Using the factorizations of suitable operators, we establish several identities that give simple and direct understandings as well provide remainders “virtual” optimizers Hardy Hardy–Rellich type inequalities.

In this article, we study an inverse problem with inhomogeneous source to determine initial data from the time fractional diffusion equation. general, is ill‐posed in sense of Hadamard, so quasi‐boundary value method proposed solve problem. theoretical results, propose a priori and posteriori parameter choice rules analyze them. Finally, two numerical results one‐dimensional two‐dimensional case show evidence used regularization method.

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, establish two local existence theorems by using Faedo–Galerkin method and standard arguments density. Next, give sufficient condition to guarantee the global exponential decay weak solutions. Finally, present numerical results.

In this work, we study the final value problem for a system of parabolic diffusion equations. which, functions are derived from random model. This is severely ill‐posed in sense Hadamard. By nonparametric estimation and truncation methods, offer new regularized solution. We also investigate an estimate error convergence rate between mild solution its solutions. Finally, some numerical experiments constructed to confirm efficiency proposed method.

Abstract In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give form of the mild solution under expression Fourier series which contains some Mittag-Leffler functions. present two new results. Firstly, show well-posedness and regularity for our problem. Secondly, ill-posedness problem in sense Hadamard. Using truncation method, construct regularized convergence rate between exact solutions.

We set up several identities that imply some versions of the Hardy type inequalities.These equalities give a straightforward understanding inequalities as well nonexistence nontrivial optimizers.These also provide "virtual" extremizers for many inequalities.

Motivated by the recent known results as regards existence and exponential decay of solutions for wave equations, this paper is devoted to study an N-dimensional nonlinear equation with a nonlocal boundary condition. We first state two local theorems. Next, we give sufficient condition guarantee global weak solutions. The main tools are Faedo-Galerkin method Lyapunov method.

Abstract In this paper, we study a sideways heat equation with nonlinear source in bounded domain, which the Cauchy data at $x = \mathcal {X}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>X</mml:mi></mml:math> are given and solution $0 \le x &lt; xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>x</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>X</mml:mi></mml:math> is sought. The problem severely ill-posed...

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary 2 N ‐point type, which connect values an unknown function u ( x , t ) at = 1, 0, η i θ ), where for all ≥ 0. First, we prove local existence unique weak solution by using density arguments applying Banach's contraction principle. Next, under suitable conditions, show that problem considered has global energy decaying exponentially as → + ∞ . Finally, present numerical results.

Abstract In this paper, we consider the existence of a solution u ( x , t ) for inverse backward problem nonlinear strongly damped wave equation with statistics discrete data. The is severely ill-posed in sense Hadamard, i.e., does not depend continuously on order to regularize unstable solution, use trigonometric method non-parametric regression associated truncated expansion method. We investigate convergence rate under some priori assumptions an exact both L 2 and H q &gt; 0) norms....

This paper is devoted to the study of a system nonlinear viscoelastic wave equations with boundary conditions. Based on Faedo-Galerkin method and standard arguments density corresponding regularity initial conditions, we first establish two local existence theorems weak solutions. By construction suitable Lyapunov functional, next prove blow up result decay global

This note presents corrections to our paper.