1,938 results
Search Results
152. A modified variable‐order fractional SIR model to predict the spread of COVID‐19 in India
- Author
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Abhishek Kumar Singh, Mani Mehra, and Samarth Gulyani
- Subjects
Variable (computer science) ,Lakh ,Coronavirus disease 2019 (COVID-19) ,Order (exchange) ,General Mathematics ,Statistics ,General Engineering ,Fraction (mathematics) ,Epidemic model ,Differential evolution algorithm ,Economic consequences ,Mathematics - Abstract
The first case of COVID-19 in India detected on January 30, 2020, after its emergence in Wuhan, China, in December 2019. The lockdown was imposed as anemergency measure by the Indian government to prevent the spread of COVID-19 but gradually eased out due to its vast economic consequences. Just 15?days after the relaxation of lockdown restrictions, Delhi became India's worst city in terms of COVID-19 cases. In this paper, we propose a variable-order fractional SIR (susceptible, infected, removed) model at state-level scale. We introduce a algorithm that uses the differential evolution algorithm in combination with Adam?Bashforth?Moulton method to learn the parameters in a system of variable-order fractional SIR model. The model can predict the confirm COVID-19 cases in India considering the effects of nationwide lockdown and the possible estimate of the number of infliction inactive cases after the removal of lockdown on June 1, 2020. A new parameter p is introduced in the classical SIR model representing the fraction of infected people that get tested and are thereby quarantined. The COVID-19 trajectory in Delhi, as per our model, predicts the slowing down of the spread between January and February 2021, touching a peak of around 5 lakh confirmed cases.
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- 2021
153. Global strong solution to a thermodynamic compressible diffuse interface model with temperature‐dependent heat conductivity in 1D
- Author
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Yazhou Chen, Qiaolin He, Xiaoding Shi, and Bin Huang
- Subjects
Shock wave ,Interface model ,General Mathematics ,Degenerate energy levels ,General Engineering ,Mechanics ,Physics::Fluid Dynamics ,Nonlinear system ,Mathematics - Analysis of PDEs ,Thermal conductivity ,Flow (mathematics) ,Phase (matter) ,FOS: Mathematics ,35Q30, 76T30, 35C20 ,Compressibility ,Current (fluid) ,Compressible navier stokes equations ,Allen–Cahn equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we investigate the wellposedness of the non-isentropic compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity proportional to a positive power of the temperature. This system describes the flow of a two-phase immiscible heat-conducting viscous compressible mixture. The phases are allowed to shrink or grow due to changes of density in the fluid and incorporates their transport with the current. We established the global existence and uniqueness of strong solutions for this system in 1-D, which means no phase separation, vacuum, shock wave, mass or heat or phase concentration will be developed in finite time, although the motion of the two-phase immiscible flow has large oscillations and the interaction between the hydrodynamic and phase-field effects is complex. Our result can be regarded as a natural generalization of the Kazhikhov-Shelukhin's result ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressible single-phase flow with constant heat conductivity to the non-isentropic compressible immiscible two-phase flow with degenerate and nonlinear heat conductivity.
- Published
- 2021
154. Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise
- Author
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Jean Daniel Mukam and Antoine Tambue
- Subjects
General Mathematics ,Numerical analysis ,finite element method ,General Engineering ,White noise ,Exponential integrator ,VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 ,Noise (electronics) ,Finite element method ,strong convergence ,Stochastic partial differential equation ,Galerkin projection method ,Nonlinear system ,symbols.namesake ,Wiener process ,symbols ,Applied mathematics ,stochastic convection–reaction–diffusion equations ,additive noise ,exponential integrators ,Mathematics - Abstract
In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided
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- 2021
155. On asymptotically statistical equivalent functions on time scales
- Author
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Selma Altundağ and Bayram Sözbir
- Subjects
Modulo operation ,General Mathematics ,General Engineering ,Applied mathematics ,Statistical convergence ,Mathematics - Abstract
In this paper, we introduce the concepts of asymptotically f-statistical equivalence, asymptotically f-lacunary statistical equivalence, and strong asymptotically f-lacunary equivalence for non-negative two delta measurable real-valued functions defined on time scales with the aid of modulus function f. Furthermore, the relationships between these new concepts are investigated. We also present some inclusion theorems.
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- 2021
156. Grüss type inequalities via generalized fractional operators
- Author
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Ahmet Ocak Akdemir, Saad Ihsan Butt, Malik Ali Raza, Muhammad Nadeem, and Belirlenecek
- Subjects
generalized fractional integral operators ,Integral-Inequalities ,Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,General Engineering ,Gruss type inequalities ,Type (model theory) ,Chebyshev Type Inequalities ,Mathematics ,media_common - Abstract
One of the main motivation points in studies on inequalities is to obtain generalizations and to introduce new approaches. In this direction, the generalized fractional integral operators defined within the scope of fractional analysis are quite functional. In this paper, some new integral inequalities have been proved by using generalized fractional integral operators and some classical inequalities for integrable functions. In the proofs of the main findings, the definitions of the generalized fractional integral operator, certain classical relations, and some classical inequalities are used., H.E.C. Pakistan under NRPU [7906]; H.E.C. Pakistan [7906], H.E.C. Pakistan under NRPU project 7906; H.E.C. Pakistan, Grant/Award Number: 7906
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- 2021
157. Global existence to a three-dimensional non-linear thermoelasticity system arising in shape memory materials
- Author
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Irena Paw ow and Wojciech M. Zaj czkowski
- Subjects
Nonlinear system ,Mechanical dissipation ,General Mathematics ,General Engineering ,Calculus ,Elastic energy ,Fixed-point theorem ,Applied mathematics ,Shape-memory alloy ,Thermal conduction ,Energy (signal processing) ,Viscoelasticity ,Mathematics - Abstract
This paper is concerned with the unique global solvability of a three-dimensional (3-D) non-linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non-convex elastic energy and non-linear heat conduction with mechanical dissipation. The present paper extends the previous 2-D existence result of the authors Reference [1] to 3-D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.
- Published
- 2005
158. Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators
- Author
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Bernd Hofmann
- Subjects
Semi-elliptic operator ,Pseudo-monotone operator ,Operator (computer programming) ,Multiplication operator ,General Mathematics ,Mathematical analysis ,General Engineering ,Finite-rank operator ,Operator theory ,Compact operator ,Shift operator ,Mathematics - Abstract
The object of this paper is threefold. First, we investigate in a Hilbert space setting the utility of approximate source conditions in the method of Tikhonov–Phillips regularization for linear ill-posed operator equations. We introduce distance functions measuring the violation of canonical source conditions and derive convergence rates for regularized solutions based on those functions. Moreover, such distance functions are verified for simple multiplication operators in L2(0, 1). The second aim of this paper is to emphasize that multiplication operators play some interesting role in inverse problem theory. In this context, we give examples of non-linear inverse problems in natural sciences and stochastic finance that can be written as non-linear operator equations in L2(0, 1), for which the forward operator is a composition of a linear integration operator and a non-linear superposition operator. The Frechet derivative of such a forward operator is a composition of a compact integration and a non-compact multiplication operator. If the multiplier function defining the multiplication operator has zeros, then for the linearization an additional ill-posedness factor arises. By considering the structure of canonical source conditions for the linearized problem it could be expected that different decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely different ill-posedness situations. As third we apply the results on approximate source conditions to such composite linear problems in L2(0, 1) and indicate that only integrals of multiplier functions and not the specific character of the decay of multiplier functions in a neighbourhood of a zero determine the convergence behaviour of regularized solutions. Copyright © 2005 John Wiley & Sons, Ltd.
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- 2005
159. Long time behaviour of a flow in infinite pipe conforming to slip boundary conditions
- Author
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Witold Sadowski and Piotr B. Mucha
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General Mathematics ,Mathematical analysis ,General Engineering ,Geometry ,Slip (materials science) ,Inflow ,Upper and lower bounds ,Pipe flow ,Physics::Fluid Dynamics ,Hausdorff dimension ,Attractor ,Boundary value problem ,Navier–Stokes equations ,Mathematics - Abstract
The paper analyses long time behaviour of solutions of the Navier–Stokes equations in a two-dimensional pipe-like domain. The system is studied with perfect slip boundary conditions with arbitrary inflow conditions at infinity. The main results show the existence of global in time solutions and of an attractor for the dynamical system generated by the model. The paper also establishes an upper bound for the Hausdorff dimension of the attractor. Copyright © 2005 John Wiley & Sons, Ltd.
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- 2005
160. Spectral convergence for vibrating systems containing a part with negligible mass
- Author
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Eugenia Pérez
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General Mathematics ,Bounded function ,Dimension (graph theory) ,Mathematical analysis ,General Engineering ,Boundary value problem ,Half-space ,Space (mathematics) ,Laplace operator ,Domain (mathematical analysis) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SUMMARY We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domainR of R n , with n =2 ; 3, which contains axed bounded domain B where the density takes the value 1 and 0 outside. � R has a diameter depending on a parameter R, with R?1, diam(� R) →∞ as R →∞ and the union of these sets is the whole space R n (the half space {x ∈ R n =x ni0}, respectively). Depending on the dimension of the space n, and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R →∞ . We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright ? 2005 John Wiley & Sons, Ltd. This paper deals with the approach to spectral problems for the Laplace operator in unbounded domains via problems in bounded domains. These kinds of spectral problems can appear in dierentelds of mechanics. This is the case, for instance, of the study of local behaviours of eigenmodes for vibrating systems containing either parts with negligible mass or concen- trated masses. We also apply the approach in this paper to obtain certain remarkable spectral properties for vibrating systems with concentrated masses. As is well known, the study of the vibrations of certain mechanical systems containing a part with negligible small mass leads us to the study of the spectral problem: Findand
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- 2005
161. Mathematical models of therapeutical actions related to tumour and immune system competition
- Author
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Pierre Emmanuel Jabin and Elena De Angelis
- Subjects
Immune defense ,Competition (economics) ,Immune system ,Qualitative analysis ,Angiogenesis ,General Mathematics ,General Engineering ,Mathematical economics ,Neuroscience ,Analyse qualitative ,Mathematics - Abstract
This paper deals with the qualitative analysis of a model related to the description of two medical therapies which have been intensively developed in recent years. In particular, we refer to the modelling of the actions applied by proteins, to activate the immune defense, and to the control of angiogenesis, to contrast the growth of tumour cells by preventing the feeding actions of endothelial cells. The therapeutical actions which are object of the modelling process developed in this paper have to be regarded as applied within the framework of the competition between the immune system and tumour cells. We prove the existence of solutions to the Cauchy problem related to the model. The efficiency of the therapies and the asymptotic behaviour in time of our solutions is also investigated.
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- 2005
162. Higher order non-resonance for differential equations with singularities
- Author
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Ping Yan and Meirong Zhang
- Subjects
Differential equation ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,General Engineering ,Infinity ,Resonance (particle physics) ,law.invention ,symbols.namesake ,Invertible matrix ,Mathieu function ,law ,symbols ,Order (group theory) ,Gravitational singularity ,Eigenvalues and eigenvectors ,Mathematical physics ,Mathematics ,media_common - Abstract
In this paper we prove an existence result of positive periodic solutions to second order differential equations with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. Different from the nonsingular case, the result in this paper shows that both of the periodic and the antiperiodic eigenvalues play the same role in such an existence result. Copyright © 2003 John Wiley & Sons, Ltd.
- Published
- 2003
163. Asymptotic and spectral analysis of non-selfadjoint operators generated by a filament model with a critical value of a boundary parameter
- Author
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Marianna A. Shubov
- Subjects
Operator (computer programming) ,General Mathematics ,Spectrum (functional analysis) ,Mathematical analysis ,General Engineering ,Boundary (topology) ,Boundary value problem ,Mixed boundary condition ,Operator theory ,Eigenfunction ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We consider a class of non-selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non-conservative boundary conditions at one end and a heavy load at the other end. The filament has a non-constant density and is subject to a viscous damping with a non-constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24(15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non-selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24(15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.
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- 2003
164. Multiplicity of solutions for a class of non-symmetric eigenvalue hemivariational inequalities
- Author
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Claudiu Ciulcu, Dumitru Motreanu, and Vicenţiu D. Rădulescu
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Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Non symmetric ,General Engineering ,Perturbation (astronomy) ,Multiplicity (mathematics) ,Natural topology ,Symmetric case ,Variational inequality ,Eigenvalues and eigenvectors ,Mathematics ,media_common - Abstract
SUMMARY 7 The aim of this paper is to establish the in;uence of a non-symmetric perturbation for a symmetric hemivariational eigenvalue inequality with constraints. The original problem was studied by Goeleven 9 et al. (Math. Methods Appl. Sci. 1997; 20:548) who deduced the existence of in%nitely many solutions for the symmetric case. In this paper it is shown that the number of solutions of the perturbed problem 11 becomes larger and larger if the perturbation tends to zero with respect to a natural topology. The approach relies on topological methods in non-smooth critical point theory leading to this new multiplicity 13 information. Copyright ? 2002 John Wiley & Sons, Ltd.
- Published
- 2003
165. An iterative method for optimal control of bilateral free boundaries problem
- Author
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Youness El Yazidi and Abdellatif Ellabib
- Subjects
Computer science ,Iterative method ,General Mathematics ,010102 general mathematics ,General Engineering ,Inverse problem ,Optimal control ,01 natural sciences ,Regularization (mathematics) ,Finite element method ,010101 applied mathematics ,Robustness (computer science) ,Conjugate gradient method ,Applied mathematics ,Gravitational singularity ,Shape gradient ,Shape optimization ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to construct a numerical scheme for solving a class of bilateral free boundaries problem. First, using a shape functional and some regularization terms, an optimal control problem is formulated, in addition, we prove its solution existence's. The first optimality conditions and the shape gradient are computed. the proposed numerical scheme is a genetic algorithm guided conjugate gradient combined with the finite element method, at each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results, and to prove the robustness of the proposed scheme.
- Published
- 2021
166. The number of Dirac‐weighted eigenvalues of Sturm–Liouville equations with integrable potentials and an application to inverse problems
- Author
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Xiao Chen and Jiangang Qi
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Dirac (software) ,General Engineering ,Dirac delta function ,Sturm–Liouville theory ,Mathematics::Spectral Theory ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Dirichlet eigenvalue ,Distribution (mathematics) ,Mathematics - Classical Analysis and ODEs ,Dirichlet boundary condition ,34A06, 34A55, 34B09 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Complex number ,Eigenvalues and eigenvectors ,Mathematics ,Characteristic polynomial - Abstract
In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages
- Published
- 2021
167. Dealing with variability in ecological modelling: An analysis of a random non‐autonomous logistic population model
- Author
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Marc Jornet, Julia Calatayud, Fabio Antonio Dorini, and Juan Carlos Cortés
- Subjects
General Mathematics ,Ecological modelling ,Logistic growth model ,logistic growth model ,General Engineering ,Probability density function ,Random parameters ,92D25 ,Time-varying carrying capacity ,Population model ,34F05 ,time-varying carrying capacity ,Statistics ,92D40 ,random parameters ,probability densityfunction ,Logistic function ,MATEMATICA APLICADA ,Mathematics - Abstract
[EN] This paper presents a methodology to deal with the randomness associated to ecological modelling. Data variability makes it necessary to analyse the impact of random perturbations on the fitted model parameters. We conduct such analysis for the logistic growth model with a certain sigmoid functional form of the carrying capacity, which was proposed in the literature for the study of parasite growth during infection. We show how the probability distributions of the parameters are set via the maximum entropy principle. Then the random variable transformation method allows for computing the density function of the population., Spanish Ministerio de Economia y Competitividad, Grant/Award Number: PID2020-115270GB-I00; Agencia Estatal de Investigacion
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- 2021
168. Mathematical models for the improvement of detection techniques of industrial noise sources from acoustic images
- Author
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Gianluca Vinti, Giorgio Baldinelli, Francesco Bianchi, Francesco D'Alessandro, Marco Seracini, Danilo Costarelli, Francesco Asdrubali, Flavio Scrucca, Asdrubali, Francesco, Baldinelli, Giorgio, Bianchi, Francesco, Costarelli, Danilo, D'Alessandro, Francesco, Scrucca, Flavio, Seracini, Marco, and Vinti, Gianluca
- Subjects
Beamforming ,acoustic images, applied mathematics, beamforming, image reconstruction, industrial noise, sam-pling Kantorovich algorithm ,Mathematical model ,business.industry ,General Mathematics ,industrial noise ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,General Engineering ,acoustic images ,Industrial noise ,Iterative reconstruction ,image reconstruction ,sampling Kantorovich algorithm ,beamforming ,applied mathematics ,Computer vision ,Artificial intelligence ,business ,Mathematics - Abstract
In this paper, a procedure for the detection of the sources of industrial noise and the evaluation of their distances is introduced. The above method is based on the analysis of acoustic and optical data recorded by an acoustic camera. In order to improve the resolution of the data, interpolation and quasi interpolation algorithms for digital data processing have been used, such as the bilinear, bicubic, and sampling Kantorovich (SK). The experimental tests show that the SK algorithm allows to perform the above task more accurately than the other considered methods.
- Published
- 2021
169. On the parabolic‐elliptic Keller–Segel system with signal‐dependent motilities: A paradigm for global boundedness and steady states
- Author
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Zhi-An Wang
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,35A01, 35B44, 35K57, 35Q92, 92C17 ,Space (mathematics) ,01 natural sciences ,Signal ,Domain (mathematical analysis) ,Quantitative Biology::Cell Behavior ,Quantitative Biology::Subcellular Processes ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Chemical signal ,Bounded function ,FOS: Mathematics ,Neumann boundary condition ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal-dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non-constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller-Segel system with signal-dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.
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- 2021
170. Asymptotic approach to anti‐plane dynamic problem of asymmetric three‐layered composite plate
- Author
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Rab Nawaz, Rahmatullah Ibrahim Nuruddeen, and Q. M. Zaigham Zia
- Subjects
Shear (sheet metal) ,Dynamic problem ,Plane (geometry) ,Composite plate ,General Mathematics ,Dispersion relation ,Mathematical analysis ,General Engineering ,Range (statistics) ,Motion (geometry) ,Material properties ,Mathematics - Abstract
In this paper, the anti-plane shear motion of an asymmetric three-layered inhomogeneous elastic plate has been examined. An asymptotic approach is employed for the present investigation. Both the generalized and unified dispersion relations within the long-wave low-frequency range have been determined. The obtained unified dispersion relation is investigated taking into account the recently analyzed material contrast for layered plate with mixed stiff-soft layers of different material properties. Finally, we make comparison with symmetric plate being a special case of the asymmetric plate under consideration in the end.
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- 2021
171. A generalized fractional ( q , h )–Gronwall inequality and its applications to nonlinear fractional delay ( q , h )–difference systems
- Author
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Feifei Du and Baoguo Jia
- Subjects
010101 applied mathematics ,Nonlinear system ,Uniqueness theorem for Poisson's equation ,Stability criterion ,General Mathematics ,Gronwall's inequality ,010102 general mathematics ,General Engineering ,Applied mathematics ,Uniqueness ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, a generalized fractional $(q,h)$-Gronwall inequality is investigated. Based on this inequality, we derive the uniqueness theorem and the finite-time stability criterion of nonlinear fractional delay $(q,h)$-difference systems. Several examples are given to illustrate our theoretical result.
- Published
- 2021
172. On the binomial transforms of the Horadam quaternion sequences
- Author
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Arzu Özkoç Öztürk, Faruk Kaplan, and [Belirlenecek]
- Subjects
Sequence ,Pure mathematics ,quaternion ,Recurrence relation ,Binomial (polynomial) ,General Mathematics ,010102 general mathematics ,General Engineering ,Generating function ,Order (ring theory) ,Horadam quaternions ,01 natural sciences ,binomial transform ,010101 applied mathematics ,Iterated function ,Binomial transform ,iterated binomial transform ,0101 mathematics ,Quaternion ,Mathematics - Abstract
The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform. WOS:000638679800001 2-s2.0-85104113068
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- 2021
173. Constraint minimizers of inhomogeneous mass subcritical minimization problems
- Author
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Shuai Li and Yongshuai Gao
- Subjects
Physics ,010101 applied mathematics ,Combinatorics ,Constraint (information theory) ,General Mathematics ,010102 general mathematics ,General Engineering ,Nabla symbol ,Minification ,0101 mathematics ,01 natural sciences ,Mathematics ,Energy functional - Abstract
This paper considers minimizers of the following inhomogeneous $L^2$-subcritical energy functional \[E(u):=\int_{\R^N}|\nabla u|^{2}dx-\frac{2}{p+1}\int_{\R^N}m(x)|u|^{p+1}dx,%\ u\in H^{1}(\R^N), \] under the mass constraint $\|u\|^{2}_{2}=M$. Here $N\geq1$, $p\in(1,1+\frac{4}{N})$, $M>0$ and the inhomogeneous term $m(x)$ satisfies $0
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- 2021
174. Modeling and optimal control analysis of COVID‐19: Case studies from Italy and Spain
- Author
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Mini Ghosh, Xue-Zhi Li, Liming Cai, and Akhil Kumar Srivastav
- Subjects
Computer science ,Download ,General Mathematics ,Population ,Control (management) ,Disease ,Interval (mathematics) ,Permission ,34a34 ,01 natural sciences ,law.invention ,optimal control ,sensitivity analysis ,basic reproduction number ,law ,Pandemic ,Statistics ,Sensitivity (control systems) ,0101 mathematics ,education ,Research Articles ,Mathematics ,education.field_of_study ,Actuarial science ,COVID‐19 model ,49j15 ,010102 general mathematics ,Warranty ,General Engineering ,92b05 ,Optimal control ,010101 applied mathematics ,Transmission (mechanics) ,parameter estimation ,Basic reproduction number ,Research Article - Abstract
Coronavirus disease 2019 (COVID‐19) is a viral disease which is declared as a pandemic by WHO This disease is posing a global threat, and almost every country in the world is now affected by this disease Currently, there is no vaccine for this disease, and because of this, containing COVID‐19 is not an easy task It is noticed that elderly people got severely affected by this disease specially in Europe In the present paper, we propose and analyze a mathematical model for COVID‐19 virus transmission by dividing whole population in old and young groups We find disease‐free equilibrium and the basic reproduction number (R0) We estimate the parameter corresponding to rate of transmission and rate of detection of COVID‐19 using real data from Italy and Spain by least square method We also perform sensitivity analysis to identify the key parameters which influence the basic reproduction number and hence regulate the transmission dynamics of COVID‐19 Finally, we extend our proposed model to optimal control problem to explore the best cost‐effective and time‐dependent control strategies that can reduce the number of infectives in a specified interval of time [ABSTRACT FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use This abstract may be abridged No warranty is given about the accuracy of the copy Users should refer to the original published version of the material for the full abstract (Copyright applies to all Abstracts )
- Published
- 2021
175. Existence and stability for a nonlinear hybrid differential equation of fractional order via regular Mittag–Leffler kernel
- Author
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Ibrahim Slimane, Juan J. Nieto, Thabet Abdeljawad, and Zoubir Dahmani
- Subjects
Mathematics::Functional Analysis ,Nonlinear system ,Differential equation ,General Mathematics ,Kernel (statistics) ,Mathematics::Classical Analysis and ODEs ,General Engineering ,Order (group theory) ,Applied mathematics ,Contraction (operator theory) ,Stability (probability) ,Mathematics ,Fractional calculus - Abstract
This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed-point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem.
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- 2021
176. Nonoscillation of half‐linear dynamic equations on time scales
- Author
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Michal Veselý, Petr Hasil, Michal Pospíšil, and Jozef Kiselak
- Subjects
Ideal (set theory) ,Oscillation ,General Mathematics ,010102 general mathematics ,General Engineering ,Qualitative theory ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Riccati equation ,Applied mathematics ,0101 mathematics ,Dynamic equation ,Linear equation ,Mathematics - Abstract
The research contained in this paper belongs to the qualitative theory of dynamic equations on time scales. Via the detailed analysis of solutions of the associated Riccati equation and an advanced averaging technique, we provide the description of domain of nonoscillation of very general equations. The results are formulated and proved for half-linear equations (i.e., equations connected to PDEs with one dimensional p-Laplacian) on time scales. Nevertheless, we obtain new results also for linear difference equations. Moreover, the combination of the presented results with previous ones shows that many useful equations are conditionally oscillatory. Such equations are ideal as testing and comparison equations in real-world models which are beyond capabilities of known oscillation and nonoscillation criteria often.
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- 2021
177. Phase-field systems for multi-dimensional Prandtl-Ishlinskii operators with non-polyhedral characteristics
- Author
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Pavel Krejčí and Jürgen Sprekels
- Subjects
Series (mathematics) ,Measurable function ,General Mathematics ,Mathematical analysis ,General Engineering ,Regular polygon ,parabolic systems ,Field (mathematics) ,hysteresis operators ,Prandtl-Ishlinskii operators ,74N30 ,34C55 ,35K60 ,Phase-field systems ,phase transitions ,Hysteresis ,Polyhedron ,Compact space ,Convergence (routing) ,Applied mathematics ,80A22 ,47J40 ,Mathematics - Abstract
Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase-field models in which hysteresis non-linearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so-called Prandtl–Ishlinskii operators. For these operators, the corresponding phase-field systems are in the multi-dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi-dimensional Prandtl–Ishlinskii operators having non-polyhedral convex characteristicsets. Copyright © 2002 John Wiley & Sons, Ltd.
- Published
- 2002
178. Non-standard Stokes and Navier-Stokes problems: existence and regularity in stationary case
- Author
-
J. M. Bernard
- Subjects
General Mathematics ,Viscosity (programming) ,Stationary case ,Mathematical analysis ,General Engineering ,Stokes problem ,Boundary (topology) ,Boundary value problem ,Navier stokes ,Bilinear form ,Mathematics - Abstract
This paper is devoted to Stokes and Navier–Stokes problems with non-standard boundary conditions: we consider, in particular, the case where the pressure is given on a part of the boundary. These problems were studied by Begue, Conca, Murat and Pironneau. They proved the existence of variational solutions, indicating that these were solutions of the initial non-standard problems, if they are regular enough, but without specifying the conditions on the data which would imply this regularity. In this paper, first we show that the variational solutions, on supposing pressure on the boundary Γ2 of regularity H1/2 instead of H−1/2, have their Laplacians in L2 and, therefore, are solutions of non-standard Stokes problem. Next, we give a result of regularity H2, which we generalize, obtaining regularities Wm, r, m∈ℕ, m⩾2, r⩾2. Finally, by a fixed-point argument, we prove analogous results for the Navier–Stokes problem, in the case where the viscosity νis large compared to the data. Copyright © 2002 John Wiley & Sons, Ltd.
- Published
- 2002
179. Asymptotics of eigenfrequencies and eigenmodes of non-homogeneous inextensible filament with an end load
- Author
-
Clyde F. Martin, Boris P. Belinskiy, and Marianna A. Shubov
- Subjects
Controllability ,Normal mode ,General Mathematics ,Mathematical analysis ,General Engineering ,Boundary (topology) ,Natural frequency ,Boundary value problem ,Eigenfunction ,Integral equation ,Self-adjoint operator ,Mathematics - Abstract
We consider a class of non-selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non-conservative boundary conditions at one end and a heavy load at the other end. The filament has a non-constant density and is subject to a viscous damping with a non-constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two-parameter family of non-selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.
- Published
- 2001
180. Microbiological fluid mechanics: a tribute to Sir James Lighthill
- Author
-
John Blake
- Subjects
Filter feeding ,General Mathematics ,General Engineering ,Tribute ,Fluid mechanics ,Engineering mathematics ,Mathematics ,Epistemology - Abstract
The choice of title of this presentation comes from two papers that James Lighthill published in the issue of Journal of Engineering Mathematics commemorating H. A. Lorentz's famous 1896 paper on fluid mechanics [1]. This paper will discuss Lighthill's significant contribution to microbiological fluid mechanics, both through his own work and his postgraduate and postdoctoral students, of which this author is one! The paper introduces the wide range of interests that Lighthill had in biofluiddynamics and then moves to concentrate on contributions to microbiological fluid mechanics. The presentation successively considers eukaryotic flagellar propulsion, prokaryotic flagellar propulsion, filter feeding and ciliary propulsion. In the Spirit of Lighthill, the paper concludes with a list of topics of research in the area that have been identified for further study.
- Published
- 2001
181. Characteristics of rogue waves on a soliton background of the vector Lakshmanan‐Porsezian‐Daniel equation
- Author
-
Lixin Tian and Dong Min-Jie
- Subjects
Breather ,General Mathematics ,010102 general mathematics ,General Engineering ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Interference (wave propagation) ,01 natural sciences ,010101 applied mathematics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Transformation (function) ,Classical mechanics ,Soliton ,0101 mathematics ,Rogue wave ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.
- Published
- 2021
182. Quantum Ostrowski‐type integral inequalities for functions of two variables
- Author
-
Muhammad Ali, Hüseyin Budak, Tuba Tunç, and [Belirlenecek]
- Subjects
convex function ,Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,General Engineering ,Quantum calculus ,Type (model theory) ,quantum calculus ,q‐ ,Ostrowski inequality ,Convex function ,Quantum ,integral ,Mathematics ,media_common - Abstract
In this study, we established some new inequalities of Ostrowski type for the functions of two variables by using the concept of newly defined double quantum integrals. We also revealed that the results presented in this paper are the consolidation and generalization of some existing results on the literature of Ostrowski inequalities. WOS:000616064200001 2-s2.0-85100764477
- Published
- 2021
183. An accelerated hybrid projection method with a self‐adaptive step‐size sequence for solving split common fixed point problems
- Author
-
Songxiao Li, Zheng Zhou, and Bing Tan
- Subjects
Sequence ,General Mathematics ,010102 general mathematics ,General Engineering ,Hilbert space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Operator (computer programming) ,Robustness (computer science) ,Convergence (routing) ,Projection method ,symbols ,0101 mathematics ,Focus (optics) ,Algorithm ,Dykstra's projection algorithm ,Mathematics - Abstract
This paper attempts to focus on the split common fixed point problem for demicontractive mappings. We give an accelerated hybrid projection algorithm which combines the hybrid projection method and the inertial technique. The strong convergence theorems of this algorithm are obtained under mild conditions by a self-adaptive step-size sequence, which does not need prior knowledge of operator norms. Some numerical experiments in infinite Hilbert space are provided to illustrate the reliability and robustness of the algorithm and also to compare it with existin
- Published
- 2021
184. Dynamics of an infinite age‐structured particle system
- Author
-
Yuri Kozitsky and Dominika Jasinska
- Subjects
Particle system ,education.field_of_study ,Pure mathematics ,Markov chain ,General Mathematics ,010102 general mathematics ,Population ,General Engineering ,Zero (complex analysis) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Quantitative Biology::Populations and Evolution ,Fokker–Planck equation ,0101 mathematics ,education ,Stationary state ,Mathematics ,Probability measure - Abstract
The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $\mu_0 \to \mu_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $\mu$ existing if the emigration rate is separated away from zero. It is then shown that $\mu_t$ weakly converges to $\mu$ as $t\to +\infty$.
- Published
- 2021
185. Steady solutions to a model of compressible chemically reacting fluid with high density
- Author
-
Milan Pokorný and Šimon Axmann
- Subjects
Strong solutions ,Entropy inequality ,Mathematics - Analysis of PDEs ,General Mathematics ,FOS: Mathematics ,General Engineering ,Compressibility ,High density ,Mechanics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider a model describing the steady flow of compressible heat-conducting chemically-reacting multi-component mixture. We show the existence of strong solutions under the additional assumption that the mixture is sufficiently dense. We work in the $L^p$-setting combining the methods for the weak solutions with the method of decomposition. The result is a generalization of our previous papers, where the case of single-constituted fluid was studied.
- Published
- 2021
186. Magnetic confinement at a boundary approximates specular reflection
- Author
-
Katherine Zhiyuan Zhang
- Subjects
Field (physics) ,General Mathematics ,010102 general mathematics ,35Q83, 76X05 ,General Engineering ,FOS: Physical sciences ,Magnetic confinement fusion ,Boundary (topology) ,Mathematical Physics (math-ph) ,Plasma ,01 natural sciences ,Domain (mathematical analysis) ,Magnetic field ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Quantum electrodynamics ,Bounded function ,FOS: Mathematics ,Specular reflection ,0101 mathematics ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We conjecture that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near-boundary region. In this paper, we verify this conjecture for the 1.5D relativistic Vlasov-Maxwell system (RVM) on a bounded domain $\Omega = (0, 1)$ with an external confining magnetic field.
- Published
- 2021
187. New approach to uniformly quasi circular motion of quasi velocity biharmonic magnetic particles in the Heisenberg space
- Author
-
Talat Körpinar
- Subjects
General Mathematics ,Electric potential energy ,General Engineering ,uniformly quasi circular motion ,Space (mathematics) ,quasi plasma ,Magnetic field ,Heisenberg space ,Classical mechanics ,Circular motion ,magnetic flux density ,Biharmonic equation ,Magnetic nanoparticles ,electrical energy ,Mathematics - Abstract
Korpinar, Talat/0000-0003-4000-0892 In this paper, we define concept of the uniformly quasi circular motion (UQCM) with biharmonicity condition in the Heisenberg space. That is, we aim to define a new class of UQCM in the three-dimensional Heisenberg space. We further improve an alternative method to find uniformly quasi circular potential electric energy of biharmonic velocity magnetic particles in the Heisenberg space. We also give the relationships between physical and geometrical characterizations of uniformly quasi circular potential electric energy. Finally, we illustrate important figures for uniformly quasi circular potential electric energy with respect to its electric field in the radial direction.
- Published
- 2021
188. Existence and non-existence of global solutions for a class of non-linear wave equations
- Author
-
Yang Zhijian and Chen Guowang
- Subjects
Class (set theory) ,Partial differential equation ,General Mathematics ,Mathematical analysis ,General Engineering ,Boundary value problem ,Non linear wave ,Finite time ,Wave equation ,Mathematics - Abstract
This paper studies the existence and the non-existence of global solutions to the initial boundary value problems for the non-linear wave equation utt + uxxxx = σ(ux)x + f(x, t) and the Boussinesq-type equation utt + uxxxx = σ(u)xx + f(x, t). The paper proves that every above-mentioned problem has a unique global solution under rather mild confining conditions, and arrives at some sufficient conditions of blow-up of the solutions in finite time. Finally, a few examples are given. Copyright © 2000 John Wiley & Sons, Ltd.
- Published
- 2000
189. Riesz basis property of root vectors of non-self-adjoint operators generated by aircraft wing model in subsonic airflow
- Author
-
Marianna A. Shubov
- Subjects
Laplace transform ,General Mathematics ,Mathematical analysis ,Spectrum (functional analysis) ,General Engineering ,Hilbert space ,Differential operator ,symbols.namesake ,Operator (computer programming) ,symbols ,Boundary value problem ,Self-adjoint operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper is the third in a series of several works devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first two papers (see [33, 34]) and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches, and have derived their precise spectral asymptotics with respect to the eigenvalue number. In the second paper, we have derived the asymptotical approximations for the mode shapes. Based on the asymptotical results of the first two papers, in the present paper, we (a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space; (b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when theremight be not only eigenvectors but associate vectors as well; and (c) prove that the set of the generalized eigenvectors forms a Riesz basis in the energy space. To prove the main result of the paper, we made use of the Nagy-Foias functional model for non-self-adjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform in the forthcoming papers.
- Published
- 2000
190. Existence and blow‐up studies of a p ( x )‐Laplacian parabolic equation with memory
- Author
-
Gnanavel Soundararajan and Lakshmipriya Narayanan
- Subjects
General Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Type (model theory) ,01 natural sciences ,Upper and lower bounds ,010101 applied mathematics ,symbols.namesake ,Dirichlet boundary condition ,symbols ,0101 mathematics ,Finite time ,Laplace operator ,Differential inequalities ,Mathematics - Abstract
In this paper, we establish existence and finite time blow up of weak solutions of a parabolic equation of p(x)-Laplacian type with the Dirichlet boundary condition. Moreover, we obtain upper and lower bounds for the blow up time of solutions, by employing concavity method and differential inequality technique respectively.
- Published
- 2020
191. The gradient descent method from the perspective of fractional calculus
- Author
-
Joel A. Rosenfeld and Pham Viet Hai
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Order (ring theory) ,Unconstrained optimization ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Perspective (geometry) ,Optimization and Control (math.OC) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Gradient descent ,Mathematics - Optimization and Control ,Gradient method ,Mathematics - Abstract
Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $\psi$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $\alpha$ and weight $\psi$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods., Comment: 27 pages
- Published
- 2020
192. Translation‐invariant generalized P ‐adic Gibbs measures for the Ising model on Cayley trees
- Author
-
Otabek Khakimov and Farrukh Mukhamedov
- Subjects
Phase transition ,General Mathematics ,010102 general mathematics ,General Engineering ,Fixed point ,Invariant (physics) ,01 natural sciences ,010101 applied mathematics ,Singularity ,Probability theory ,Physical phenomena ,Applied mathematics ,Ising model ,0101 mathematics ,Mathematics ,p-adic number - Abstract
Main aim of the present paper is explore certain physical phenomena by means of $p$-adic probability theory. To overcome this study, we deal with a more general setting to define $p$-adic Gibbs measures. For the sake of simplicity of explanations, we restrict ourselves to the Ising model on the Cayley tree, since such a model has broad theoretical and practical applications. To study $p$-adic quasi Gibbs measures, we reduce the problem to the description of the fixed points of the Ising-Potts mapping. Finding fixed points is not an easy job as in the real setting. Furthermore, the phase transition for the model is established. In the real case, the phase transition yields the the singularity of the limiting Gibbs measures. However, we show that the $p$-adic quasi Gibbs measures do not exhibit the mentioned type of singularity, such kind of phenomena is called strong phase transition. Finally, we deal with the solvability and the number of solutions of ceratin $p$-adic equation depending on several parameters. Such a description allows us to find all possible translation-invariant $p$a-adic quasi Gibbs measures.
- Published
- 2020
193. Generalized approximate boundary synchronization for a coupled system of wave equations
- Author
-
Yanyan Wang
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Boundary (topology) ,State (functional analysis) ,Kalman filter ,Wave equation ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Matrix (mathematics) ,symbols.namesake ,Synchronization (computer science) ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.
- Published
- 2020
194. Optimal control for nonlocal reaction‐diffusion system describing calcium dynamics in cardiac cell
- Author
-
Mostafa Bendahmane, Fahd Karami, Elmahdi Erraji, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Modélisation et calculs pour l'électrophysiologie cardiaque (CARMEN), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-CHU Bordeaux [Bordeaux], Ecole Supérieure de Technologie d'Essaouira, Université Cadi Ayyad [Marrakech] (UCA), and the research program PPR supported by CNRST (Morocco), project 'Modèles Mathématiques appliquées l’environnement, à l’imagerie médicale et aux biosystèmes'
- Subjects
calcium model ,Computer simulation ,[SDV]Life Sciences [q-bio] ,General Mathematics ,Weak solution ,finite element method ,General Engineering ,nonlocal diffusion ,weak solution ,Mechanics ,Optimal control ,first order optimality conditions ,Cardiac cell ,Finite element method ,Calcium dynamics ,numerical simulation ,Reaction–diffusion system ,[MATH]Mathematics [math] ,Mathematics - Abstract
International audience; The purpose of this paper is to introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction-diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well-posedness result of the adjoint problem using Faedo-Galerkin approximation, a priori estimates and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the 2-norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme.
- Published
- 2020
195. On the transport limit of singularly perturbed convection–diffusion problems on networks
- Author
-
Herbert Egger and Nora Philippi
- Subjects
Asymptotic analysis ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Boundary (topology) ,Order (ring theory) ,Type (model theory) ,Coupling (probability) ,01 natural sciences ,010101 applied mathematics ,Limit (mathematics) ,0101 mathematics ,Convection–diffusion equation ,Conservation of mass ,Mathematics - Abstract
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(\sqrt{\epsilon})$ in the $L^\infty(L^2)$-norm with diffusion $\epsilon \to 0$. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.
- Published
- 2020
196. Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity
- Author
-
Ting Zhang, Daoyuan Fang, and Hui Chen
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Directional derivative ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Vector field ,0101 mathematics ,Navier–Stokes equations ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{u} \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}
- Published
- 2020
197. Local well‐posedness of compressible radiation hydrodynamic equations with density‐dependent viscosities and vacuum
- Author
-
Hao Li and Yachun Li
- Subjects
Viscosity ,Isentropic process ,Density dependent ,General Mathematics ,Radiation hydrodynamics ,Mathematical analysis ,General Engineering ,Compressibility ,Initial value problem ,Radiation ,Well posedness ,Mathematics - Abstract
In this paper, we consider the Cauchy problem for three-dimensional isentropic compressible radiation hydrodynamic equations with density-dependent viscosity coefficients. When the viscosity coefficients are given as power of density ($\rho^\delta$ with $\delta>1$), we establish the local-in-time existence of classical solutions containing a vacuum for large initial data. Here, we point out that the initial layer compatibility conditions are not necessary.
- Published
- 2020
198. Regularity of the solutions of the steady-state Boussinesq equations with thermocapillarity effects on the surface of the liquid
- Author
-
Luc Paquet
- Subjects
Hölder's inequality ,Surface (mathematics) ,Dirichlet problem ,General Mathematics ,Mathematical analysis ,General Engineering ,Hilbert space ,Space (mathematics) ,Sobolev inequality ,symbols.namesake ,symbols ,Heat equation ,Boundary value problem ,Mathematics - Abstract
In this paper we show that every variational solution of the steady-state Boussinesq equations (u, p, θ) with thermocapillarity effect on the surface of the liquid has the following regularity: u ∈ H2(Ω)2, p ∈ H1(Ω), θ ∈ H2(Ω) under appropriate hypotheses on the angles of the ‘2-D’ container (a cross-section of the 3-D container in fact) and of the horizontal surface of the liquid with the inner surface of the container. The difficulty comes from the boundary condition on the surface of the liquid (e.g. water) which modelizes the thermocapillarity effect on the surface of the liquid (equation (68.10) of Levich [7]). More precisely we will show that u ∈ P22(Ω)2 and that θ ∈ P22(Ω), where P22(Ω) denotes the usual Kondratiev space. This result will be used in a forthcoming paper to prove convergence results for finite element methods intended to compute approximations of a non-singular solution [1] of this problem. Copyright © 1999 John Wiley & Sons, Ltd.
- Published
- 1999
199. The Nyström method for solving a class of singular integral equations and applications in 3D-plate elasticity
- Author
-
Andreas Kirsch and Stefan Ritter
- Subjects
Dirichlet problem ,General Mathematics ,Mathematical analysis ,General Engineering ,Hilbert space ,Fredholm integral equation ,Singular integral ,Integral equation ,Sobolev space ,symbols.namesake ,Collocation method ,symbols ,Nyström method ,Mathematics - Abstract
The paper consists of two parts. In the first part we investigate a Nystrom- or product integration method for second kind singular integral equations. We prove an asymptotically optimal error estimate in the scale of Sobolev Hilbert spaces. Although the result can also be obtained as a special case of a discrete iterated collocation method our proof is more direct and uses the Nystrom interpolation. In the second part of this paper we consider the Dirichlet problem for thin elastic plates with transverse shear deformation. The boundary value problem is transformed into a 3 x 3 system of singular Fredholm integral equations of second kind. After discussing existence and uniqueness of the solution to the integral equations in a Sobolev space setting, we apply the Nystrom method to solve the integral equations numerically.
- Published
- 1999
200. Control of the thermoelastic model of a plate activated by shape memory alloy reinforcements
- Author
-
A. Żochowski and Karl-Heinz Hoffmann
- Subjects
Partial differential equation ,Thermoelastic damping ,General Mathematics ,Mathematical analysis ,Plate theory ,General Engineering ,Rigidity (psychology) ,Geometry ,Differentiable function ,Shape-memory alloy ,Uniqueness ,System of linear equations ,Mathematics - Abstract
In the paper we study the problem of control by means of a heat source g for a thermoelastic system of equations U tt - ρ⊇.p(θ, ⊇u) - vΔu t + DΔ 2 = f, c v (θ,⊇u)θ t - kΔθ - ρθ[p θ (θ,⊇u).-⊇u t ] - v|⊇vu t | 2 = g, in a two-dimensional domain, where both viscosity v and rigidity D are positive. Such a system has been considered in our former papers, and existence of solutions as well as uniqueness have been obtained. Here we prove the continuity and differentiability of solutions under somewhat stronger assumptions. An example of a control problem and necessary optimality conditions are presented. The system has an interpretation as a plate reinforced with shape memory alloy (SMA) wire mesh.
- Published
- 1998
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