A nonlinear fractional Rayleigh–Stokes equation under nonlocal integral conditions

AbstractIn this paper, we study the fractional nonlinear Rayleigh–Stokes equation under nonlocal integral conditions, and the existence and uniqueness of the mild solution to our problem are considered. The ill-posedness of the mild solution to the problem recovering the initial value is also investigated. To tackle the ill-posedness, a regularized solution is constructed by the Fourier truncation method, and the convergence rate to the exact solution of this method is demonstrated.

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On a time fractional diffusion with nonlocal in time conditions

Advances in Difference Equations ◽

10.1186/s13662-021-03365-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nguyen Hoang Tuan ◽

Nguyen Anh Triet ◽

Nguyen Hoang Luc ◽

Nguyen Duc Phuong

Keyword(s):

Fourier Series ◽

Diffusion Equation ◽

Convergence Rate ◽

Exact Solutions ◽

Mild Solution ◽

Integral Condition ◽

Fractional Diffusion ◽

Well Posedness ◽

Nonlocal Integral Condition ◽

Regularized Solution

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

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Regularity of a Stochastic Fractional Delayed Reaction-Diffusion Equation Driven by Lévy Noise

Abstract and Applied Analysis ◽

10.1155/2013/807459 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Tianlong Shen ◽

Jianhua Huang ◽

Jin Li

Keyword(s):

Fixed Point ◽

Diffusion Equation ◽

Mild Solution ◽

Existence And Uniqueness ◽

Reaction Diffusion ◽

Reaction Diffusion Equation ◽

Banach Fixed Point Theorem ◽

Fractional Operator ◽

Initial Value ◽

Delayed Reaction

The current paper is devoted to the regularity of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. By the Banach fixed point theorem, the existence and uniqueness of the mild solution are proved in the proper working function space which is affected by the delays. Furthermore, the time regularity and space regularity of the mild solution are established respectively. The main results show that both time regularity and space regularity of the mild solution depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the regularity of initial value with delays.

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The Unique Positive Solution for Singular Hadamard Fractional Boundary Value Problems

Journal of Function Spaces ◽

10.1155/2019/5923490 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6 ◽

Cited By ~ 6

Author(s):

Jinxiu Mao ◽

Zengqin Zhao ◽

Chenguang Wang

Keyword(s):

Exact Solution ◽

Convergence Rate ◽

Boundary Value Problems ◽

Positive Solution ◽

Existence And Uniqueness ◽

Boundary Value ◽

Approximation Error ◽

Iterative Solution ◽

Unique Positive Solution ◽

Fractional Boundary Value Problems

In this paper, we investigate singular Hadamard fractional boundary value problems. The existence and uniqueness of the exact iterative solution are established only by using an iterative algorithm. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have also been derived.

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Existence, uniqueness and numerical solution of a fractional PDE with integral conditions

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2019.3.4 ◽

2019 ◽

Vol 24 (3) ◽

pp. 368-386

Author(s):

Jesus Martín-Vaquero ◽

Ahcene Merad

Keyword(s):

Existence And Uniqueness ◽

Recent Literature ◽

Numerical Approach ◽

Integral Conditions ◽

Fractional Partial Differential Equation ◽

One Dimensional ◽

Fractional Pde ◽

Uniqueness Of The Solution ◽

Nonlocal Integral Conditions ◽

Hereditary Properties

This paper is devoted to the solution of one-dimensional Fractional Partial Differential Equation (FPDE) with nonlocal integral conditions. These FPDEs have been of considerable interest in the recent literature because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances. Existence and uniqueness of the solution of this FPDE are demonstrated. As for the numerical approach, a Galerkin method based on least squares is considered. The numerical examples illustrate the fast convergence of this technique and show the efficiency of the proposed method.

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Regularization of a Cauchy problem for the heat equation

Science and Technology Development Journal - Natural Sciences ◽

10.32508/stdjns.v1it5.552 ◽

2018 ◽

Vol 1 (T5) ◽

pp. 184-192

Author(s):

Au Van Vo ◽

Tuan Hoang Nguyen

Keyword(s):

Cauchy Problem ◽

Exact Solution ◽

Heat Equation ◽

Error Estimates ◽

A Priori ◽

Cauchy Data ◽

Linear Source ◽

Truncation Method ◽

Ill Posed ◽

Regularized Solution

In this paper, we study a Cauchy problem for the heat equation with linear source in the form ut(x,t)= uxx(x,t)+f(x,t), u(L,t)= φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data φε and Ψε satisfying ‖ φε - φ ‖+‖ Ψε - Ψ ‖ ≤ ε and that fε satisfying ‖ fε(x,. ) - f(x,.) ‖ ≤ ε . We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.

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