Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Received: 20 February 2020

DOI: 10.1002/mma.7102

RESEARCH ARTICLE

Recovering the initial value for a system of nonlocal

diffusion equations with random noise on the

measurements

Nguyen Anh Triet1 Tran Thanh Binh2 Nguyen Duc Phuong3,4,5

Dumitru Baleanu6,7 Nguyen Huu Can8

1Institute of Research and Development,

Duy Tan University, Da Nang, 550000,

Vietnam

2Faculty of Mathematics and

Applications, Sai Gon University, Ho Chi

Minh City, Vietnam

3Department of Mathematics and

Computer Science, University of Science,

Ho Chi Minh City, Vietnam

4Vietnam National University, Ho Chi

Minh City, Vietnam

5Faculty of Fundamental Science,

Industrial University of Ho Chi Minh City,

Ho Chi Minh City, Vietnam

6Department of Mathematics, Cankaya

University, Ankara, Turkey

7Institute of Space Sciences, Magurele,

Bucharest, Romania

8Applied Analysis Research Group,

Faculty of Mathematics and Statistics, Ton

Duc Thang University, Ho Chi Minh City,

Vietnam

Correspondence

Nguyen Huu Can, Applied Analysis

Research Group, Faculty of Mathematics

and Statistics, Ton Duc Thang University,

Ho Chi Minh City, Vietnam.

Email: [email protected]

Communicated by: J. Muñoz Rivera

Funding information

On the Cauchy problem for elliptic and

parabolic equations, Grant/Award

Number: CS2019-26

In this work, we study the final value problem for a system of parabolic diffusion

equations. In which, the final value functions are derived from a random model.

This problem is severely ill-posed in the sense of Hadamard. By nonparametric

estimation and truncation methods, we offer a new regularized solution. We

also investigate an estimate of the error and a convergence rate between a mild

solution and its regularized solutions. Finally, some numerical experiments are

constructed to confirm the efficiency of the proposed method.

KEYWORDS

Ill-posed problem, Nonlocal diffusion, Random noise, Regularized solution

MSC CLASSIFICATION

35K05; 35K99; 47J06; 47H10

1 INTRODUCTION

Denote  = (0, ��) and for T>0. In this paper, we consider that the system describes the interaction of two species which

is modeled as diffusion equations with time diffusion coefficient as follows:

Math Meth Appl Sci. 2020;1–22. wileyonlinelibrary.com/journal/mma © 2020 John Wiley & Sons, Ltd. 1

2 TRIET ET AL.

{ ��

��t

u1 = 1(u)Δu1 + ��(x, t), (x, t) ∈  × (0, T),

��

��t

u2 = 2(u)Δu2 + g(x, t), (x, t) ∈  × (0, T), (1)

subject to the final conditions

u1(x, T) = ��(x), u2(x, T) = ��(x), x ∈ , (2)

and homogeneous Dirichlet boundary condition

u1(x, t) = u2(x, t) = 0, (x, t) ∈ �� × (0, T), (3)

where u = (u1, u2) is the vector of states or density of species. The functions f and g are the external sources. The diffusion

coefficients

l(u) = l(u)(t) = l

(

��

u1

hl

(t), ��u2

kl

(t)

)

, l = 1, 2, (4)

where ��

u1

hl

(t), ��u2

kl

(t) are defined as the means of the weight function hl(x), kl(x) ∈ L2() with respect to a density of states

u1, u2 respectively,

��

u1

hl

(t) = ∫

hl(x)u1(x, t)dx, ��u2

kl

(t) = ∫

kl(x)u2(x, t)dx, l = 1, 2. (5)

Let us assume that the nonlocal terms l(·, ·) satisfy the following assumptions:

G1. There exist positive constants Gmin and Gmax such that

0 < Gmin ≤ i(x, ��) ≤ Gmax, (x, ��) ∈ R × R, i = 1, 2. (6)

G2. The functions l(·, ·) ∶ R × R → R, i = 1, 2, are Lipschitz continuous relative to two variables with the same

Lipschitz constant Lip. This means, given x1, x2, ��1, ��2 ∈ R, then

|l(x1, ��1) − l(x2, ��2)| ≤ Lip(|x1 − x2| + |��1 − ��2|). (7)

Some work related to the problem which is studied such as that of Hapuarachchi and Xu1 constructed an approximation

problem and use a quasi-boundary value method to regularize nonlinear heat equation by using a small parameter. In

addition, in 2019, the Liu et al.2 considered that the boundary value problem of the partial differential equations can be

transformed into an equivalent system of nonlinear and ill-posed integral equations for the unknown boundary. Then,

they applied the regularized Newton iterative method to reconstruct the boundary for the linearized system. A parabolic

equation with nonlocal diffusion is widely applied in physical and biological studies3–6 and the references therein. Among

them, the most successful model is used to describe population density in biology or particle density in material science.

In order to study the interaction of two or more biological species, systems of parabolic equations have been proposed;

see Hao et al.7 Problem (1–3) brought significant question that is “If we have the density of the two species at the present

time, how to determine their density at a previous time?” In the case of constant or time-dependent diffusion coefficient,

the classical backward heat conduction equation has been studied in many works, see for example, other studies.8–10

However, to the best of our knowledge, there are not any result on (1–3). Our paper is the first study on this direction for

parabolic systems with random model.

In real-life applications, we don't have the true final value functions �� and ��. Instead, we only have their observed value

at a certain number of points. However, observations always contain errors, and errors can come from many sources such

as inaccurate measuring machines and instruments; the low level of measuring skills, the ability of the senses is limited;

external influences, including changing weather, wind, rain, heat, etc. In this work, we assume that the observed values

of them are conducted at discrete points and spaced evenly, xk = �� 2k−1

2n , k = 1, … , n, and errors follow a random noise

model

��̃k = ��(xk) + k,

��

̃k = ��(xk) + k, (8)