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1.
Nguyen Huy Tuan Nguyen Hoang Tuan Dumitru Baleanu Tran Ngoc Thach 《Mathematical Methods in the Applied Sciences》2020,43(3):1292-1312
In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L2 and Hτ norms is considered and illustrated by numerical example. 相似文献
2.
L. Von Wolfersdorf 《Mathematical Methods in the Applied Sciences》2001,24(14):1073-1088
The paper deals with the auto‐correlation equation and its regularization by means of a Lavrent'ev regularization procedure in L2. The solution of this quadratic integral equation of the first kind and of the regularized equation of the second kind are obtained by reduction to a boundary value problem for the Fourier transform of the solution. We prove convergence of the approximate solution to the exact solution and derive a stability estimate for the error. Copyright © John Wiley & Sons, Ltd. 相似文献
3.
Nguyen Duc Phuong Nguyen Huy Tuan Dumitru Baleanu Nguyen Hoang Luc 《Mathematical Methods in the Applied Sciences》2019,42(18):6829-6848
The aim of this paper is to study the Cauchy problem of determining a solution of nonlinear elliptic equations with random discrete data. A study showing that this problem is severely ill posed in the sense of Hadamard, ie, the solution does not depend continuously on the initial data. It is therefore necessary to regularize the in‐stable solution of the problem. First, we use the trigonometric of nonparametric regression associated with the truncation method in order to offer the regularized solution. Then, under some presumption on the true solution, we give errors estimates and convergence rate in L2‐norm. A numerical example is also constructed to illustrate the main results. 相似文献
4.
On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem
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Nina Ovcharova 《Mathematical Methods in the Applied Sciences》2017,40(1):60-77
In this paper, we couple regularization techniques of nondifferentiable optimization with the h‐version of the boundary element method (h‐BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an h‐BEM. We prove convergence of the h‐BEM Galerkin solution of the regularized problem in the energy norm, provide an a priori error estimate and give a numerical examples. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
5.
In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution. 相似文献
6.
We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H1‐norm. For the simplified friction problem, we establish an abstract H1‐error estimate, which implies the convergence if the exact solution u∈H1(Ω) and the optimal error estimate if u∈H1 + α(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
7.
The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L 2 norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014 相似文献
8.
A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates
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Nguyen Huy Tuan Dang Duc Trong Dinh Nguyen Duy Hai Duong Dang Xuan Thanh 《Mathematical Methods in the Applied Sciences》2017,40(11):4040-4064
In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
Zhi Qian Chu-Li Fu Zhen-Ping Li 《Journal of Mathematical Analysis and Applications》2008,338(1):479-489
A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates. 相似文献
10.
Nguyen Huy Tuan Le Nhat Huynh Dumitru Baleanu Nguyen Huu Can 《Mathematical Methods in the Applied Sciences》2020,43(6):2858-2882
In this paper, we consider an inverse problem of recovering the initial value for a generalization of time-fractional diffusion equation, where the time derivative is replaced by a regularized hyper-Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules. 相似文献
11.
We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ?n and presenting a linear growth for |x | → ∞. We prove well‐posedness in the Schwartz space ?? (?n ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
We study the unique solvability of the Dirichlet problem for the biharmonic equation in the exterior of a compact set under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a,a we prove uniqueness theorems or present exact formulas for the dimension of the solution space of the Dirichlet problem. 相似文献
13.
We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth‐order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro‐polar elasticity. 相似文献
14.
Kab Seok Kang Do Young Kwak 《Numerical Methods for Partial Differential Equations》2006,22(1):165-179
We investigate an L2‐error estimate of a covolume scheme for the Stokes problem recently introduced by Chou (Math Comp 66 (1997), 85–104). We show the error in L2 norm is of second order provided the exact velocity is in H 3 and the exact pressure is in H2. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
15.
Lixin Yu 《Mathematical Methods in the Applied Sciences》2012,35(13):1505-1517
In this paper, by means of a constructive method based on the theory of the existence and the uniqueness of the C1 solution to the Cauchy problem and the Goursat problem, the global exact boundary observability for the first‐order quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics is obtained. In the case that the system has no zero characteristics, we realize the two‐sided and one‐sided global exact boundary observability by the boundary observed values and obtain the observability inequality. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
16.
We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted ??p‐penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such ??p‐penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc. 相似文献
17.
We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non‐linear Darcy's law and the energy balance equation with second member in L1. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
18.
The initial/Neumann boundary-value enthalpy formulation for the two-phase Stefan problem is regularized by smoothing. Known estimates predict a convergence rate of 1/2, and this result is extended in this paper to include the case of a (nonzero) residual in the regularized problem. A modified Newton Kantorovich framework is established, whereby the exact solution of the regularized problem is replaced by one Newton iteration. It is shown that a consistent theory requires measure-theoretic hypotheses on the starting guess and the Newton iterate, otherwise residual decrease is not expected. The circle closes in one spatial dimension, where it is shown that the residual decrease of Newton's method correlates precisely with the 1/2 convergence theory. 相似文献
19.
We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ? u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L3 ∕ 2‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
20.
We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one. 相似文献