Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction

Tuan, Nguyen Huy; Au, Vo Van; Khoa, Vo Anh; Lesnic, D.

doi:10.1088/1361-6420/aa635f

Cited by 33 publications

(37 citation statements)

References 35 publications

(123 reference statements)

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“…In this paper, we shall make use of the following notation: Λ p , q will denote the abstract Gevrey class of functions of order p >0 and index q , see, eg, Tuan et al, defined by

Λ_{p, q} = ({}, ξ \in L^{2} false(0, π false) : {(‖‖, ξ)}_{p, q}^{2} = true \sum_{n = 1}^{\infty} \frac{e^{2 p n ℓ}}{n^{q}} c_{n}^{2} false(ξ false) < + \infty) p \geq 0, q \in double-struckR,

where

c_{n} false(ξ false) = {(⟨⟩, ξ, ψ_{n})}_{L^{2} false(0, π false)} = \int_{0}^{π} ξ false(x false) ψ_{n} false(x false) d x = \sqrt{\frac{2}{π}} \int_{0}^{π} ξ false(x false) \sin false(n x false) d x

is the Fourier coefficient of the function ξ . The norm in Λ p , q and L 2 (0, π ) will be denoted respectively by

{(‖‖ .)}_{p, q}

and

(‖‖ .) .

…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

Benrabah

Boussetila

Rebbani³

2019

Math Methods in App Sciences

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In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well-posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method. KEYWORDS auxiliary boundary conditions, biharmonic equation, convergence estimate, ill-posed problems, plates, regularization MSC CLASSIFICATION 35J40; 31B30; 35B45; 74K20; 47A52 Abbreviations: BHP, biharmonic problem; MABCM, modified auxiliary boundary conditions method; QBVM, quasi-boundary value method. wileyonlinelibrary.com/journal/mma

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“…In this paper, we shall make use of the following notation: Λ p , q will denote the abstract Gevrey class of functions of order p >0 and index q , see, eg, Tuan et al, defined by

Λ_{p, q} = ({}, ξ \in L^{2} false(0, π false) : {(‖‖, ξ)}_{p, q}^{2} = true \sum_{n = 1}^{\infty} \frac{e^{2 p n ℓ}}{n^{q}} c_{n}^{2} false(ξ false) < + \infty) p \geq 0, q \in double-struckR,

where

c_{n} false(ξ false) = {(⟨⟩, ξ, ψ_{n})}_{L^{2} false(0, π false)} = \int_{0}^{π} ξ false(x false) ψ_{n} false(x false) d x = \sqrt{\frac{2}{π}} \int_{0}^{π} ξ false(x false) \sin false(n x false) d x

is the Fourier coefficient of the function ξ . The norm in Λ p , q and L 2 (0, π ) will be denoted respectively by

{(‖‖ .)}_{p, q}

and

(‖‖ .) .

…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

Benrabah

Boussetila

Rebbani³

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Most selected models are the fractional diffusion equation using the Riemann-Liouville or Cauputo derivatives. Many applications of these models in fields of science such as [3,[5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In recent times, the mathematical model of time-fractional diffusion equation is a class of important physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

Determination of source term for fractional heat equation on the sphere

et al. 2020

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“…Another related problem is the inverse problem of reconstructing the initial condition for parabolic equation. This problem is very important and interesting, see [31,36,33,27,38] for theoretical results and numerical methods. In the current paper, we introduce the following approach to solve Problem 1.1.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

Nguyen

2019

Journal of Inverse and Ill-Posed Problems

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AbstractTwo main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

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Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction

Cited by 33 publications

References 35 publications

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

Determination of source term for fractional heat equation on the sphere

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

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