A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation

Ahsan, Muhammad; Lin, Shanwei; Ahmad, M. Omair; Nisar, Muhammad; Ahmad, Imtiaz; Ahmed, Hijaz; Liu, Xuan

doi:10.1515/phys-2021-0080

Cited by 25 publications

(6 citation statements)

References 39 publications

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“…For this purpose, partial differential equations (PDEs) are used to describe the physical behavior [ 6 – 9 ]. The solution of the PDEs is an active area of research, and various techniques are applied for the solutions of PDEs [ 10 – 16 ]. Forty years ago, it was a faith that medical science has made tremendous progress in reducing the mortality rate of humans because it is due to improvements in nutrition, drugs, and vaccines.…”

Section: Introductionmentioning

confidence: 99%

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate

et al. 2023

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The current study deals with the stochastic reaction–diffusion epidemic model numerically with two proposed schemes. Such models have many applications in the disease dynamics of wildlife, human life, and others. During the last decade, it is observed that the epidemic models cannot predict the accurate behavior of infectious diseases. The empirical data gained about the spread of the disease shows non-deterministic behavior. It is a strong challenge for researchers to consider stochastic epidemic models. The effect of the stochastic process is analyzed. So, the SIR epidemic model is considered under the influence of the stochastic process. The time noise term is taken as the stochastic source. The coefficient of the stochastic term is a Borel function, and it is used to control the random behavior in the solutions. The proposed stochastic backward Euler scheme and the proposed stochastic implicit finite difference scheme (IFDS) are used for the numerical solution of the underlying model. Both schemes are consistent in the mean square sense. The stability of the schemes is proven with Von-Neumann criteria and schemes are unconditionally stable. The proposed stochastic backward Euler scheme converges toward a disease-free equilibrium and does not converge toward an endemic equilibrium but also possesses negative behavior. The proposed stochastic IFD scheme converges toward disease-free equilibrium and endemic equilibrium. This scheme also preserves positivity. The graphical behavior of the stochastic SIR model is much similar to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The efficacy of the proposed scheme is shown in the graphical behavior of the test problem for the various values of the parameters.

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Section: Introductionmentioning

confidence: 99%

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate

et al. 2023

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“…In Ahsan et al, 51 HWCM is implemented to calculate the space‐dependent heat source without regularization technique. The nonlinear inverse Cauchy problem with unknown right side boundary condition and unknown source is calculated via HWCM, 57 while the unknown left side boundary condition is calculated using Tikhonov regularization HWCM in Pourgholi et al 58 The unknown coefficient as a source control parameter in the inverse problem is also obtained using HWCM along with the solution

S\left(x,\tau \right)

in Ahsan et al 59 Apart from these findings, the current paper is focused on solving nonlinear inverse problems influenced by the heat source with given final time information.…”

Section: Introductionmentioning

confidence: 99%

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

Ahsan

Haq

et al. 2022

Math Methods in App Sciences

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In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill‐posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill‐posed problem into well‐conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill‐posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.

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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 non-linear Cauchy equation has been accurately evaluated by HWCM, where the unknown space-depending heat source and the unknown solutions are approximated with two various Haar series converting it into well-conditioned algebraic equations [14]. Other source functions like unknown control parameters are also calculated using HWCM [19]. The right side unknown boundary condition is accurately measured by HWCM in a nonlinear inverse problem [14].…”

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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A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation

Cited by 25 publications

References 39 publications

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

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

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