Article
Thermodynamics
Suman Bala, Aarti Khurana, S. K. Tomar
Summary: A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. The study establishes Lagrange identity and proves four theorems regarding uniqueness of solution, continuous dependence of solution on loads and initial data, and reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is avoided in the analysis.
JOURNAL OF THERMAL STRESSES
(2021)
Article
Mathematics, Applied
Frederick Maes, Karel Van Bockstal
Summary: The study investigates an isotropic thermoelastic system with dual-phase-lag heat conduction and formulates the variational formulation for the associated coupled system. By using a time discretization method, it shows the existence of a unique weak solution under lower regularity assumptions on the data.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mechanics
Md Abul Kashim Molla, Sadek Hossain Mallik
Summary: This study proposes a novel multi-field coupled mathematical model for thermoelastic diffusion, considering the modified Fourier's law of heat conduction, the modified Fick's law of mass diffusion, and the concept of phase lag. The model includes variational principle and proof of uniqueness of solution, and a reciprocity theorem is established.
MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES
(2023)
Article
Mathematics, Applied
Jose Vanterler da C. Sousa, Daniela S. Oliveira, Gastao S. F. Frederico, Delfim F. M. Torres
Summary: We present a new version of ?-Hilfer fractional derivative on arbitrary time scales and investigate its fundamental properties. We derive an integration by parts formula and propose a ?-Riemann-Liouville fractional integral using Laplace transform. We demonstrate the applicability of these new operators by studying a fractional initial value problem and prove the existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial, suggesting new directions for further research. The article concludes with comments and future work.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mechanics
Jose R. Fernandez, Marta Pellicer, Ramon Quintanilla
Summary: In this article, we investigate how the solutions of the Moore-Gibson-Thompson thermoelasticity vary after a change of the relaxation parameter or the conductivity rate parameter, although, in the second case, only for radial solutions. The results focus on the structural stability. We also obtain the convergence of the Moore-Gibson-Thompson thermoelasticity to the type III thermoelasticity and the convergence of the Moore-Gibson-Thompson thermoelasticity to the Lord-Shulman thermoelasticity in the case of radial solutions.
MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES
(2023)
Article
Mathematics
Allaberen Ashyralyev, Sa'adu Bello Mu'azu
Summary: This work studies the initial boundary value problem (IBVP) for a semi-linear delay differential equation in a Banach space with unbounded positive operators. The main theorem establishes the uniqueness and existence of a bounded solution (BS) for this problem. Applications of the main theorem to four different semi-linear delay parabolic differential equations are presented. The paper also considers the first- and second-order accuracy difference schemes (FSADSs) for solving a one-dimensional semi-linear time-delay parabolic equation, presenting new numerical results and their discussion.
Article
Engineering, Mechanical
Huanting Li, Yunfei Peng, Kuilin Wu
Summary: This paper investigates the existence and uniqueness of solutions for nonlinear on-off switched systems with switching at variable times, as well as the continuous dependence and differentiability of the solutions with respect to the initial state. Additionally, the switching phenomenon and periodic solutions of the systems are also discussed.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Huanting Li, Yunfei Peng, Kuilin Wu
Summary: This paper deals with the qualitative theory for a class of nonlinear differential equations with switching at variable times, including the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters, and the stability of the system. The global exponential stability is discussed, with necessary and sufficient conditions for SSVT. The paper also investigates the continuous dependence and differentiability of the solution with respect to the initial state and the switching line.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics, Applied
Maria Carmela De Bonis, Donatella Occorsio
Summary: The paper discusses the approximate solution of Prandtl's type of integro-differential equations using quadrature methods with optimal Lagrange interpolation processes. It proves the stability and convergence of these methods in suitable weighted spaces of continuous functions. The method's efficiency has been validated through numerical experiments, including comparisons with other numerical procedures. As an application, the method has been successfully implemented to solve Prandtl's equation governing circulation air flow around the contour of elliptic or rectangular wing profiles.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematical & Computational Biology
Apassara Suechoei, Parinya Sa Ngiamsunthorn, Waraporn Chatanin, Somchai Chucheepsakul, Chainarong Athisakul, Danuruj Songsanga, Nuttanon Songsuwan
Summary: This article investigates the equilibrium configurations of a cantilever beam, discussing the minimization of a generalized total energy functional and the solution to the boundary value problem. It also explores the dependence of solutions on the parameters of the problem and presents the Adomian decomposition method for approximating the solution. Numerical results are provided to support the theoretical analysis.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2022)
Article
Mathematics, Applied
Marek Kryspin, Janusz Mierczynski
Summary: This paper demonstrates the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions made in this study are very weak, only requiring convergence in the weak-* topology of delay coefficients. The results are important for the application of Lyapunov exponents theory in investigating PDEs with delay.
JOURNAL OF EVOLUTION EQUATIONS
(2023)
Article
Mathematics, Applied
Sedigheh Sabermahani, Yadollah Ordokhani
Summary: This manuscript presents a numerical method for solving delay differential equations with piecewise constant delays, using GLHFs and collocation method. The accuracy and validity of the proposed method are demonstrated through numerical examples, such as tumor growth in mice and HIV infection of CD4(+) T-cells. Additionally, a bibliometric analysis of delay differential equations is conducted in this study.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mechanics
Md Abul Kashim Molla, Sadek Hossain Mallik
Summary: This paper derives a variational principle for a higher order time-fractional four-phase-lag generalized thermoelastic diffusion model, proves uniqueness of solutions for the governing field equations of the model under suitable conditions, and obtains a reciprocity theorem for the model.
MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES
(2023)
Article
Mathematics
N. Bazarra, J. R. Fernandez, R. Quintanilla
Summary: This paper investigates the application of multi-dimensional dual-phase-lag problem in porous thermoelasticity with microtemperatures. The existence and uniqueness results are proved using the theory of semigroup of linear operators. The numerical study using finite element method and Euler scheme provides a fully discrete approximation, with demonstrated discrete stability property and a priori error estimates. Numerical simulations are performed to validate the accuracy of the approximation and the behavior of the solution in one- and two-dimensional problems.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics
Kevin E. M. Church
Summary: This paper proves that under natural conditions, an impulsive differential equation with state-dependent delay typically exhibits non-uniqueness of solutions. However, uniqueness of solutions can be recovered by applying a specific condition. Additionally, the paper provides a result on linear stability when the uniqueness of solutions is uncertain, and applies it to a specific equation model.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)