Determination of a term in the right-hand side of parabolic equations

Hào, Ðinh Nho; Huong, Bui Viet; Oanh, Nguyen Thi Ngoc; Thanh, Phan Xuan

doi:10.1016/j.cam.2016.05.022

Cited by 22 publications

(3 citation statements)

References 45 publications

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“…and the initial and boundary conditions (2.2) and (2.3). Remark that this formulation has recently been adopted by Hào et al (2017) for solving the corresponding inverse source problem for parabolic PDE's.…”

Section: The Mathematical Formulationmentioning

confidence: 99%

Identification of the forcing term in hyperbolic equations

Alosaimi

Lesnic

Hào

2020

International Journal of Computer Mathematics

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We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite difference methods. Numerical results are presented and discussed for one and two-dimensional problems.

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Section: The Mathematical Formulationmentioning

confidence: 99%

Identification of the forcing term in hyperbolic equations

Alosaimi

Lesnic

Hào

2020

International Journal of Computer Mathematics

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“…The unknown term to be identified in the inverse problem may be a source term depending either only on the spatial variable [11][12][13][14] or on the time variable [15][16][17][18] or maybe an unknown coefficient called source control parameters [19]. Recently, the inverse source problem with fractional types is presented in [20,21].…”

Section:  Introductionmentioning

confidence: 99%

A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems

et al. 2022

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This paper is devoted to an inverse problem of determining discontinuous space-wise dependent heat source in a linear parabolic equation from the measurements at the ﬁnal moment. In the existing literature, a considerable accurate solution to the inverse problems with unknown space-wise dependent heat source is impossible without introducing any type of regularization method but here we have determine the unknown discontinuous space-wise dependent heat source acutely using Haar wavelet collocation method (HWCM) without applying the regularization technique. This HWCM is based on ﬁnite-diﬀerence and Haar wavelets approximation to the inverse problem. In contrast to other numerical techniques, in HWCM, we use Haar functions that create a well-conditioned system of algebraic equations. The proposed method is stable and convergent because the numerical solution converges to the exact solution without observing any diﬃculty. Finally, some numerical examples are presented to verify the validity of the HWCM for diﬀerent cases of source term.

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“…To overcome this situation, those parameters can be estimated by solving the corresponding inverse problem. Besides, it is now possible to work out these type of problems using diverse global optimization algorithms (Hào et al, 2017;Wang et al, 2017;Cebo-Rudnicka et al, 2016;Nedin et al, 2016;Chen et al, 2016;Strongin & Sergeyev, 2000;Zhigljavsky & Žilinskas, 2008;Kvasov & Sergeyev, 2014). Previous works related to heat generation proposed estimating heat generation in the case of a rotatory friction welding system (Yang et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

Internal volumetric heat generation and heat capacity prediction during a material electromagnetic treatment process using hybrid algorithms

2018

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This work considers the estimation of internal volumetric heat generation, as well as the heat capacity of a solid spherical sample, heated by a homogeneous, time-varying electromagnetic field. To that end, the numerical strategy solves the corresponding inverse problem. Three functional forms (linear, sinusoidal, and exponential) for the electromagnetic field were considered. White Gaussian noise was incorporated into the theoretical temperature profile (i.e. the solution of the direct problem) to simulate a more realistic situation. Temperature was pretended to be read through four sensors. The inverse problem was solved through three different kinds of approach: using a traditional optimizer, using modern techniques, and using a mixture of both. In the first case, we used a traditional, deterministic Levenberg-Marquardt (LM) algorithm. In the second one, we considered three stochastic algorithms: Spiral Optimization Algorithm (SOA), Vortex Search (VS), and Weighted Attraction Method (WAM). In the final case, we proposed a hybrid between LM and the metaheuristics algorithms. Results show that LM converges to the expected solutions only if the initial conditions (IC) are within a limited range. Oppositely, metaheuristics converge in a wide range of IC but exhibit low accuracy. The hybrid approaches converge and improve the accuracy obtained with the metaheuristics. The difference between expected and obtained values, as well as the RMS errors, are reported and compared for all three methods.

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Determination of a term in the right-hand side of parabolic equations

Cited by 22 publications

References 45 publications

Identification of the forcing term in hyperbolic equations

Identification of the forcing term in hyperbolic equations

A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems

Internal volumetric heat generation and heat capacity prediction during a material electromagnetic treatment process using hybrid algorithms

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