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1.
Nguyen Huy Tuan Vo Au Le Nhat Huynh Yong Zhou 《Mathematical Methods in the Applied Sciences》2020,43(8):5450-5463
In this paper, we consider a backward problem for an inhomogeneous time-fractional wave equation in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The existence and regularity for the backward problem are investigated. The backward problem is ill-posed, and we propose a regularizing scheme by using a modified regularization method. We also prove the convergence rate for the regularized solution by using some a priori regularization parameter choice rule. 相似文献
2.
In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach. 相似文献
3.
In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem. 相似文献
4.
《Mathematical Methods in the Applied Sciences》2018,41(1):336-352
This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long‐term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work. 相似文献
5.
In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative Dαo+is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel’skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results. 相似文献
6.
Yong Zhou Jia Wei He Bashir Ahmad Nguyen Huy Tuan 《Mathematical Methods in the Applied Sciences》2019,42(18):6775-6790
In this paper, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of Mittag‐Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions as well as the classical solutions of the proposed problem in a weighted Hölder continuous function space. 相似文献
7.
Numerical solution of the one phase 1D fractional Stefan problem using the front fixing method
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In this paper, we present a novel numerical scheme to solve a 1D, one‐phase extended Stefan problem with fractional Caputo derivative with respect to time. The proposed method is based on a suitable choice of the new space coordinate for the subdiffusion equation and extends the front‐fixing method to the subdiffusion case. In the final part, examples of numerical results are discussed. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
8.
Ngoc Tran Vo Van Au Yong Zhou Nguyen Huy Tuan 《Mathematical Methods in the Applied Sciences》2020,43(6):3086-3098
In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag-Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained. 相似文献
9.
田元生 《应用泛函分析学报》2012,14(3):274-281
应用凸锥上的不动点定理,讨论了一类分数阶微分方程m点边值问题正解的存在性,得到了这类边值问题至少存在一个正解的充分条件,并给出了一个实例. 相似文献
10.
The aim of this article is to seek some adequate conditions via a prior estimate method (topological degree method) to derive the existence of solution to a nonlinear boundary value problem of fractional differential equations (FDEs). With the help of topological degree method which has been applied in many articles, we establish the required results for existence and uniqueness of solution to a class of FDEs. Moreover, we also formulate sufficient conditions for Hyers-Ulam stability to the solution of the considered problem. Finally, an appropriate example is provided to justify the relevant results. 相似文献
11.
韦忠礼 《应用泛函分析学报》2011,13(3):274-284
第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质.第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性.第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性.第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性.第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性(次线性)条件下C310,11正解存在的充分必要条件.最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性. 相似文献
12.
13.
Zhousheng Ruan 《Applicable analysis》2017,96(10):1638-1655
In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method. 相似文献
14.
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
15.
Identification of an inverse source problem for time‐fractional diffusion equation with random noise
Tran Ngoc Thach Tuan Nguyen Huy Pham Thi Minh Tam Mach Nguyet Minh Nguyen Huu Can 《Mathematical Methods in the Applied Sciences》2019,42(1):204-218
In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method. 相似文献
16.
Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions 相似文献
17.
Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method. 相似文献
18.
Satoru Hashizume Masao Fukushima Naoki Katoh Toshihide Ibaraki 《Mathematical Programming》1987,37(3):255-267
We are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function.
This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function.
Because of its combinatorial nature, however, it is often difficult to solve the auxiliary problem exactly. In this paper,
we propose an algorithm which assumes that the auxiliary problems are solved only approximately, and prove that it gives an
approximate solution to the original problem, of which the accuracy is at least as good as that of approximate solutions to
the auxiliary problems. It is also shown that the time complexity is bounded by the square of the computation time of the
approximate algorithm for the auxiliary problem. As an example of the proposed algorithm, we present a fully polynomial time
approximation scheme for the fractional 0–1 knapsack problem. 相似文献
19.
Ricardo Almeida 《Applied Mathematics Letters》2012,25(2):142-148
In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem. 相似文献
20.
本文利用Jacobi谱配置方法数值求解了一类分数阶多项延迟微分方程,并证明了该方法是收敛的,通过若干数值算例验证了相应的理论结果,结果表明Jacobi谱配置方法求解这类方程是非常高效的,同时也为这类分数阶延迟微分方程的数值求解提供了新的选择,对分数阶泛函方程的数值方法的研究有一定的指导意义. 相似文献