for the one-dimensional bipolar quantum Euler-Poisson system with time-dependent damping effects $\frac{J_i}{(1+t)^{\lambda}}(i=1,2)$ for $-1<\lambda<1$. Applying the technical time-weighted energy method, we prove that the classical solutions to the Cauchy problem exist uniquely and globally, and time-algebraically converge to the nonlinear diffusion waves. This study generalizes the results in [Y.-P. Li, Nonlinear Anal., 74(2011), 1501-1512] which considered the bipolar quantum Euler-Poisson system with constant coefficient damping.

We consider nonlocal problems in which the leading operator contains a sign-changing weight which can be unbounded. We begin studying the existence and the properties of the first eigenvalue. Then we study a nonlinear problem in which the nonlinearity does not satisfy the usual Ambrosetti-Rabinowitz condition. Finally, we study a problem with general concave-convex nonlinearities.

In this paper, an output-based event-triggered control problem of discrete-time networked control systems (NCSs) subject to bilateral data packet dropouts is investigated. In view of the stochastic sequences of packet dropouts in measurement channels (from sensors to controller) and control channels (from controller to actuators), the NCS is converted into a closed-loop stochastic parameter system. In the aid of a Lyapunov functional based on stochastic variables, sufficient conditions on co-design of event-triggering strategy and exponentially mean-square stability of NCSs are derived. Furthermore, an improved iterative algorithm is given to obtain the dynamic output feedback control law and event-triggering parameters from the nonconvex inequalities. Finally, a numerical example and the corresponding simulation results are given to show the validity and applicability of the developed techniques.

In this paper, we study different ways for introducing memory to a parametric family of optimal two-step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave.

{In the present paper, a $\mu$-delayed Mittag-Leffler type function is introduced as a fundamental function. By means of $\mu$-delayed Mittag-Leffler type function, an exact analytical solution formula to non-homogeneous linear delayed Langevin equations involving two distinct $\mu$-Caputo type fractional derivatives of general orders is given. Also, a global solution of nonlinear version of delayed Langevin equations is inferred from the findings on hand and is verified with the aid of the functional(substitutional) operator. In terms of exponential function, we estimate $\mu$-delayed Mittag-Leffler type function. Existence uniqueness of solutions to nonlinear delayed Langevin fractional differential equations are obtained with regard to the weighted norm defined in accordance with exponential function. The notion of stability analysis in the sense of solutions to the described Langevin equations is discussed on the grounds of the fixed point approach. Numerical and simulated examples are shared to exemplify the theoretical findings. This paper provides novel results.

In this paper, we consider a one-dimensional porous thermoelastic system with microtemperatures effects and past history term acting only on the porous equation. We show that the system is well-posed in the sens of semigroup and based on the energy method we establish a general decay result for the solutions of the system under an appropriate assumptions on the kernel. Furthermore, our result depends on the stability number that was first considered in [25]. The result of this paper improves some results in the literature.

We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non-degenerate simplex contracting to that point. Such result allows to fix a issue with a defective definition of pseudo-scalar field in Sobczyck’s Simplicial Calculus. Then, we provide some consequences and conjectures implied by the foregoing results.

From decomposition method for operators, we consider Newton-like iterative processes for approximating solutions of nonlinear operators in Banach spaces. These iterative processes maintain the quadratic convergence of Newton's method. Since the operator decomposition method has its highest degree of application in non-differentiable situations, we construct Newton-type methods using symmetric divided differences, which allow us to improve the accessibility of the methods. Experimentally, by studying the basins of attraction of these methods, observe an improvement in the accessibility of derivative-free iterative processes that are normally used in these non-differentiable situations, such as the classic Steffensen's method. In addition, we study both the local and semi-local convergence of the Newton-type methods considered.

In this paper, we study Aronson-B\’{e}nilan gradient estimates for positive solutions of weighted porous medium equations $$\partial_{t}u(x,t)=\Delta_{\phi}u^{p}(x,t),\,\,\,\,(x,t)\in M\times[0,T]$$ coupled with the geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$ on a complete measure space $(M^{n},g,e^{-\phi}dv)$. As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock-out call options is presented. With regard to the well-known Black-Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, that makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm.

In this note, we show that the Hausdorff operator $H_{\Phi}$ is unbounded on a large family of Quasi-Banach spaces, unless $H_{\Phi}$ is a zero operator.

In this paper, an identical approximate regularization method is extended to the Cauchy problem of two-dimensional heat conduction equation, this kind of problem is severely ill-posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries, and compared the identical approximate regularization solutions which are displayed in paper. The numerical results show that our method is effective, accurate and stable to solve the ill-posed Cauchy problems.

In this paper, we show that any displacement of a plate is the sum of a Kirchhoff-Love displacement and two terms, one for shearing and one for warping. Then, the plate is loaded in order to obtain that the bending and shearing contribute the same order of magnitude to the fiber rotations.

In this work, we investigated the invariance analysis of fractional-order Hirota-Satsoma coupled Korteveg-de-Vries (HSC-KdV) system of equations based on Riemann-Liouville (RL) derivatives. The Lie Symmetry analysis is considered to obtain infinitesimal generators; we reduced the system of coupled equations into nonlinear fractional ordinary differential equations (FODEs) with the help of Erdelyi’s-Kober (EK) fractional differential and integral operators. The reduced system of FODEs solved by means of the power series technique with its convergence. The conservation laws of the system constructed by Noether’s theorem.