# Nguyen Huy TuanVăn Lang University · Faculty of Technology

Nguyen Huy Tuan

Professor

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Vietnam

## About

203

Publications

31,763

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2,076

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Citations since 2017

## Publications

Publications (203)

In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are establis...

In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable b...

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data u(T,x)=φ(x)$$ u\left(T,x\right)=\varphi (x) $$ and the spectrum of the fractional Laplacian operator (−Δ)s/2$$...

In this paper, we discuss a class of nonlinear Schr\"odinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + \Delta ) \psi = \lambda |\psi|^{2\eta}\psi$ in $\mathbf R^N \times \mathbf R^+$. Based on the Gagliardo-Nirenberg interpolation inequality, we prove the local existence and long-time behavior (continuat...

In this paper, we study a non-autonomous damped wave equation with a nonlinear memory term. By using the theory of evolution process and sectorial operators, we ensure sufficient conditions for well-posedness and spatial regularity to the problem.

In this work, we study two final value problems for fractional reaction equation with standard Brownian motion W(t) and fractional Brownian motion BH(t), for H∈(14,12)∪(12,1). Firstly, the well-posedness of each problem is investigated under strongly choices of data. We aim to find spaces where we obtain the existence of a unique solution of each p...

In this work, the following stochastic Rayleigh-Stokes equations are considered
\begin{document}$ \begin{align*} \partial_t \big[ x(t)+f(t,x_\rho(t)) \big] = \big( A +\vartheta &\partial_t^\beta A \big) \big[ x(t)+f(t,x_\rho(t)) \big] \\ &+ g(t,x_\tau(t)) + B(t,x_\xi(t)) \dot{W}(t), \end{align*} $\end{document}
which involve the Riemann-Liouville f...

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators...

In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy...

In this paper, we aim to study a time-fractional Cauchy problem for a heat equation with a nonlocal nonlinearity driven by simulation problems arising in populations, and biological mathematics. Using the Banach fixed-point argument, we investigate the existence and uniqueness of mild solutions in Besov spaces defined on an open subset of RN. The k...

In this work, we ponder on a Cauchy problem for the Rayleigh–Stokes equation accompanied by polynomial and gradient nonlinearities. We particularly concern about the behavior of mild solutions for the different instances of the nonlinear source term. In the case of polynomial nonlinearities, we present the local‐in‐time existence and uniqueness of...

In this article, we are interested in investigating the nonlocal nonlinear reaction - diffusion system with final conditions. This problem is called backward in time problem, or terminal value problem which is understood as redefining the previous distributions when the distribution data at the terminal observation are known. There are three main g...

This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solu...

In this paper, we consider a class of pseudoparabolic equations with the nonlocal condition and the Caputo derivative. Two cases of problems (1–2) will be studied, which are linear case and nonlinear case. For the first case, we establish the existence, the uniqueness, and some regularity results by using some estimates technique and Sobolev embedd...

In this study, we investigate a Cauchy problem for the stochastic elliptic equation driven by Wiener noise. We show this problem is not well‐posed by proposing a simple illustrative example. To regularize the instable solution, we apply a regularization method called Fourier truncated expansion method. Furthermore, the convergence rate of the regul...

In this paper, we study the backward problem for the stochastic parabolic heat equation driven by a Wiener process. We show that the problem is ill-posed by violating the continuous dependence on the input data. In order to restore stability, we apply a filter regularization method which is completely new in the stochastic setting. Convergence rate...

An initial value problem for space-time fractional stochastic heat equations driven by colored noise \(\partial _t^s u_t + \frac{\nu }{2} (- \Delta )^{\alpha /2} u_t = \sigma (u_t) \dot{W}\) has been discussed in this work. Here, \(\partial _t^s\) and \((- \Delta )^{\alpha /2}\) stand for the Caputo’s fractional derivative of order \(s\in (0,1)\) a...

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild...

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of d...

In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1 2 , 1 ) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guarantee...

In this paper, we study a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we give a mild solution of our problem. Our main tool here is using some Sobolev embeddings.

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The continuous dependence of the solution on the given data is shown and the ill-posedness of the mild solution at t = 0 is also considered. In order to avoi...

In this study, fractional stochastic pseudo-parabolic equations driven by fractional Brownian motion are investigated. This work aims at establishing existence, uniqueness, regularity results for mild solutions to an initial value problem for considered equations in two cases of H that are H>12 and H<12. In addition, the continuities of mild soluti...

This paper is concerned with a solution of backward pseudo-Parabolic equationut−αΔut+G(∥∇u∥L2(Ω))(−Δ)βu=K(t,x)subject to the final data which is blurred by random Gaussian white noise. The primary aim of this work is to obtain the solution u. This problem is severely ill-posed (the solution’s behavior does not change continuously with the final con...

In this paper, we consider the existence of a solution u ( x , t ) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we u...

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefle...

In this paper, we consider a backward problem for a nonlinear diffusion equation with a conformable derivative in the case of multidimensional and discrete data. We show that this problem is ill‐posed and then we establish stable approximate solutions by two different regularization methods: the Fourier truncated method and the quasi‐boundary value...

In this paper, we consider a pseudo‐parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data u0 ∈ L2. In the case of initial data u0 ∈ Lq, q ≠ 2, we obtain the local existen...

In this paper, a time-fractional integrodifferential equation with the Caputo–Fabrizio type derivative will be considered. The Banach fixed point theorem is the main tool used to extend the results of a recent paper of Tuan and Zhou [ J. Comput. Appl. Math. 375 (2020) 112811]. In the case of a globally Lipschitz source terms, thanks to the L p − L...

With every passing day, one comes to know that cases of the corona virus disease are increasing. This is an alarming situation in many countries of the globe. So far, the virus has attacked as many as 188 countries of the world and 5 549 131 (27 May 2020) human population is affected with 348 224 deaths. In this regard, public and private health au...

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, w...

Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in \cite{Nguyen2019}, it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction-diff...

In this paper, we study a pseudo-parabolic equation with the Caputo fractional derivative. By applying the properties of Mittag–Leffler functions and the method of eigenvalue expansion, under a suitable definition of mild solution of our problem, we obtain the existence result and \(L^p\) regularity of the mild solution by using some Sobolev embedd...

Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and con...

In this work, a stochastic Rayleigh–Stokes equation driven by fractional Brownian motion is considered in both cases h ∈ 0 , 1 2 and h ∈ 1 2 , 1 . The existence and uniqueness of mild solution in each case are established separately by applying a standard method that is Banach fixed point theorem. The required results are obtained by stochastic ana...

This paper is concerned with the asymptotic behaviour of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space $\mathbb {R}^n$ . The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustn...

In this paper, we discuss an initial value problem for the semilinear time-fractional diffusion equation. The local well-posedness (existence and regularity) is presented when the source term satisfies a global Lipschitz condition. The unique continuation of solution and finite time blowup result are presented when the reaction terms are logarithmi...

In this paper we consider the Cauchy problem for the pseudo-parabolic equation: $$\begin{aligned} \dfrac{\partial }{\partial t} \left( u + \mu (-\Delta )^{s_1} u\right) + (-\Delta )^{s_2} u = f(u),\quad x \in \Omega ,~ t>0. \end{aligned}$$Here, the orders \(s_1, s_2\) satisfy \(0<s_1 \ne s_2 <1\) (order of diffusion-type terms). We establish the lo...

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atan...

We study the semilinear strongly damped plate equation by considering its two different problems. For initial value problem, we prove the local well-posedness and blow-up results of solution for the problem with polynomial nonlinear source terms. For terminal value problem, given the ill-posedness in the sense of Hadamard we propose a regularizatio...

In this paper, we consider a multi-dimensional fractional pseudo-parabolic problem with nonlinear source in case the input data is measured on a discrete set of points instead of the whole domain. For any number of dimensions, the solution is not stable. This makes the problem we are interested in be ill-posed. Here, we construct regularized soluti...

In this paper, we consider the problem of finding the source distribution for the linear inhomogeneous biparabolic equation when the measured output data is given in the form of the final overdetermination. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous pro...

In this paper, we study a problem of finding the solution for the nonlinear biharmonic equation Δ2u=f(x,t,u(x,t)) from the final data. By using a simple example, the ill-posedness of the present problem with random noise is demonstrated. The Fourier method is conducted in order to establish an estimator for the mild solution (called regularized sol...

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 pro...

In this work, we study an inverse problem to determine an unknown source term for fractional diffusion equation with Riemann–Liouville derivative. In general, the problem is severely ill posed in the sense of Hadamard. To regularize the unstable solution of the problem, we have applied the quasi-boundary value method. In the theoretical result, we...

In this paper, we study an initial value problem for the time fractional diffusion equationC∂β∂tβu+Au=F,0<β≤1,on Ω × (0, T), where the time fractional derivative is the conformable derivative. We study the existence and regularity of mild solutions in the following three cases with source term F:
•F=F(x,t), i.e., linear source term;
•F=F(u) is nonl...

In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of...

We consider two final value problem of a stochastic strongly damped wave equation driven by white noise (Section 3) and fractional noise (Section 4). We show that a stochastic integral in the solution is not stable and the problem is not well posed. To regularize the problem in two cases of noise, we apply the Fourier truncated method to control th...

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,...

In this article, we study an inverse problem with inhomogeneous source to determine an initial data from the time fractional diffusion equation. In general, this problem is ill‐posed in the sense of Hadamard, so the quasi‐boundary value method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori paramet...

We study a boundary value problem for a 2-D fractional differential equation (FDE) with random noise. This problem is not well-posed. Hence, we use truncated regularization method to establish regularized solutions for the such problem. Finally, the convergence rate of this approximate solution and a numerical example are investigated.

In this paper, we consider the final boundary value problem for non-local Kirchhoff’s model of parabolic type with discrete random noise. We first discuss the instability of solutions. Then we present the regularized solution by the trigonometric method in non-parametric regression associated with the truncated expansion method. In addition, under...

In this paper, we consider the terminal value problem for pseudo-parabolic equations with Riemann–Liouville fractional derivatives, from a given final value and we investigate the existence (and regularity) of mild solutions.

For an inverse nonlinear diffusion equation with conformable time derivative, we study the ill-posed property in the sense of Hadamard. To obtain a stable numerical solution, we propose two regularization methods. The results of existence and uniqueness, regularity and stability of the regularized problem are obtained. We also show that the corresp...

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reaction–diffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the q...

This article is devoted to the study of final value problems for biparabolic equation with discrete data in two cases as the linear source and the nonlinear source, respectively. In each of the cases, we show the instability of the solutions and then establish approximate solutions by applying some regularization methods. In addition, the convergen...

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 approxi...

We study the backward problem of determining the initial condition for a system of parabolic diffusion equations, which is severely ill-posed in the sense of Hadamard. To stabilize the solution, we develop the quasi-reversibility (QR) and Fourier truncation methods to construct the regularized solutions. We also investigate the error estimates and...

In this paper, we consider the following Cauchy problem of space–time fractional diffusion equations CFD0,tαu+(−L)su=G(t,x;u),in(0,T]×Ω,u(t,x)=0,on(0,T]×∂Ω,u(0,x)=u0(x),inΩ,The time fractional derivative is taken in the sense of Caputo-Fabizzio type. We derive representation of solutions by using Laplace transform and we further establish the exist...

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order \begin{document}$ \sigma,\; \; 0<\sigma<1 $\end{document} and the space fractional derivative is of order \begin{document}$ \alpha,\beta >0 $\end{document}. In the first part, we o...

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 spac...

In 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 n...

This paper is concerned with finding the solution u(x, t) of the Cauchy problem for nonlinear fractional elliptic equation with perturbed input data. This study shows that our forward problem is severely ill-posed in sense of Hadamard. For this ill-posed problem, the trigonometric of non-parametric regression associated with the truncation method i...

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in uniformly with respect...

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...

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reactiondiffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the qu...

The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the existence and regularity of mild solutions in two cases: \begin{itemize} \item[--] The first case: The nonlin...

In 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 {\color{black} a} priori assumptions on the sought solution,...

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining an initial condition appearing in a general class of time fractional wave equations with Caputo derivative from knowledge of solution at the final time. We are concerned with the existence, regularity upon the ter...

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^\beta$, $0<\beta\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions...

Lotka–Volterra systems are used to describe the dynamics of biological systems. We study the backward problem for the Lotka–Volterra system to determine the population density of species at preceding times. The problem is ill-posed in the sense that if the solution exists it does not depend continuously on the given data. We propose two stable regu...

In this paper, we deal with the backward problem of determining initial condition for Rayleigh‐Stokes where the data are given at a fixed time. The problem has many applications in some non‐Newtonian fluids. We give some regularity properties of the solution to backward problem.

We study for the first time the ill-posed backward problem for a contaminated nonlinear predator-prey system whose velocities of migration depend on the total average populations in the considered space domain. We propose a new regularized problem for which we are able to prove its unique solvability in Theorem 1. Moreover, under some mild assumpti...

In this paper, we consider the initial inverse problem (backward problem) for an inhomogeneous time-fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized sol...

In this study, we investigate a problem of finding the function u(x, y, t) for the fractional Rayleigh-Stokes equation with nonlinear source as follows(1){∂tu−(1+α∂tβ)Δu=f(x,y,t,u),(x,y,t)∈Ω×(0,T),u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T),u(x,y,T)=v(x,y),(x,y)∈Ω,where Ω=(0,π)×(0,π). The values of the final data v at n × m points (xp, yq) of Ω are contaminated by...

We consider a Cauchy semilinear problem for a time-fractional diffusion system∂αu∂tα+Au=F(u,v),∂αv∂tα+Bv=G(u,v),which involves symmetric uniformly elliptic operators A,B on a bounded domain Ω in Rd with sufficiently smooth boundary. The problem is equipped with final value conditions (FVCs), i.e., (u;v)|t=T are given. We derive a spectral represent...

In this paper, we study initial value problems for nonlinear fractional elliptic equations. In general, the problem is not well-posed, and herein the Hadamard-instability occurs. Under some weak a priori assumptions on the sought solution, we propose two new regularization methods to stabilize the problem when the source term is a globally or local...

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approx...

In this paper, we deal with an inverse problem of giving the terminal condition for time-fractional diffusion equations. This problem is ill-posed, i.e. the solution does not depend continuously on the data. To regularize the instable solution, we use the spectral method (or called the Fourier truncation method) combined with some techniques in non...

We investigate a backward problem for the Rayleigh‐Stokes problem, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well‐known to be ill‐posed because of the rapid decay of the forward process. We construct a regularized solution using the fil...

In this paper, we consider a final value problem for time-fractional diffusion equation with inhomogeneous source. The main goal of our paper is to determine an approximated initial data from the observation data at final time by constructing a regularized solution using a mollification method. Under appropriate regularity assumptions of the exact...

Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in \cite{Nguyen2019}, it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction-diff...

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...