In this paper, we consider the stabilization question for a nonlinear model of an axially moving string. The model is assumed to undergo the variable tension and variable disturbances. The Hamilton principle is used to describe the dynamic of transverse vibrations. We establish the well-posedness by means of the Faedo-Galerkin method. A boundary control with a time-varying delay is designed to stabilize uniformly the string. Then, we derive a decay rate of the solution assuming that the retarded term be dominated by the damping one. Some examples are given to clarify when the rate is exponential or polynomial.

In this article, we consider a swelling porous-heat system with viscous damping and distributed delay term. It’s well-known in the literature that the thermal effect only lacks exponential stability. For that, we need to add another damping mechanism to stabilize exponentially the system. However, we prove, based on the energy method, that the combination of viscous damping and thermal effect provokes an exponential stability in the presence of distributed delay irrespective of the wave speeds of the system.

We show that Cauchy-Schwarz inequality for shifted quantum integral operator does not hold in general and we prove the conditions under which it is valid. We apply it to Ostrowski type inequalities for shifted quantum integral operator.

The achievement of this paper is to propose a new kind of fractional derivative which is called New Constant Proportional Caputo (NCPC) operator and to construct the solution of time-fractional initial value problem (TFIVPs) with NCPC derivative by taking the combination of Laplace transform (LT) and Homotopy Analysis method (HAM) into account. Later, the obtained solution is compared with the solutions of TFIVPs with Caputo and Constant Proportional Caputo (CPC) derivatives. The gained results reveal that the combination of LT and HAM together form an efficient method to build the approximate results of TFIVPs in NCPC sense.

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

In this paper, as a part of uncertainty theory, we explore the concepts of lacunary I and I∗-convergence almost surely in complex uncertain sequences and study some of their properties and identify the relationships between them. Also, we introduced the notions of lacunary I and lacunary I∗-Cauchy sequence almost surely of complex uncertain sequences and analyze a few of their characteristics and try to determine how they relate to one another.

The aim of this work is to develop a numerical tool for computing the weak periodic solution for a class of parabolic equations with nonlinear boundary conditions. We formulate our problem as a minimization problem by introducing a least-squares cost function. With the help of the Lagrangian method, we calculate the gradient of the cost function. We build an iterative algorithm to simulate numerically the weak periodic solution to the considered problem. To illustrate our approach, we present some numerical examples.

The aim of this paper is to present some applications of the Neumann Laplacian in image processing, along with the necessary mathematical background. We prove weak and strong versions of the maximum principle for weak solutions of elliptic and parabolic problems and apply them to a Fisher K.P.P.-type equation. The original contribution lies in the application of this equation in image processing, where various diffusion-like effects can be achieved. Additionally, a review of the basics of linear and nonlinear PDEs with Neumann boundary conditions is provided, along with updated bibliography and recent qualitative results. There are also some new theoretical results developed in this work.

The present paper is concerned with exploiting an iterative decoupling algorithm to address the problem of third-order tensor deblurring. The regularized deblurring problem, which is mathematically given by the sum of a fidelity term and a regularization term, is decoupled into an observation fidelity and a denoiser model steps. One basic advantage of the iterative decoupling algorithm is that the deblurring problem is supervised by the efficiency of the denoiser model. Thus, we consider a patch-based weighted low-rank tensor with sparsity prior. Numerical tests to image deblurring are given to demonstrate the efficiency of the proposed decoupling based algorithm.

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method.