Time periodic solutions for strongly nonlinear parabolic systems with p(x)-growth conditions

Alaa, Hamza; Alaa, Nour Eddine; Charkaoui, Abderrahim

doi:10.1007/s41808-021-00118-9

Cited by 11 publications

(7 citation statements)

References 28 publications

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“…Thanks to the result of Theorem 1 from Section 3, we deduce the existence of a weak solution u n to the approximate problem (38) satisfying…”

Section: Approximate Problemmentioning

confidence: 94%

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An improved nonlinear anisotropic model with p(x)‐growth conditions applied to image restoration and enhancement

Alaa,

Eddine Alaa,

Bouchriti

et al. 2024

Math Methods in App Sciences

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This work proposes a novel nonlinear parabolic equation with ‐growth conditions for image restoration and enhancement. Based on the generalized Lebesgue and Sobolev spaces with variable exponent, we demonstrate the well‐posedness of the proposed model. As a first result, we prove the existence of a weak solution to our model when the reaction term is bounded by a suitable function. Secondly, we use the approximations method to establish the existence of a nonnegative weak SOLA (Solution Obtained as Limit of Approximations) solution to the proposed model. We illustrate our theoretical results in the context of image denoising and enhancement. Specifically, we present various numerical implementations on a set of grayscale images. To further enrich these simulations, we assess the efficiency of the proposed model on several color images. The obtained numerical results strongly suggest that our model outperforms existing state‐of‐the‐art methods in terms of both visual and quantitative comparison, demonstrating superior efficiency and robustness in noise removal and contrast enhancement.

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“…Thanks to the result of Theorem 1 from Section 3, we deduce the existence of a weak solution u n to the approximate problem (38) satisfying…”

Section: Approximate Problemmentioning

confidence: 94%

“…There are numerous approaches to solve PDEs of this alike. The earlier studies [34,38,[58][59][60][61], for example, provide various techniques. And since we are dealing with pixels in image processing, finite differences approaches and explicit schemes are the best options.…”

Section: Numerical Discretizationmentioning

confidence: 99%

An improved nonlinear anisotropic model with p(x)‐growth conditions applied to image restoration and enhancement

Alaa,

Eddine Alaa,

Bouchriti

et al. 2024

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…As well known, theoretical analysis of PDEs involving ( )-growth conditions need the use of some complex spaces called Lebesgue and Sobolev spaces with variable exponents (see for example 3,7,16,17,24,32,40,52,53 ). Therefore, the variable exponent (⋅) appears in problem (1) requires the consideration of these types of spaces.…”

Section: Mathematical Backgrounds and Assumptionsmentioning

confidence: 99%

“…There are numerous approaches to solve partial differential equations of this alike. The references 8,3,30,31,33,44 , for example, provide various techniques. And since we are dealing with pixels in image processing, finite differences approaches and explicit schemes are the best options.…”

Section: Proofmentioning

confidence: 99%

An improved nonlinear anisotropic PDE with p(x)-growth conditions applied to image restoration and enhancement

Alaa

Bouchriti

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

This work proposes a novel nonlinear parabolic equation with p(x)-growth conditions for image restoration and enhancement. Based on the generalized Lebesgue and Sobolev spaces with variable exponent, we demonstrate the well-posedness of the proposed model. As a first result, we prove the existence of a weak solution to our model when the reaction term is bounded by a suitable function. Secondly, we use the approximations method to establish the existence of a nonnegative weak SOLA solution (Solution Obtained as Limit of Approximations) to the proposed model. Finally, numerical experiments illustrate that the proposed model performs better for image enhancement and denoising.

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“…A large literature exists on periodic equations and several researchers have been interested in the subject, we quote in particular [8,6,11,12,13,15,14,18,19,23,24]. We start by referring the readers to see the book [24] for a major and comprehensive introduction to periodic parabolic equations with regular data.…”

Section: Introductionmentioning

confidence: 99%

Periodic parabolic problem with discontinuous coefficients: Mathematical analysis and numerical simulation

Alaa

Charkaoui²,

Elaassri³

2022

Proyecciones (Antofagasta)

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This work presents a new approach for the mathematical analysis and numerical simulation of a class of periodic parabolic equations with discontinuous coefficients. Our technique is based on the minimization of a least squares cost function. By the means of variational calculus, we prove that the considered optimization problem admits an optimal solution. Using the Lagrangian method, we compute the gradient of the cost function associated with our problem. Finally, we give several numerical simulations that show the efficiency and robustness of our method.

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Time periodic solutions for strongly nonlinear parabolic systems with p(x)-growth conditions

Cited by 11 publications

References 28 publications

An improved nonlinear anisotropic model with p(x)‐growth conditions applied to image restoration and enhancement

An improved nonlinear anisotropic model with p(x)‐growth conditions applied to image restoration and enhancement

An improved nonlinear anisotropic PDE with p(x)-growth conditions applied to image restoration and enhancement

Periodic parabolic problem with discontinuous coefficients: Mathematical analysis and numerical simulation

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