472 results
Search Results
2. Solutions of fractional gas dynamics equation by a new technique
- Author
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Alicia Cordero Barbero, Juan Ramón Torregrosa Sánchez, and Ali Akgül
- Subjects
Fractional gas dynamics equation ,General Mathematics ,Operators ,010102 general mathematics ,Hilbert space ,General Engineering ,Gas dynamics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,MATEMATICA APLICADA ,Mathematics ,Mathematical physics - Abstract
[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.
- Published
- 2019
3. Fractional powers of the noncommutative Fourier's law by theS‐spectrum approach
- Author
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Stefano Pinton, Samuele Mongodi, Marco M. Peloso, Fabrizio Colombo, Colombo, F, Mongodi, S, Peloso, M, and Pinton, S
- Subjects
S-spectrum ,General Mathematics ,fractional diffusion processe ,General Engineering ,fractional diffusion processes ,Spectrum (topology) ,Noncommutative geometry ,fractional Fourier's law ,symbols.namesake ,Engineering (all) ,Fourier transform ,the S-spectrum approach ,symbols ,Mathematics (all) ,fractional powers of vector operators ,fractional powers of vector operator ,Mathematical physics ,Mathematics - Abstract
Let e ℓ , for ℓ = 1,2,3, be orthogonal unit vectors in R 3 and let Ω ⊂ R 3 be a bounded open set with smooth boundary ∂Ω. Denoting by x a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: T = a(x)∂xe1 + b(x)∂ye2 + c(x)∂ze3, into the conservation of energy law, here a, b, c ∶ Ω → R are given functions. With the S-spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, c ∶ Ω → R the fractional powers of T exist in the sense of the S-spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.
- Published
- 2019
4. Time-harmonic and asymptotically linear Maxwell equations in anisotropic media
- Author
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Xianhua Tang and Dongdong Qin
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Lipschitz domain ,Maxwell's equations ,Bounded function ,Homogeneous space ,symbols ,Tensor ,Boundary value problem ,0101 mathematics ,Perfect conductor ,Nehari manifold ,Mathematics - Abstract
This paper is focused on following time-harmonic Maxwell equation: ∇×(μ−1(x)∇×u)−ω2e(x)u=f(x,u),inΩ,ν×u=0,on∂Ω, where Ω⊂R3 is a bounded Lipschitz domain, ν:∂Ω→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as |u|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor e∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.
- Published
- 2017
5. Computation of periodic orbits in three-dimensional Lotka-Volterra systems
- Author
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Rubén Poveda and Juan F. Navarro
- Subjects
Series (mathematics) ,General Mathematics ,Computation ,Mathematical analysis ,General Engineering ,Periodic sequence ,010103 numerical & computational mathematics ,Systems modeling ,Symbolic computation ,01 natural sciences ,Poincaré–Lindstedt method ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,symbols ,Periodic orbits ,0101 mathematics ,Mathematics - Abstract
This paper deals with an adaptation of the Poincare-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.
- Published
- 2017
6. A data assimilation process for linear ill-posed problems
- Author
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X.-M. Yang and Z.-L. Deng
- Subjects
Well-posed problem ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,Bayesian probability ,Posterior probability ,General Engineering ,Markov chain Monte Carlo ,Inverse problem ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Data assimilation ,symbols ,Applied mathematics ,Ensemble Kalman filter ,0101 mathematics ,Randomness ,Mathematics - Abstract
In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
7. Some inequalities involving Hadamard-type k -fractional integral operators
- Author
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Praveen Agarwal
- Subjects
Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,General Engineering ,Riemann integral ,Type (model theory) ,01 natural sciences ,Fourier integral operator ,Fractional calculus ,010101 applied mathematics ,Algebra ,symbols.namesake ,Hadamard transform ,Improper integral ,symbols ,Daniell integral ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2017
8. Controllability of a class of heat equations with memory in one dimension
- Author
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Xiuxiang Zhou and Hang Gao
- Subjects
0209 industrial biotechnology ,General Mathematics ,010102 general mathematics ,Null (mathematics) ,Mathematical analysis ,General Engineering ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Volterra integral equation ,Controllability ,symbols.namesake ,020901 industrial engineering & automation ,Dimension (vector space) ,symbols ,Initial value problem ,State space ,Heat equation ,0101 mathematics ,Mathematics - Abstract
This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
9. Asymptotic behavior of solutions of a model derived from the 1‐D Keller–Segel model on the half line
- Author
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Renkun Shi
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Half-space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Green's function ,symbols ,Boundary value problem ,Half line ,0101 mathematics ,Exponential decay ,Stationary solution ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line x > as follows: ut−lux−uxx=−β(uvx)x,x>0,t>0,λv−vxx=u,x>0,t>0,lu(0,t)+ux(0,t)=vx(0,t)=0,t>0,u(x,0)=u0(x),x>0, where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l
- Published
- 2016
10. On existence of solutions of differential-difference equations
- Author
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Hai-chou Li
- Subjects
Independent equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,Euler equations ,010101 applied mathematics ,Stochastic partial differential equation ,Examples of differential equations ,Theory of equations ,symbols.namesake ,Simultaneous equations ,symbols ,Applied mathematics ,0101 mathematics ,C0-semigroup ,Differential algebraic equation ,Mathematics - Abstract
This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
11. On global stability of an HIV pathogenesis model with cure rate
- Author
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Yoshiaki Muroya and Yoichi Enatsu
- Subjects
Lyapunov function ,Mathematical optimization ,General Mathematics ,General Engineering ,Human immunodeficiency virus (HIV) ,medicine.disease_cause ,Stability (probability) ,Upper and lower bounds ,Pathogenesis ,symbols.namesake ,Monotone polygon ,Stability theory ,medicine ,symbols ,Applied mathematics ,Logistic function ,Mathematics - Abstract
In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
12. Spectral analysis of non-compact symmetrizable operators on Hilbert spaces
- Author
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Mustapha Mokhtar-Kharroubi and Yahya Mohamed
- Subjects
Pure mathematics ,General Mathematics ,Essential spectrum ,General Engineering ,Hilbert space ,Interval (mathematics) ,Characterization (mathematics) ,Injective function ,Algebra ,symbols.namesake ,Operator (computer programming) ,symbols ,Eigenvalues and eigenvectors ,Mathematics ,Complement (set theory) - Abstract
This paper revisits and complement in different directions the classical work by W. T. Reid on symmetrizable completely continuous transformations in Hilbert spaces and a more recent paper by one of the authors. More precisely, we deal with spectral properties of % non-compact operators G on a complex Hilbert space H such that SG is self-adjoint where S is a (not necessarily injective) nonnegative operator. We study the isolated eigenvalues of G outside its essential spectral interval and provide variational characterization of them as well as stability estimates. We compare them also to spectral objects of SG. Finally, we characterize the Schechter essential spectrum of strongly symmetrizable operators in terms singular Weyl sequences; in particular, we complement J. I. Nieto's paper on the essential spectrum of symmetrizable. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
13. Strong convergence of the split-stepθ-method for stochastic age-dependent population equations
- Author
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Hong-li Wang, Yongfeng Guo, and Jianguo Tan
- Subjects
education.field_of_study ,General Mathematics ,Mathematical analysis ,Population ,General Engineering ,Age dependent ,Stochastic partial differential equation ,Euler method ,symbols.namesake ,Convergence (routing) ,symbols ,Order (group theory) ,education ,Mathematics - Abstract
In this paper, we constructed the split-step θ (SSθ)-method for stochastic age-dependent population equations. The main aim of this paper is to investigate the convergence of the SS θ-method for stochastic age-dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS θ-method. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
14. An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials
- Author
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A. Sami Bataineh and Zaid Odibat
- Subjects
Series (mathematics) ,General Mathematics ,Homotopy ,General Engineering ,Algebra ,Nonlinear system ,n-connected ,symbols.namesake ,Difference polynomials ,Taylor series ,symbols ,Adomian decomposition method ,Homotopy analysis method ,Mathematics - Abstract
In this paper, a new adaption of homotopy analysis method is presented to handle nonlinear problems. The proposed approach is capable of reducing the size of calculations and easily overcome the difficulty arising in calculating complicated integrals. Furthermore, the homotopy polynomials that decompose the nonlinear term of the problem as a series of polynomials are introduced. Then, an algorithm of calculating such polynomials, which makes the solution procedure more straightforward and more effective, is constructed. Numerical examples are examined to highlight the significant features of the developed techniques. The algorithms described in this paper are expected to be further employed to solve nonlinear problems in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
15. On sparse representation of analytic signal in Hardy space
- Author
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Tao Qian and Shuang Li
- Subjects
K-SVD ,Function space ,General Mathematics ,General Engineering ,Sparse approximation ,Hardy space ,Unit disk ,Algebra ,symbols.namesake ,Fourier transform ,Compressed sensing ,Square-integrable function ,symbols ,Mathematics - Abstract
This paper is concerned with the sparse representation of analytic signal in Hardy space , where is the open unit disk in the complex plane. In recent years, adaptive Fourier decomposition has attracted considerable attention in the area of signal analysis in . As a continuation of adaptive Fourier decomposition-related studies, this paper proves rapid decay properties of singular values of the dictionary. The rapid decay properties lay a foundation for applications of compressed sensing based on this dictionary. Through Hardy space decomposition, this program contributes to sparse representations of signals in the most commonly used function spaces, namely, the spaces of square integrable functions in various contexts. Numerical examples are given in which both compressed sensing and l1-minimization are used. Copyright © 2013 John Wiley & Sons, Ltd.
- Published
- 2013
16. Scattering theory for the cubic nonlinear Klein-Gordon system
- Author
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Junyong Zhang
- Subjects
Nonlinear system ,symbols.namesake ,General Mathematics ,General Engineering ,symbols ,Scalar (physics) ,Point (geometry) ,Scattering theory ,Klein–Gordon equation ,Mathematical physics ,Mathematics - Abstract
In this paper, we consider the scattering theory of a nonlinear Klein–Gordon system, which describes the interaction of two scalar fields. The analysis in this paper is an adaptation of the technique used by Nakanishi, which is originally due to Bourgain. The new technical point appears in the localization argument of proving a concentration phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.
- Published
- 2013
17. Paley-Wiener theorems and uncertainty principles for the windowed linear canonical transform
- Author
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Rui-Hui Xu, Yan‐Hui Zhang, and Kit Ian Kou
- Subjects
Uncertainty principle ,Paley–Wiener theorem ,General Mathematics ,Mathematical analysis ,Poisson summation formula ,General Engineering ,Sampling (statistics) ,Inverse ,Fractional Fourier transform ,symbols.namesake ,symbols ,Applied mathematics ,Series expansion ,Mathematics ,Interpolation - Abstract
In a recent paper, the authors have introduced the windowed linear canonical transform and shown its good properties together with some applications such as Poisson summation formulas, sampling interpolation, and series expansion. In this paper, we prove the Paley–Wiener theorems and the uncertainty principles for the (inverse) windowed linear canonical transform. They are new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
18. Impulsive control for differential systems with delay
- Author
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Changming Ding and Huali Wang
- Subjects
Lyapunov function ,Differential equation ,General Mathematics ,General Engineering ,Impulse (physics) ,Differential systems ,Upper and lower bounds ,symbols.namesake ,Exponential stability ,Control theory ,Stability theory ,Control system ,symbols ,Mathematics - Abstract
This paper deals with the impulsive control for a class of differential systems with delay. Using Lyapunov functions and the comparison principle, we present some sufficient conditions for the asymptotic stability and exponential stability of impulsive control systems with delay. Moreover, we give an estimate of the upper bound of impulse interval. The results in this paper extend and improve the earlier publications. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
19. On Fourier series for higher order (partial) derivatives of functions
- Author
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Weiming Sun and Zimao Zhang
- Subjects
General Mathematics ,Fourier inversion theorem ,Mathematical analysis ,Fourier sine and cosine series ,General Engineering ,02 engineering and technology ,Trigonometric polynomial ,01 natural sciences ,symbols.namesake ,020303 mechanical engineering & transports ,Generalized Fourier series ,0203 mechanical engineering ,Fourier analysis ,Discrete Fourier series ,0103 physical sciences ,Conjugate Fourier series ,symbols ,010301 acoustics ,Fourier series ,Mathematics - Abstract
This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
20. Numerical iterative method for Volterra equations of the convolution type
- Author
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Rani W. Sullivan, Jutima Simsiriwong, and Mohsen Razzaghi
- Subjects
Iterative method ,General Mathematics ,Numerical analysis ,Constitutive equation ,Mathematical analysis ,Linear system ,General Engineering ,Volterra integral equation ,Integral equation ,Convolution ,symbols.namesake ,Singularity ,symbols ,Mathematics - Abstract
The objective of this paper is to present an algorithm from which a rapidly convergent solution is obtained for Volterra integral equations of Hammerstein type. Such equations are often encountered when describing the response of viscoelastic materials where the time dependency of the material properties is often expressed in the form of a convolution integral. Frequently, singularity is encountered and often ignored when dealing with the constitutive equations of viscoelastic materials. In this paper, the singularity is incorporated in the solution and the iterative scheme used to solve the equation converges within six iterations to a typical toleration error of 10−5. The algorithm is applied to the strain response of a polymer under impulsive (constant) loading and the results show excellent correlation between the experimental and analytical solution. Copyright © 2010 John Wiley & Sons, Ltd.
- Published
- 2010
21. Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids
- Author
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Sergey Repin and Martin Fuchs
- Subjects
Convex analysis ,General Mathematics ,Mathematical analysis ,General Engineering ,Function (mathematics) ,generalized Newtonian fluids ,viscous incompressible fluids ,Domain (mathematical analysis) ,symbols.namesake ,Variational method ,Dirichlet boundary condition ,Bounded function ,Variational inequality ,symbols ,Boundary value problem ,variational inequalities ,Mathematics - Abstract
This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary-value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn- and Friedrichs-type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L2 norm of a vector-valued function from H1(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L2 norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.
- Published
- 2009
22. Analysis for the identification of an unknown diffusion coefficient via semigroup approach
- Author
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Ebru Ozbilge and Ali Demir
- Subjects
Source function ,Pure mathematics ,Semigroup ,General Mathematics ,Probleme inverse ,Mathematical analysis ,General Engineering ,Inverse ,Inverse problem ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Boundary value problem ,Diffusion (business) ,Mathematics - Abstract
This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the inhomogenenous quasi-linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:C1[0, T], Ψ[·]:C1[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.
- Published
- 2009
23. Double-wall nanotube as vibrational system: Mathematical approach
- Author
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Miriam Rojas-Arenaza and Marianna A. Shubov
- Subjects
General Mathematics ,Mathematical analysis ,General Engineering ,Compact operator ,Differential operator ,symbols.namesake ,Relatively compact subspace ,symbols ,Boundary value problem ,van der Waals force ,Hyperbolic partial differential equation ,Self-adjoint operator ,Mathematics ,Resolvent - Abstract
In this paper, we present a recently developed mathematical model for short double-wall carbon nanotubes. The model is governed by a system of four hyperbolic equations representing the two Timoshenko beams coupled through the Van der Waals forces. The system is equipped with a four-parameter family of the boundary conditions and can be reduced to an evolution equation. This equation defines a strongly continuous semi-group. Spectral properties of the semi-group generator are presented in the paper. We show that it is an unbounded non-selfadjoint operator with compact resolvent. Moreover, this operator is a relatively compact perturbation of a certain selfadjoint operator. Copyright © 2008 John Wiley & Sons, Ltd.
- Published
- 2008
24. Global stability and the Hopf bifurcation for some class of delay differential equation
- Author
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Urszula Foryś and Marek Bodnar
- Subjects
Hopf bifurcation ,education.field_of_study ,Steady state (electronics) ,Differential equation ,General Mathematics ,Population ,Mathematical analysis ,General Engineering ,Delay differential equation ,Stability (probability) ,Unimodality ,symbols.namesake ,symbols ,education ,Numerical stability ,Mathematics - Abstract
In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right-hand side depending only on the past. We extend the results from paper by U. Foryś (Appl. Math. Lett. 2004; 17(5):581–584), where the right-hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.
- Published
- 2007
25. Convergence rates toward the travelling waves for a model system of the radiating gas
- Author
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Masataka Nishikawa and Shinya Nishibata
- Subjects
General Mathematics ,Weak solution ,Mathematical analysis ,General Engineering ,Perturbation (astronomy) ,Sobolev space ,Elliptic curve ,Riemann hypothesis ,symbols.namesake ,Exponential stability ,Rate of convergence ,symbols ,Algebraic number ,Mathematics - Abstract
The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)−1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.
- Published
- 2007
26. Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation
- Author
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Ebru Ozbilge and Ali Demir
- Subjects
Discrete mathematics ,Integral representation ,Semigroup ,General Mathematics ,General Engineering ,Boundary (topology) ,Inverse ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Quasi linear ,Boundary value problem ,Diffusion (business) ,Mathematical physics ,Mathematics - Abstract
This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x, t)) in the quasi-linear parabolic equation u t (x, t) = (k(u(x, t))u x (x, t)) x , with Dirichlet boundary conditions u(0, t) = ψ 0 , u(1,t) = ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input-output mappings Φ[·]: K →C 1 [0, T], Ψ[·]: Κ → C 1 [0, T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[·] and Ψ[·] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0, t))u x (0, t) or/and h(t):=k (u(1,t))u x (1, t), the values k(ψ 0 ) and k(ψ 1 ) of the unknown diffusion coefficient k(u(x,t)) at (x, t) = (0,0) and (x,t) = (1,0), respectively, can be determined explicitly. In addition to these, the values k u (ψ 0 ) and k u (ψ 1 ) of the unknown coefficient k(u(x, t)) at (x,t)=(0,0) and (x, t) = (1, 0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input-output mappings Φ[·]:Κ→ C 1 [0, T], Ψ[·]:K→ C 1 [0, T] are given explicitly in terms of the semigroup.
- Published
- 2007
27. Non-linear stability for convection with quadratic temperature dependent viscosity
- Author
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Ashuwin Vaidya and Rachmadian Wulandana
- Subjects
Convection ,Continuum (measurement) ,General Mathematics ,General Engineering ,Thermodynamics ,Mechanics ,Rayleigh number ,Critical value ,Physics::Fluid Dynamics ,symbols.namesake ,Quadratic equation ,Maximum principle ,Newtonian fluid ,symbols ,Rayleigh scattering ,Mathematics - Abstract
In this paper, we study the non-linear stability of convection for a Newtonian fluid heated from below, where the viscosity of the fluid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Rac, the rest state of the fluid and the steady temperature distribution remains non-linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16:347–352). The central contribution of this paper lies in a simpler proof of non-linear stability, than the ones in the current literature, by use of a suitable maximum principle argument. Copyright © 2006 John Wiley & Sons, Ltd.
- Published
- 2006
28. On Cauchy estimates and growth orders of entire solutions of iterated Dirac and generalized Cauchy–Riemann equations
- Author
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R. De Almeida, Denis Constales, and R. S. Kraußhar
- Subjects
Partial differential equation ,General Mathematics ,Weak solution ,Mathematical analysis ,General Engineering ,Cauchy–Riemann equations ,Dirac operator ,symbols.namesake ,Dirac equation ,symbols ,Biharmonic equation ,Heat equation ,Klein–Gordon equation ,Mathematics - Abstract
In this paper, we study the growth behaviour of entire Clifford algebra-valued solutions to iterated Dirac and generalized Cauchy–Riemann equations in higher-dimensional Euclidean space. Solutions to this type of systems of partial differential equations are often called k-monogenic functions or, more generically, polymonogenic functions. In the case dealing with the Dirac operator, the function classes of polyharmonic functions are included as particular subcases. These are important for a number of concrete problems in physics and engineering, such as, for example, in the case of the biharmonic equation for elasticity problems of surfaces and for the description of the stream function in the Stokes flow regime with high viscosity. Furthermore, these equations in turn are closely related to the polywave equation, the poly-heat equation and the poly-Klein–Gordon equation. In the first part we develop sharp Cauchy-type estimates for polymonogenic functions, for equations in the sense of Dirac as well as Cauchy–Riemann. Then we introduce generalizations of growth orders, of the maximum term and of the central index in this framework, which in turn then enable us to perform a quantitative asymptotic growth analysis of this function class. As concrete applications we develop some generalizations of some of Valiron's inequalities in this paper. Copyright © 2006 John Wiley & Sons, Ltd.
- Published
- 2006
29. On the integral representation formula for a two-component elastic composite
- Author
-
Miao-Jung Ou and Elena Cherkaev
- Subjects
Integral representation ,General Mathematics ,Mathematical analysis ,Composite number ,General Engineering ,Hilbert space ,Special class ,Homogenization (chemistry) ,Viscoelasticity ,symbols.namesake ,symbols ,Elasticity (economics) ,Elastic modulus ,Mathematics - Abstract
The aim of this paper is to derive, in the Hilbert space setting, an integral representation formula for the effective elasticity tensor for a two-component composite of elastic materials, not necessarily well-ordered. This integral representation formula implies a relation which links the effective elastic moduli to the N-point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for inverse-homogenization. The analysis presented in this paper can be generalized to an n-component composite of elastic materials. The relations developed here can be applied to the inverse-homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2005 John Wiley & Sons, Ltd.
- Published
- 2006
30. Identification of objects in an acoustic wave guide inversion II: Robin-Dirichlet conditions
- Author
-
Robert P. Gilbert and Doo-Sung Lee
- Subjects
Conjecture ,Dirichlet conditions ,General Mathematics ,Mathematical analysis ,General Engineering ,Ranging ,Inversion (meteorology) ,Acoustic wave ,Inverse problem ,symbols.namesake ,symbols ,Rayleigh scattering ,Electrical impedance ,Mathematics - Abstract
SUMMARY 9 In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for 11 solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body 13 radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of 15 arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct 17 solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown
- Published
- 2006
31. Spatiotemporal patterns in a diffusive predator–prey system with Leslie–Gower term and social behavior for the prey
- Author
-
Abdelkader Lakmeche and Fethi Souna
- Subjects
Equilibrium point ,Hopf bifurcation ,General Mathematics ,General Engineering ,Stability (probability) ,Term (time) ,symbols.namesake ,Nonlinear Sciences::Adaptation and Self-Organizing Systems ,Bifurcation theory ,symbols ,Neumann boundary condition ,Quantitative Biology::Populations and Evolution ,Statistical physics ,Constant (mathematics) ,Center manifold ,Mathematics - Abstract
In this paper, we deal with a new approximation of a diffusive predator--prey model with Leslie--Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator-prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator-prey interaction in the presence of Leslie--Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.
- Published
- 2021
32. Higher order non-resonance for differential equations with singularities
- Author
-
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
33. Strong instability of solitary waves for inhomogeneous nonlinear Schrödinger equations
- Author
-
Jian Zhang and Chenglin Wang
- Subjects
symbols.namesake ,Nonlinear system ,Classical mechanics ,General Mathematics ,General Engineering ,symbols ,Instability ,Schrödinger equation ,Mathematics - Abstract
This paper studies the inhomogeneous nonlinear Schrödinger equations, which may model the propagation of laser beams in nonlinear optics. Using the cross-constrained variational method, a sharp condition for global existence is derived. Then, by solving a variational problem, the strong instability of solitary waves of this equation is proved.
- Published
- 2021
34. Stability and Hopf bifurcation analysis of fractional‐order nonlinear financial system with time delay
- Author
-
Sunita Chand, Sundarappan Balamuralitharan, and Santoshi Panigrahi
- Subjects
Hopf bifurcation ,Nonlinear system ,symbols.namesake ,Computer simulation ,Laplace transform ,General Mathematics ,General Engineering ,symbols ,Order (ring theory) ,Financial system ,Stability (probability) ,Mathematics - Abstract
In this paper, we study a fractional order time delay for nonlinear financial system. By using Laplace transformation, stability and Hopf bifurcation analysis have been done for the model. Furthermore, numerical simulation has been carried out for better understanding of our results.
- Published
- 2021
35. Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise
- Author
-
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
- Published
- 2021
36. 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
37. 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
38. 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
39. The number of Dirac‐weighted eigenvalues of Sturm–Liouville equations with integrable potentials and an application to inverse problems
- Author
-
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
40. Spectral theory for the wave equation in two adjacent wedges
- Author
-
Felix Ali Mehmeti, Erhard Meister, and Krešo Mihalinčić
- Subjects
Constant coefficients ,Spectral theory ,General Mathematics ,Mathematical analysis ,General Engineering ,Hilbert space ,Spectral theorem ,Eigenfunction ,Wave equation ,symbols.namesake ,Dirichlet boundary condition ,symbols ,spectral theory ,time-dependent wave and Klein Gordon equations ,Mathematics ,Resolvent - Abstract
Consider the two adjacent rectangular wedges K1, K2 with common edge in the upper halfspace of ℝ3 and the operator A (=−Laplacian multiplied by different constant coefficients a1, a2 in K1, K2, respectively) acting on a subspace of ∏2j=1L2(Kj). This subspace should consist of those sufficiently regular functions u=(u1,u2) satisfying the homogeneous Dirichlet boundary condition on the bottom of the upper halfspace. Moreover, the coincidence of u1 and u2 along the interface of the two wedges is prescribed as well as a transmission condition relating their first one-sided derivatives. We interpret the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. In this paper it is shown (by Friedrichs' extension) that A is selfadjoint in a suitable Hilbert space. Applying the Fourier (-sine) transformations we reduce our problem with singularities along the z-axis to a non-singular Klein–Gordon equation in one space dimension with potential step. The resolvent, the limiting absorption principle and expansion in generalized eigenfunctions of A are derived (by Plancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theorem for unbounded selfadjoint operators on Hilbert spaces yields the solution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physical geometry of the problem and its spectral properties. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.
- Published
- 1997
41. Well-posedness of the Hydrodynamic Model for Semiconductors
- Author
-
Li-Ming Yeh
- Subjects
Dirichlet problem ,Conservation law ,Dirichlet conditions ,General Mathematics ,Mathematical analysis ,General Engineering ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,Parabolic partial differential equation ,symbols.namesake ,symbols ,Dissipative system ,Boundary value problem ,Convection–diffusion equation ,Hyperbolic partial differential equation ,Mathematics - Abstract
This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasilinear elliptic-parbolic-hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L2 sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L2 sense. The well-posedness of the model under the boundary conditions is demonstrated. This paper addresses the well-posedness of the hydrodynamic model for semiconductors. The model is derived from moments of the Boltzmann’s equation, taken over group velocity space. When coupled with the charge conservation equation, it describes the behaviour of small semiconductor devices and accounts for special features such as hot electrons and velocity overshoots. The model consists of a set of non-linear conservation laws for particle number, momentum, and energy, coupled to Poisson’s equation for the electric potential. It is a perturbation of the drift diffusion model [7]. We consider a ballistic diode problem which models the channel of a MOSFET, so the effect of holes in the model can be neglected. The model is [4,11]
- Published
- 1996
42. 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
43. The double layer potential method for a boundary transmission problem for the Laplace operator in an infinite wedge
- Author
-
Dirk Mirschinka
- Subjects
General Mathematics ,Mathematical analysis ,General Engineering ,Microlocal analysis ,Operator theory ,Wedge (geometry) ,Fourier integral operator ,symbols.namesake ,Fourier transform ,symbols ,Double layer potential ,Boundary value problem ,Laplace operator ,Mathematics - Abstract
This paper is concerned with the solution of a boundary transmission problem in an infinite wedge. We treat this problem by a boundary integral method using Green's contact function for two half-spaces. The integral operators are studied via a harmonic analysis approach which goes back to a paper of Fabes et al. We improve their results studying the Fourier symbol of the associated integral operators on the half-plane. This leads to invertibility criteria for the boundary integral operators.
- Published
- 1995
44. 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
45. 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
46. Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems
- Author
-
Miguel Lobo and M. Eugenia Pérez
- Subjects
General Mathematics ,Mathematical analysis ,General Engineering ,Mathematics::Spectral Theory ,Eigenfunction ,Wave equation ,Homogenization (chemistry) ,Dirichlet distribution ,symbols.namesake ,Harmonic function ,Bounded function ,symbols ,Boundary value problem ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low-frequency and high-frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter e, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov-type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ℝ2, and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments Te of size O(e) periodically distributed along the boundary; e also measures the periodicity of the structure. We consider associated second-order evolution problems on spaces of traces that depend on e, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions ue(t) as e0. Copyright © 2009 John Wiley & Sons, Ltd.
- Published
- 2009
47. Global existence of weak solutions for a system of non-linear Boltzmann equations in semiconductor physics
- Author
-
Francisco José Mustieles
- Subjects
business.industry ,General Mathematics ,Mathematical analysis ,General Engineering ,Boltzmann equation ,Nonlinear system ,symbols.namesake ,Distribution function ,Compact space ,Semiconductor ,Electric field ,Regularization (physics) ,Boltzmann constant ,symbols ,Statistical physics ,business ,Mathematics - Abstract
In this paper we give a proof of the global existence of weak solutions for the semiconductor Boltzmann equation. This equation rules the evolution of the distribution function of carriers in the kinetic model of semiconductors. The main tool for the proof consists of a recent compactness result on velocity averages of solutions of transport equations. This result needs a L2-estimate of the electric field, which is obtained from the energy estimate, using the original regularization procedure of the problem, proposed in this paper.
- Published
- 1991
48. The rigorous derivation of unipolar Euler–Maxwell system for electrons from bipolar Euler–Maxwell system by infinity‐ion‐mass limit
- Author
-
Liang Zhao
- Subjects
Thermodynamic equilibrium ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Electron ,Decoupling (cosmology) ,01 natural sciences ,Local convergence ,010101 applied mathematics ,symbols.namesake ,Convergence (routing) ,Euler's formula ,symbols ,Convergence problem ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
In the paper, we consider the local-in-time and the global-in-time infinity-ion-mass convergence of bipolar Euler-Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi→∞. We use the method of asymptotic expansions to handle the local-in-time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling, but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data is sufficiently close to the constant equilibrium state, we establish the global-in-time infinity-ion-mass convergence.
- Published
- 2020
49. Metamaterial acoustics on the Poincaré disk
- Author
-
Tung, Michael Ming-Sha
- Subjects
Acoustic analogue models of gravity ,Applications of local differential geometry to the sciences ,General Mathematics ,Poincaré disk model ,General Engineering ,Metamaterial ,Relativity and gravitational theory ,Variational principles of mathematical physics ,Traveling wave solutions ,symbols.namesake ,Classical mechanics ,Poincaré disk ,symbols ,MATEMATICA APLICADA ,Mathematics - Abstract
[EN] Historically, the Poincare disk model has taken a pioneering role in the development of non-Euclidean geometry. But still today, this model is critical as a playground for simulations and new theories. In this paper, we discuss how to implement and simulate acoustic wave phenomena on the Poincare disk with the help of so-called metamaterials. After formally developing a theory based on the acoustic potential for the curved spacetime of the Poincare model, we also provide practical instructions on how to manufacture the appropriate metadevice in a laboratory environment. Finally, as an example, analytical results for the nontrivial radial contributions in concentric wave propagation are derived, and the corresponding numerical predictions are presented., Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF), Grant/Award Number: TIN2017-89314-P; Programa de Apoyo a la Investigacion y Desarrollo 2018, Grant/Award Number: PAID-06-18
- Published
- 2020
50. Minimal-energy splines: I. Plane curves with angle constraints
- Author
-
Weimin Han and Emery D. Jou
- Subjects
Box spline ,Plane curve ,General Mathematics ,General Engineering ,Geometry ,Mathematics::Numerical Analysis ,symbols.namesake ,Spline (mathematics) ,Smoothing spline ,Computer Science::Graphics ,Lagrange multiplier ,symbols ,Applied mathematics ,Spline interpolation ,Thin plate spline ,Mathematics - Abstract
This is the first in a series of papers on minimal-energy splines. The paper is devoted to plane minimal-energy splines with angle constraints. We first consider minimal-energy spline segments, then general minimal-energy spline curves. We formulate problems for minimal-energy spline segments and curves, prove the existence of solutions, justify the Lagrange multiplier rules, and obtain some nice properties (e.g., the infinite smoothness). Finally, we report our computational experience on minimal-energy splines.
- Published
- 1990
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