Article
Mathematics, Interdisciplinary Applications
Y. S. Liang, H. X. Wang
Summary: This paper investigates the fractal dimension of fractional calculus for certain continuous functions. It has been shown that the upper Box dimension of Riemann-Liouville fractional integral decreases linearly for fractal functions with a dimension greater than a certain positive order. The fractal dimension of Riemann-Liouville fractional integral for one-dimensional continuous functions with unbounded variation remains one-dimensional. An example of fractal linear interpolation functions demonstrates optimal estimation.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Yanan Li, Zhijian Yang
Summary: This paper investigates the existence of strong global and exponential attractors for a class of semilinear wave equations with fractional damping. By introducing a new approach, it improves the existence and continuity of strong solutions, and further deepens the study of attractors on the dissipative index theta.
ADVANCES IN DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
M. M. Freitas, A. J. A. Ramos, A. O. Ozer, D. S. Almeida Junior
Summary: In this study, vibrations on a piezoelectric beam with fractional damping dependent on a parameter v ∈ (0,1/2) were modeled using a variational approach. Magnetic and thermal effects were taken into account, and the existence and uniqueness of solutions, as well as the existence of smooth global attractors with finite fractal dimension and exponential attractors, were proven. Additionally, the upper-semicontinuity of global attractors as v -> 0(+) was demonstrated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Interdisciplinary Applications
Wei Xiao
Summary: This paper investigates the relationship between box dimension of a continuous fractal function and its Riemann-Liouville fractional integral. It is proven that the fractal dimension of the integral does not exceed that of the original function, indicating that fractional integration does not reduce the smoothness of the integrand. This result partially addresses the conjecture in fractal calculus.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Ailing Qi, Die Hu, Mingqi Xiang
Summary: This paper examines the asymptotic behavior of solutions to the fractional p-Kirchhoff equation's initial-boundary value problem, proving the existence of global attractors and showing that the fractal dimension of global attractors is infinite under certain conditions.
BOUNDARY VALUE PROBLEMS
(2021)
Article
Mathematics, Applied
Yanxia Qu, Zhijian Yang
Summary: This paper establishes the upper semicontinuity of the strong global attractors A(0) on the dissipative index for the Kirchhoff wave model with structural nonlinear damping, extending previous research on the topic. The result presented in this study improves and deepens those found in recent literature, enhancing our understanding of strong attractors in such models.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Interdisciplinary Applications
Liu Xuan, Shabir Ahmad, Aman Ullah, Sayed Saifullah, Ali Akguel, Haidong Qu
Summary: This paper studies the chaotic behavior of a cancer model and analyzes different bifurcations and hidden attractors using the fractal fractional operator. The existence and stability of the solution are examined using nonlinear analysis. Numerical simulations validate the results and the sensitivity of the system to initial conditions is also studied.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
Moncef Aouadi
Summary: This paper investigates the long-time dynamics of the perturbed system of suspension bridge equations. Global well-posedness and the existence of global attractors with finite fractal dimension are established under general assumptions on source terms. The upper semicontinuity of global attractors with respect to the perturbed parameter epsilon and the upper semicontinuity of the family of global attractors with respect to the fractional exponent gamma are demonstrated.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Interdisciplinary Applications
H. B. Gao, Y. S. Liang, W. Xiao
Summary: This paper investigates the relationship between the fractal dimension of continuous functions and the orders of Weyl fractional integrals, including properties of functions with bounded variation defined on closed intervals, estimation of the fractal dimension of Weyl fractional integrals for functions satisfying Milder condition, and general estimations of upper Box dimension of Weyl fractional integrals for functions continuous on R. It is also proven that the upper Box dimension of Weyl fractional integrals of continuous functions is not greater than the upper Box dimension of the original functions.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Alexei Ilyin, Anna Kostianko, Sergey Zelik
Summary: This study investigates the dependence of the fractal dimension of global attractors for the damped 3D Euler-Bardina equations on the regularization parameter and Ekman damping coefficient. Explicit upper bounds are presented for different boundary conditions. The sharpness of these estimates is demonstrated in the 3D Kolmogorov flows on a torus in the limit.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Manil T. Mohan
Summary: This work investigates the dynamics of three-dimensional viscoelastic fluid flow equations with fading memory. The existence of an absorbing ball and a global attractor for the semigroup are established. Estimates for determining modes on the attractor are provided. The finite dimensions of the global attractor are shown by demonstrating the differentiability of the semigroup. The existence of an exponential attractor and the quasi-stability of the semigroup are also proved.
EVOLUTION EQUATIONS AND CONTROL THEORY
(2022)
Article
Mathematics, Applied
Wenhua Yang, Jun Zhou
Summary: This article discusses the degenerate fractional Kirchhoff wave equation with structural damping or strong damping, and proves the well-posedness and existence of a global attractor in the natural energy space using the Faedo-Galerkin method and energy estimates. It is worth noting that the results also cover the case of possible degeneration or negativity of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite using Z(2) index theory.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics, Interdisciplinary Applications
Wei Xiao
Summary: This paper investigates the change in box dimension of an arbitrary fractal continuous function after Hadamard fractional integration, partially addressing the related problem.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Mathematics, Applied
Sonal Jain, Youssef El-Khatib
Summary: This paper discusses the application of differential operators based on convolution in depicting and capturing chaotic behaviors, as well as the introduction of new classes of differential and integral operators. The newly defined operators with fractional order, fractal dimension, and strange attractors are aimed to address the limitations of classical operators in depicting chaotic behaviors with similar patterns. The efficiency of these models is demonstrated through simulations and illustrations of the results.
Article
Mathematics, Applied
Xiaolei Dong, Yuming Qin
Summary: This paper investigates the finite fractal dimension of pullback attractors for a nonclassical diffusion equation in H-0(1) (Omega). First, the existence of pullback attractors for a nonclassical diffusion equation with arbitrary polynomial growth condition is proven using the operator decomposition method. Then, the finite fractal dimension of pullback attractors for a nonclassical diffusion equation in H-0(1) (Omega) is proved based on the fractal dimension theorem given by [6].
Article
Physics, Mathematical
A. J. A. Ramos, T. A. Apalara, M. M. Freitas, M. L. Araujo
Summary: This work investigates the equivalence between the exponential stabilization and exact boundary observability of swelling porous elastic media, and establishes the well-posedness result.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
M. M. Freitas, A. J. A. Ramos, D. S. Almeida Junior, P. T. P. Aum, J. L. L. Almeida
Summary: This paper presents a study on the long-time dynamics of a dynamical system modeling a mixture of solids with nonlinear damping and Fourier's law. By using quasi-stability theory, the existence of a smooth finite dimensional global attractor is proven as an unstable manifold of the set of stationary solutions, achieved through an estimated stabilizability. Additionally, the existence of a generalized exponential attractor is shown.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Mathematics, Applied
A. J. A. Ramos, D. S. Almeida Junior, M. M. Freitas
Summary: In this paper, we study the porous-elastic equations with Kelvin-Voigt dissipation mechanisms and thermal effect given by Fourier's law. We show that the system lacks exponential decay property for a specific equality between damping parameters. In this direction, we prove polynomial decay and the optimal decay rate.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics
O. P. V. Villagran, C. A. Nonato, C. A. Raposo, A. J. A. Ramos
Summary: This paper investigates the stability of weakly coupled wave equations with a boundary dissipation of fractional derivative type. By using the semigroup theory and a sharp result provided by Borichev and Tomilov, we have proved well posedness and polynomial stability of the problem.
RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO
(2023)
Article
Mathematics
M. M. Freitas, A. J. A. Ramos, D. S. Almeida Junior, L. G. R. Miranda, A. S. Noe
Summary: This article explores the asymptotic dynamics of a nonlinear swelling porous-elastic system with memory term and fractional damping. The existence, smoothness, and finite dimensionality of global attractors are established using the quasi-stability approach. The existence of exponential attractors is also proven. These novel results make significant contributions to the theory of nonlinear dynamics in swelling porous elastic soils.
BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA
(2023)
Article
Mathematics, Applied
M. M. Freitas, M. L. Santos, M. J. Dos Santos, A. J. A. Ramos
Summary: In this paper, we investigate the long-term behavior of solutions to the one-dimensional porous-elasticity problem with porous dissipation and nonlinear feedback force. We prove that as the parameter J tends to zero, the porous-elasticity problem converges to a quasi-static problem for microvoids motion. By utilizing the recent quasi-stability theory, we obtain a finite dimensional global attractor with additional regularity in J. Finally, we compare the porous-elasticity problem with the quasistatic problem in terms of the upper-semicontinuity of their attractors as J -> 0.
ASYMPTOTIC ANALYSIS
(2023)
Article
Mathematics, Applied
Victor R. Cabanillas Zannini, Teofanes Quispe Mendez, Anderson J. A. Ramos
Summary: This paper studies the stabilization of a laminated beam system under a new thermal coupling. By considering factors such as interfacial slip, the study generalizes the work of Almeida Junior and obtains stability results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Materials Science, Multidisciplinary
Mirelson M. Freitas, Dilberto S. Almeida Junior, Mauro L. Santos, Anderson J. A. Ramos, Ronal Q. Caljaro
Summary: This paper studies the long-time dynamics of a Timoshenko system and proves the existence of smooth finite dimensional global attractor and exponential attractor.
MATHEMATICS AND MECHANICS OF SOLIDS
(2023)
Article
Mathematics, Applied
M. M. Freitas, A. O. Ozer, G. Liu, A. J. A. Ramos, E. R. N. Fonseca
Summary: This paper presents a study on a fully-dynamic piezoelectric beam model with a nonlinear force acting on the beam's longitudinal displacements. The motion equations follow a system of non-compactly coupled wave equations. By discarding magnetic effects and decoupling the equations to a single wave equation, the authors investigate the existence of smooth finite-dimensional global attractors. Additionally, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Furthermore, a singular limit problem is considered to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, allowing for a comparison of attractors in the two models.
EVOLUTION EQUATIONS AND CONTROL THEORY
(2023)
Article
Mathematics, Applied
A. J. A. Ramos, A. D. S. Campelo, D. S. Almeida Junior, M. M. Freitas, R. C. Barbosa
Summary: In this paper, a system composed of two parallel wave equations and a heat diffusion equation is studied. Theoretical results on the existence and uniqueness of solution are presented, and the exponential stabilization of the associated semigroup is proved. The semi-discrete problem in finite differences is analyzed, and the energy method is introduced for the first time in the literature to prove the exponential stabilization of the corresponding semi-discrete system. Finally, a fully discrete finite difference scheme is proposed, which combines explicit and implicit integration methods, and numerical simulations are conducted to illustrate the theoretical results.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
M. M. Freitas, A. J. A. Ramos, M. Aouadi, D. S. Almeida Junior
Summary: In this paper, the long-time dynamics of the Bresse system under mixed homogeneous Dirichlet-Neumann boundary conditions is studied, where the heat conduction is governed by Cattaneo's law. The damping effect is only applied to the shear angle displacement, while the vertical and longitudinal displacements are not restricted. By making general assumptions on the source term and utilizing the semigroup theory, the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy are established. Furthermore, the upper semicontinuity with respect to the relaxation time tau approaching zero is proven.
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL
(2023)
Article
Materials Science, Multidisciplinary
A. J. A. Ramos, D. S. Almelda Junior, M. Aouadi, M. M. Freitas, R. C. Barbosa
Summary: In this paper, necessary and sufficient conditions for obtaining stability properties are provided for the one-dimensional Lord-Shulman thermoelastic theory with porosity subject to microtemperature but without temperature. The microtemperature conduction equations are governed by Cattaneo-Maxwell's law. A stability number ?(0) involving all coefficients of the system is introduced based on recent results, and it is proven that the exponential decay of the corresponding semigroup holds if and only if ?(0) = 0. Otherwise, it is shown that the system loses exponential stability and its solution decays polynomially with a rate equal to 1/vt.
MATHEMATICS AND MECHANICS OF SOLIDS
(2023)
Article
Thermodynamics
A. . J. A. . Ramos, L. G. R. Miranda, M. M. Freitas, R. Kovacs
Summary: In this paper, the authors revisit the Guyer-Krumhansl heat equation and show that by satisfying thermodynamic conditions and the maximum principle, the occurrence of negative temperatures can be avoided. The study further emphasizes the thermodynamic origin of heat equations and their compatibility with the second law, and explores two different approaches to determine the initial state in a thermodynamically compatible way. Computational simulations provide support for the findings.
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2023)
Article
Mathematics, Applied
Anderson J. A. Ramos, Mauro A. Rincon, Rodrigo L. R. Madureira, Mirelson M. Freitas
Summary: In this article, the problem of heat conduction in carbon nanotubes modeled like Timoshenko beams is considered. The well-posedness of the problem and the exponential stabilization of the total energy of the system of differential equations are proven using the theory of semigroups of linear operators. The fully discrete problem is analyzed using a finite difference scheme, showing numerical energy construction and simulations that validate the theoretical results and demonstrate convergence rates.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
