scholarly journals A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

2016 ◽  
Vol 434 (2) ◽  
pp. 1376-1393 ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Le Duc Thang ◽  
Daniel Lesnic
Keyword(s):  
Elliptic Equations ◽  
Regularization Method ◽  
Cauchy Problems ◽  
Nonlinear Source ◽  
Locally Lipschitz

Stepwise regularization method for a nonlinear Riesz–Feller space-fractional backward diffusion problem

2019 ◽  
Vol 27 (6) ◽  
pp. 759-775
Author(s):  
Dang Duc Trong ◽  
Dinh Nguyen Duy Hai ◽  
Nguyen Dang Minh
Keyword(s):  
Exact Solution ◽  
Initial Data ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Convergence Estimate ◽  
Nonlinear Source ◽  
Space Fractional Diffusion Equation ◽  
Final Data

Abstract In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals Solutions for nonlinear variational inequalities with a nonsmooth potential

2004 ◽  
Vol 2004 (8) ◽  
pp. 635-649 ◽  
Author(s):  
Michael E. Filippakis ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Elliptic Equations ◽  
Lower Semicontinuous ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz ◽  
Proper Convex ◽  
Nonsmooth Potential

First we examine a resonant variational inequality driven by thep-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving thep-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the formφ=φ1+φ2withφ1locally Lipschitz andφ2proper, convex, lower semicontinuous.

Quasilinear Equations via Elliptic Regularization Method

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Jia-Quan Liu ◽  
Xiang-Qing Liu ◽  
Zhi-Qiang Wang
Keyword(s):  
Elliptic Equations ◽  
Regularization Method ◽  
Quasilinear Elliptic Equations ◽  
Quasilinear Equations ◽  
Elliptic Regularization ◽  
Nodal Property ◽  
Quasilinear Problems ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic

AbstractIn this paper we study a class of quasilinear problems, in particular we deal with multiple sign-changing solutions of quasilinear elliptic equations. We further develop an approach used in our earlier work by exploring elliptic regularization. The method works well in studying multiplicity and nodal property of solutions.

General uniqueness results and blow-up rates for large solutions of elliptic equations

2012 ◽  
Vol 142 (4) ◽  
pp. 825-837 ◽  
Author(s):  
Shuibo Huang ◽  
Wan-Tong Li ◽  
Qiaoyu Tian ◽  
Yongsheng Mi
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Variation Theory ◽  
Singular Boundary ◽  
Smooth Bounded Domain ◽  
Lipschitz Continuous ◽  
Large Solutions ◽  
Uniqueness Results ◽  
Locally Lipschitz ◽  
Image Position

Making use of the Karamata regular variation theory and the López-Gómez localization method, we establish the uniqueness and asymptotic behaviour near the boundary ∂Ω for the large solutions of the singular boundary-value problemwhere Ω is a smooth bounded domain in ℝN. The weight function b(x) is a non-negative continuous function in the domain, which can vanish on the boundary ∂Ω at different rates according to the point x0 ∊ ∂Ω. f(u) is locally Lipschitz continuous such that f(u)/u is increasing on (0, ∞) and f(u)/up = H(u) for sufficiently large u and p > 1, here H(u) is slowly varying at infinity. Our main result provides a sharp extension of a recent result of Xie with f satisfying limu→f(u)/up = H for some positive constants H > 0 and p > 1.

Iterative regularization method for an abstract ill-posed generalized elliptic equation

2020 ◽  
pp. 2150069
Author(s):  
Sassane Roumaissa ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia ◽  
Benrabah Abderafik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Boundary Data ◽  
Identification Problem ◽  
Infinite Domain ◽  
Iterative Regularization ◽  
Convergence Results ◽  
Ill Posed ◽  
Well Posed

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

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