Fourier collocation algorithm for identifying the spacewise-dependent source in the advection–diffusion equation from boundary data measurements

Hasanov, Alemdar; Mukanova, Balgaisha

doi:10.1016/j.apnum.2015.05.005

Cited by 11 publications

(10 citation statements)

References 20 publications

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“…In this paper, we developed appropriate adjoint functions that led to establish a constructive Detection-Identification method for solving the nonlinear inverse source problem of identifying multiple unknown time-dependent point sources occurring in 3D transport equations. In the literature, many authors have addressed similar inverse source problems in different PDEs, for instance [2,3,4,18,19,20,31]. In these works, the developed identification approaches are mainly either iterative based on the minimisation of cost functions such as least squares and Kohn-Vogelius or quasi-direct such as the algebraic method especially used for elliptic equations [1,32].…”

Section: Conclusion Discussion and Comparaisonmentioning

confidence: 99%

“…where (a i , b i , c i ) ∈ Ω i . In addition, for later use we establish the following property of ψ i : Lemma 3.1 Provided (17) holds, the scalar potential ψ i defined in (19) satisfies: For all x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ) in Ω i , ψ i (x) = ψ i (y) implies that…”

Section: Full Adjoint Functionmentioning

confidence: 99%

“…where ψ i is the scalar potential obtained in (19) and v m 0i , v n 0i are two functions from (32). Moreover, to prove the injectivity of Ψ m,n i , we establish the following technical result:…”

Section: Dispersion-current Vector Functionmentioning

confidence: 99%

“…Proof of Lemma 3.1: Let x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ) be two elements of Ω i . In view of (19), from writing ψ i (x) = ψ i (y) under the form ψ i (y) − ψ i (x) = 0 and treating in this latest equation separately each two integrals following the same axis, we obtain:…”

Section: Appendixmentioning

confidence: 99%

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Dispersion–current adjoint functions for monitoring accidental sources in 3D transport equations

Hamdi

Tonnoir

2020

Inverse Problems in Science and Engineering

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The paper deals with the identification of multiple unknown time-dependent point sources occurring in 3D dispersion-advection-reaction equations. Based on developed appropriate adjoint functions, we establish a constructive identifiability result depending on the flow nature that yields guidelines leading to a quasi-direct Detection-Identification method. In practice, assuming to be available within a monitored domain some interfaces subdividing it into suspected sections, the developed method goes throughout the entire domain to detect the presence of all active sources. If an activity is detected within a suspected section, the method identifies the total amount discharged in this section and determines whether it is done by a single or multiple unknown occurring sources. Moreover, it localizes the sought position of a detected source as the unique root of a Dispersion-Current vector function defined from the developed adjoint functions. Application to different types of flow and some numerical experiments on surface water pollution are presented.

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Section: Conclusion Discussion and Comparaisonmentioning

confidence: 99%

Section: Full Adjoint Functionmentioning

confidence: 99%

Section: Dispersion-current Vector Functionmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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Dispersion–current adjoint functions for monitoring accidental sources in 3D transport equations

Hamdi

Tonnoir

2020

Inverse Problems in Science and Engineering

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“…This problem is an interesting problem in environmental sciences and engineering. It also play an important role in transportation of biological or chemical contaminants through sub surface systems [1,2]. In subsurface hydrology the solute transport in ground water and soils have been very important topic of theoretical and experimental research.…”

Section: Introductionmentioning

confidence: 99%

Jacobi collocation method for the fractional advection‐dispersion equation arising in porous media

Singh¹

2020

Numer Methods Partial Differential Eq.

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In present paper, we solve fractional advection-dispersion equation (ADE) using Jacobi collocation method. This equation appears in the transport of solutes in ground water and soils. Using Jacobi spectral collocation method the fractional ADE is converted into systems of nonlinear algebraic equations whose solution leads the approximate solution for this equation. We provide the convergence analysis of the proposed approximate method. The applicability of proposed method is shown by testing it on some illustrative examples. Numerical results are shown using figures. Absolute error figures show the accuracy of the proposed method. CPU time is also listed in tabular form to show efficiency of the proposed method.

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Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

Alosaimi,

Tekin,

Çetin

2023

Math Methods in App Sciences

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Pseudo‐parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo‐parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time‐varying potential and source control terms, in a linear pseudo‐parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least‐squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank–Nicolson finite‐difference method to solve the direct problem. To solve the nonlinear least‐squares minimization problem, we utilize the built‐in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology.

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Fourier collocation algorithm for identifying the spacewise-dependent source in the advection–diffusion equation from boundary data measurements

Cited by 11 publications

References 20 publications

Dispersion–current adjoint functions for monitoring accidental sources in 3D transport equations

Dispersion–current adjoint functions for monitoring accidental sources in 3D transport equations

Jacobi collocation method for the fractional advection‐dispersion equation arising in porous media

Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

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