1,184 results
Search Results
2. 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
3. The Cauchy problem of a fluid-particle interaction model with external forces
- Author
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Zaihong Jiang, Ning Zhong, and Li Li
- Subjects
Cauchy problem ,Picard–Lindelöf theorem ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Interaction model ,01 natural sciences ,010101 applied mathematics ,Fluid particle ,Nonlinear system ,Decomposition (computer science) ,Initial value problem ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we consider the Cauchy problem of a fluid-particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro-micro decomposition.
- Published
- 2017
4. Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term
- Author
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Masakazu Kato and Yoshihiro Ueda
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Lower order ,Damped wave ,Space (mathematics) ,01 natural sciences ,Burgers' equation ,Term (time) ,010101 applied mathematics ,Initial value problem ,Nonlinear convection ,0101 mathematics ,Representation (mathematics) ,Mathematics - Abstract
This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.
- 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.
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- 2017
6. Monotonicity, uniqueness, and stability of traveling waves in a nonlocal reaction-diffusion system with delay
- Author
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Hai-Qin Zhao
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Monotonic function ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Transmission (telecommunications) ,Stability theory ,Reaction–diffusion system ,Traveling wave ,Uniqueness ,0101 mathematics ,Epidemic model ,Mathematics - Abstract
The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾c∗. Then we show that the traveling wave fronts with speed c>c∗ are exponentially asymptotically stable.
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- 2017
7. Two homoclinic solutions for a nonperiodic fourth-order differential equation without coercive condition
- Author
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Shiping Lu and Tao Zhong
- Subjects
Class (set theory) ,Differential equation ,General Mathematics ,Open problem ,Mathematical analysis ,General Engineering ,01 natural sciences ,010101 applied mathematics ,Fourth order ,Variational method ,0103 physical sciences ,Mountain pass theorem ,Homoclinic orbit ,0101 mathematics ,010306 general physics ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations u(4)+wu′′+a(x)u=f(x,u), where w is a constant, a∈C(R,R) and f∈C(R×R,R). By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well-known (AR)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
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
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- 2016
10. Uncertainty principles for images defined on the square
- Author
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Pei Dang and Shujuan Wang
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Uncertainty principle ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Phase (waves) ,020206 networking & telecommunications ,Torus ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,Square (algebra) ,Set (abstract data type) ,Amplitude ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
11. Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equations on the whole line
- Author
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Tian-jun Wang
- Subjects
Condensed Matter::Quantum Gases ,Laguerre's method ,General Mathematics ,Mathematical analysis ,General Engineering ,Relaxation (iterative method) ,Domain decomposition methods ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Convergence (routing) ,Laguerre polynomials ,Fokker–Planck equation ,0101 mathematics ,Spectral method ,Mathematics - Abstract
In this paper, we propose a composite Laguerre spectral method for the nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. A composite Laguerre spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. Some results on the Laguerre approximation and techniques used in this paper are also applicable to other nonlinear problems on the whole line. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
12. Global well-posedness of the nonhomogeneous incompressible liquid crystals systems
- Author
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Xiaoyu Xi and Dongjuan Niu
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Small data ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Space (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Strong solutions ,Liquid crystal ,Bounded function ,Compressibility ,Calculus ,0101 mathematics ,Well posedness ,Mathematics - Abstract
This paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
13. Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data
- Author
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Noriaki Yoshino and Tomomi Yokota
- Subjects
010101 applied mathematics ,Degenerate diffusion ,Nonlinear system ,Smoothness (probability theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Degenerate energy levels ,General Engineering ,Symmetry in biology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with local and global existence of solutions to the parabolic-elliptic chemotaxis system . Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear m-accretive operators to giving existence of local solutions to this system when 0 1), is left incomplete. This paper presents the local and global solvability of the system with non-Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
14. The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations
- Author
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Mansour Safarpoor and Mehdi Dehghan
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Partial differential equation ,General Mathematics ,Mathematical analysis ,General Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Boundary integral equations ,Reciprocity (electromagnetism) ,Time derivative ,Boundary particle method ,Radial basis function ,0101 mathematics ,Boundary element method ,Mathematics - Abstract
In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
15. Ground state solutions for asymptotically periodic coupled Kirchhoff-type systems with critical growth
- Author
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Haibo Chen and Hongxia Shi
- Subjects
010101 applied mathematics ,Nonlinear system ,Kirchhoff type ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,0101 mathematics ,Ground state ,01 natural sciences ,Mathematics - Abstract
In this paper, we consider the coupled system of Kirchhoff-type equations: where 4 0, b,d≥0 are constants and λ is a positive parameter. The main purpose of this paper is to study the existence of ground state solutions for the aforementioned system with a nonlinearity in the critical growth under some suitable assumptions on V and F. Recent results from the literature are improved and extended. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
16. The fairing andG1continuity of quartic C-Bézier curves
- Author
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Yang Yang, Xinqiang Qin, Gang Hu, and Guo Wei
- Subjects
Convex hull ,Quartic plane curve ,General Mathematics ,Mathematical analysis ,General Engineering ,Bullet-nose curve ,020207 software engineering ,Bézier curve ,02 engineering and technology ,Curvature ,01 natural sciences ,Shape parameter ,010101 applied mathematics ,Quartic function ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Quartic surface ,Mathematics - Abstract
Quartic C-Bezier curves possess similar properties with the traditional Bezier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter α decreases. In this paper, by adjusting the shape parameter α on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C-Bezier curve with G1 continuity of quartic C-Bezier curve segments, we develop a fairing and G1 continuity algorithm for any given stitching coefficients λk(k = 1,2,…,n − 1). The shape parameters αi(i=1, 2, …, n) can be adjusted by the value of control points. The curvature of the resulting quartic C-Bezier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer-aided design/computer-aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
17. Stabilization of a one-dimensional wave equation with variable coefficient under non-collocated control and delayed observation
- Author
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Kun-Yi Yang
- Subjects
0209 industrial biotechnology ,Constant coefficients ,Partial differential equation ,Observer (quantum physics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Characteristic equation ,02 engineering and technology ,Wave equation ,01 natural sciences ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,Full state feedback ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we consider stabilization of a 1-dimensional wave equation with variable coefficient where non-collocated boundary observation suffers from an arbitrary time delay. Since input and output are non-collocated with each other, it is more complex to design the observer system. After showing well-posedness of the open-loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed-loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non-collocated control.
- Published
- 2017
18. Global attractor for suspension bridge equations with memory
- Author
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Jum-Ran Kang
- Subjects
General Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,General Engineering ,Object (computer science) ,01 natural sciences ,Bridge (interpersonal) ,Domain (mathematical analysis) ,010101 applied mathematics ,Bounded function ,Attractor ,Uniqueness ,0101 mathematics ,Suspension (vehicle) ,Mathematics - Abstract
This paper is concerned with a suspension bridge equation with memory effects , defined in a bounded domain of . For the suspension bridge equation without memory, there are many classical results. Existing results mainly devoted to existence and uniqueness of a weak solution, energy decay of solution and existence of global attractors. However the existence of global attractors for the suspension bridge equation with memory was no yet considered. The object of the present paper is to provide some results on the well-posedness and long-time behavior to the suspension bridge equation in a more with past history. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
19. 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
20. Exponential stability for a one-dimensional compressible viscous micropolar fluid
- Author
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Lan Huang and Dayong Nie
- Subjects
Exponential stability ,General Mathematics ,Mathematical analysis ,General Engineering ,Compressibility ,A priori and a posteriori ,Boundary value problem ,Polytropic process ,Mathematics - Abstract
In this paper, we consider one-dimensional compressible viscous and heat-conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H1(0,1))4, and then we establish the global existence and exponential stability of solutions in (H2(0,1))4 under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
21. Maximum principle for optimal distributed control of viscous weakly dispersive Degasperis-Procesi equation
- Author
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Shan-Shan Wang and Bing Sun
- Subjects
Nonlinear system ,Partial differential equation ,Maximum principle ,General Mathematics ,Ordinary differential equation ,Mathematical analysis ,General Engineering ,Calculus of variations ,Degasperis–Procesi equation ,Optimal control ,Hamiltonian (control theory) ,Mathematics - Abstract
This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 JohnWiley & Sons, Ltd.
- Published
- 2015
22. On existence and multiplicity of similarity solutions to a nonlinear differential equation arising in magnetohydrodynamic Falkner-Skan flow for decelerated flows
- Author
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Alaeddin Malek, R. A. Van Gorder, and R. Naseri
- Subjects
Monotone polygon ,Flow (mathematics) ,General Mathematics ,Mathematical analysis ,General Engineering ,Existence theorem ,Monotonic function ,Uniqueness ,Magnetohydrodynamic drive ,Type (model theory) ,Second derivative ,Mathematics - Abstract
Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
23. Towards a quaternionic function theory linked with the Lamé's wave functions
- Author
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M. A. Pérez-de la Rosa and João Morais
- Subjects
Helmholtz equation ,General Mathematics ,Operator (physics) ,Mathematical analysis ,General Engineering ,Separation of variables ,Function (mathematics) ,Singular integral ,Wave equation ,Quaternionic analysis ,Mathematics ,Variable (mathematics) - Abstract
Over the past few years, considerable attention has been given to the role played by the Lame's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lame's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non-commutative framework. We show that the theory of the LQWFs is determined by the Moisil-Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lame's wave equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel-Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski-Plemelj formulae, the -hyperholomorphic extension of a given Holder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
24. A generalized model for relative phases based on bilinear representation of natural image series
- Author
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Bo Chen and Hongxia Wang
- Subjects
Image Series ,Deblurring ,Wrapped Cauchy distribution ,Wavelet ,General Mathematics ,Histogram ,Mathematical analysis ,General Engineering ,von Mises distribution ,Bilinear interpolation ,Image processing ,Algorithm ,Mathematics - Abstract
Local phase is now known to carry information about image features or object motions. But it is harder to use directly compared with amplitude, so far. In this paper, we propose that the relative local phase, which is a function of scale, position and time, really matters in representing the information of image structures or movements. A unified description of relative phase is given in this paper based on a bilinear representation of natural image series via multi-scale orientated dual tree complex wavelets. Then, the behaviors of nontrivial relative phase, especially for their distribution on multi-scale and multi-subband, are investigated. We propose a new generalized model, which is derived from Mobius transform, to describe various relative phases. Numerical experiments for a large amount of test images show that the new model performs best compared with the von Mises or wrapped Cauchy distribution. Especially for those with asymmetric pdf, our function fits with the histogram quite well while the other two may fail. We thus lay a groundwork for relative phase-based image processing methods, such as classification, deblurring and motion perception. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
25. On delaminated thin Timoshenko inclusions inside elastic bodies
- Author
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Hiromichi Itou and Alexander Khludnev
- Subjects
Timoshenko beam theory ,Problem Formulations ,General Mathematics ,Delamination ,Mathematical analysis ,General Engineering ,02 engineering and technology ,01 natural sciences ,Nonlinear boundary conditions ,010101 applied mathematics ,020303 mechanical engineering & transports ,Rigidity (electromagnetism) ,0203 mechanical engineering ,0101 mathematics ,Anisotropy ,Mathematics - Abstract
In the paper, we consider equilibrium problems for 2D elastic bodies with thin inclusions modeled in the frame of Timoshenko beam theory. It is assumed that a delamination of the inclusion takes place thus providing a presence of cracks between the inclusion and the elastic body. Nonlinear boundary conditions at the crack faces are imposed to prevent a mutual penetration between the faces. Different problem formulations are analyzed: variational and differential. Dependence on physical parameters characterizing the mechanical properties of the inclusion is investigated. The paper provides a rigorous asymptotic analysis of the model with respect to such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Also anisotropic inclusions with parameters are analyzed when parameters tend to zero and infinity. In particular, in the limit, we obtain the so called semi-rigid inclusions. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
26. 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
27. Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity
- Author
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Tomomi Yokota, Kentarou Fujie, and Michael Winkler
- Subjects
General Mathematics ,Bounded function ,Open problem ,Mathematical analysis ,General Engineering ,Neumann boundary condition ,Uniform boundedness ,Sensitivity (control systems) ,Special case ,Constant (mathematics) ,Domain (mathematical analysis) ,Mathematics - Abstract
This paper deals with the parabolic–elliptic Keller–Segel system with signal-dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u0 ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow-up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u0. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
28. Solutions for subquadratic fractional Hamiltonian systems without coercive conditions
- Author
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Rong Yuan and Ziheng Zhang
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,Genus (mathematics) ,Bounded function ,Mathematical analysis ,General Engineering ,Positive-definite matrix ,Critical point (mathematics) ,Mathematics ,Hamiltonian system - Abstract
In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems (FHS) where α ∈ (1 ∕ 2,1), , , is a symmetric and positive definite matrix for all , , and ∇ W is the gradient of W at u. The novelty of this paper is that, assuming L is bounded in the sense that there are constants 0
- Published
- 2014
29. A spectral element method using the modal basis and its application in solving second-order nonlinear partial differential equations
- Author
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Farhad Fakhar-Izadi and Mehdi Dehghan
- Subjects
Chebyshev polynomials ,Partial differential equation ,Differential equation ,General Mathematics ,Spectral element method ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,General Engineering ,Split-step method ,Nonlinear system ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Spectral method ,Numerical partial differential equations ,Mathematics - Abstract
We present a high-order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second-order partial differential equations in two-dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi-implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element-wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third-order in time. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
30. Variational approach to solutions for a class of fractional Hamiltonian systems
- Author
-
Ziheng Zhang and Rong Yuan
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,Genus (mathematics) ,media_common.quotation_subject ,Mathematical analysis ,General Engineering ,Positive-definite matrix ,Infinity ,Critical point (mathematics) ,Mathematics ,Hamiltonian system ,media_common - Abstract
In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: (FHS) where α ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ∇ W is the gradient of W at u. The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.
- Published
- 2013
31. On Fourier series for higher order (partial) derivatives of functions
- Author
-
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
32. Blow-up criterion for 3D viscous-resistive compressible magnetohydrodynamic equations
- Author
-
Shengquan Liu and Mingtao Chen
- Subjects
Strong solutions ,Resistive touchscreen ,General Mathematics ,Mathematical analysis ,General Engineering ,Compressibility ,Calculus ,Magnetohydrodynamic drive ,Navier–Stokes equations ,Mathematics - Abstract
In this paper, we establish a blow-up criterion of strong solutions for 3D viscous-resistive compressible magnetohydrodynamic equations, which depends only on and . Our result improves the previous criterion in Lu's paper (Journal of Mathematical Analysis and Applications 2011; 379: 425–438.) for compressible magnetohydrodynamic equations by removing a stringent condition on the viscous coefficients μ > 4λ. In addition, initial vacuum states are also allowed in our cases. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
33. Paley-Wiener theorems and uncertainty principles for the windowed linear canonical transform
- Author
-
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
34. Some basic boundary value problems of the plane thermoelasticity with microtemperatures
- Author
-
L. Bitsadze and George Jaiani
- Subjects
General Mathematics ,Bounded function ,Mathematical analysis ,Linear system ,General Engineering ,Fundamental solution ,Potential method ,Boundary value problem ,Uniqueness ,Singular integral ,Statics ,Mathematics - Abstract
The present paper is devoted to the two-dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2ϵ mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
35. Justification of the Ginzburg-Landau approximation for an instability as it appears for Marangoni convection
- Author
-
Dominik Zimmermann and Guido Schneider
- Subjects
Convection ,Amplitude ,Reflection symmetry ,Marangoni effect ,Classical mechanics ,General Mathematics ,Mathematical analysis ,General Engineering ,Scalar (physics) ,Wavenumber ,Instability ,Scaling ,Mathematics - Abstract
The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature-dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = kc and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = kc. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
36. Infinite dimensional attractors for porous medium equations in heterogeneous medium
- Author
-
Messoud Efendiev
- Subjects
General Mathematics ,Attractor ,Mathematical analysis ,Degenerate energy levels ,General Engineering ,Entropy (information theory) ,Porous medium ,Parabolic partial differential equation ,Upper and lower bounds ,Curse of dimensionality ,Mathematics - Abstract
In this paper, we give a detailed study of the global attractors for porous medium equations in a heterogeneous medium. Not only the existence but also the infinite dimensionality of the global attractors is obtained by showing that their ϵ-Kolmogorov entropy behaves as a polynomial of the variable 1 ∕ ϵ as ϵ tends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorov ϵ-entropy of infinite-dimensional attractors are also obtained. We believe that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
37. Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
- Author
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Tomomi Yokota, Sachiko Ishida, and Takashi Ono
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Degenerate energy levels ,Mathematics::Analysis of PDEs ,General Engineering ,Type (model theory) ,Mathematics - Abstract
This paper deals with the quasilinear ‘degenerate’ Keller–Segel system of parabolic–parabolic type under the super-critical condition. In the ‘non-degenerate’ case, Winkler (Math. Methods Appl. Sci. 2010; 33:12–24) constructed the initial data such that the solution blows up in either finite or infinite time. However, the blow-up under the super-critical condition is left as an open question in the ‘degenerate’ case. In this paper, we try to give an answer to the question under assuming the existence of local solutions. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
38. Local existence and blow-up criterion for the generalized Boussinesq equations in Besov spaces
- Author
-
Yi Du, Hua Qiu, and Zheng-an Yao
- Subjects
Physics::Fluid Dynamics ,Simultaneous equations ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Magnetohydrodynamics ,Boussinesq approximation (water waves) ,Fractional Laplacian ,System of linear equations ,Mathematics - Abstract
In this paper, we consider the three-dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian − Δ in the usual Boussinesq equations by a fractional Laplacian ( − Δ)α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin-type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
39. Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity
- Author
-
Daniel Sevcovic and Shigetoshi Yazaki
- Subjects
Mean curvature flow ,Tangential angle ,35K65, 35B35, 35K55, 53A10, 53C44 ,General Mathematics ,Mathematical analysis ,General Engineering ,Center of curvature ,Geometry ,Numerical Analysis (math.NA) ,Curvature ,Mathematics - Analysis of PDEs ,Fundamental theorem of curves ,Torsion of a curve ,FOS: Mathematics ,Total curvature ,Mathematics - Numerical Analysis ,Analysis of PDEs (math.AP) ,Osculating circle ,Mathematics - Abstract
In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the normal direction with a velocity which is assumed to be a function of the curvature, tangential angle and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so-called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations we prove local existence, uniqueness and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities like the length, enclosed area or total elastic energy of a curve. We also propose a stable numerical approximation scheme based on the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper., submitted to MMAS
- Published
- 2012
40. On the formulation of boundary value problems with the incompressible constituents constraint in finite deformation poroelasticity
- Author
-
Thomas J. Pence
- Subjects
Stress (mechanics) ,Classical mechanics ,Deformation (mechanics) ,Continuum mechanics ,Cauchy stress tensor ,General Mathematics ,Hyperelastic material ,Deformation theory ,Mathematical analysis ,General Engineering ,Boundary value problem ,Tensor ,Mathematics - Abstract
This paper considers the finite deformation theory of poroelasticity for the case in which a deformable solid constituent and an interpenetrating liquid constituent are each regarded as incompressible, and the mixing itself is regarded as taking place without the creation of voids. The resulting kinematical constraint gives rise to a Lagrange multiplier pressure in the resulting constitutive description. This pressure therefore enters into the separate momentum balance statements for each individual constituent. The formulation of boundary value problems in this context is well known in continuum mechanics. This paper examines how a systematic elimination of the Lagrange multiplier pressure from the mathematical formulation leads to a stress-like tensor that generalizes a stress tensor concept introduced by Rajagopal and Wineman in the late 1980s, which they called the saturation stress. Here, by providing a rather complete development, it is discussed how boundary value problems are naturally formulated in terms of a single such stress tensor, how the constitutive theory is framed in terms of this stress tensor, and how certain questions concerning the formulation of boundary conditions are naturally addressed. Connections to the small deformation linear poroelastic (biphasic) theory are also provided. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
41. Uniqueness results in the inverse scattering problem for periodic structures
- Author
-
Jiaqing Yang and Bo Zhang
- Subjects
Scattering ,General Mathematics ,Quasiperiodic function ,Reciprocity (electromagnetism) ,Mathematical analysis ,Inverse scattering problem ,General Engineering ,Inverse ,Uniqueness ,Grating ,Inverse problem ,Mathematics - Abstract
This paper is concerned with the inverse electromagnetic scattering by a 2D (impenetrable or penetrable) smooth periodic curve. Precisely, we establish global uniqueness results on the inverse problem of determining the grating profile from the scattered fields corresponding to a countably infinite number of quasiperiodic incident waves. For the case of an impenetrable and partially coated perfectly reflecting grating, we prove that the grating profile and its physical property can be uniquely determined from the scattered field measured above the periodic structure. For the case of a penetrable grating, we show that the periodic interface can be uniquely recovered by the scattered field measured only above the interface. A key ingredient in our proofs is a novel mixed reciprocity relation that is derived in this paper for the periodic structures and seems to be new. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
42. Multiphase double-porosity homogenization for perimeter functionals
- Author
-
Margherita Solci
- Subjects
Perimeter ,Connected component ,General Mathematics ,Mathematical analysis ,General Engineering ,Asymptotic formula ,Disjoint sets ,Boundary value problem ,Porosity ,Homogenization (chemistry) ,Finite set ,Mathematics - Abstract
In this paper, we study a homogenization problem for perimeter energies in highly contrasted media; the analysis of the previous paper is carried out by removing the hypothesis that the perforated medium Rn ∖ E is composed of disjoint compact components. Assuming E to be the union of a finite number N of connected components E1, … ,EN, the Γ-limit F is a multiphase energy with a ‘decoupled’ surface part, obtained by homogenization from the surface tensions in each E j, a trivial bulk term obtained as a weak limit, and a further interacting term between the phases, involving an asymptotic formula for a family minimum problems on invading an asymptotic formula for a family of minimum problems on invading domains with prescribed boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
43. Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness
- Author
-
Neeri Agarwal, Arun Kumar Gupta, and Harvinder Kaur
- Subjects
Differential equation ,business.industry ,General Mathematics ,Logarithmic decrement ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Separation of variables ,Fundamental frequency ,Orthotropic material ,Viscoelasticity ,Optics ,Normal mode ,Deflection (engineering) ,business ,Mathematics - Abstract
An analysis of the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plates of parabolically varying thickness in the radial direction is presented. The thickness of a circular plate varies parabolically in a radial direction. For nonhomogeneity of the circular plate material, density is assumed to vary linearly in a radial direction. This paper used the method of separation of variables in solving the governing differential equation. In this paper, an approximate but quite convenient frequency equation is derived by using the Rayleigh–Ritz technique with a two-term deflection function. Deflection, time period and logarithmic decrement for the first two modes of vibration are computed for the nonhomogeneous orthotropic viscoelastic circular plates of varying parabolic thickness with clamped edge conditions for various values of nonhomogeneity constants and taper constants and these are shown in tabular form for the Voigt–Kelvin model. Copyright © 2011 John Wiley & Sons, Ltd.
- Published
- 2011
44. Global solutions and blow-up problems to a porous medium system with nonlocal sources and nonlocal boundary conditions
- Author
-
Haihua Lu
- Subjects
General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Nonlocal boundary ,Boundary value problem ,Diffusion (business) ,Finite time ,Porous medium ,Mathematics - Abstract
This paper deals with a porous medium system with nonlocal sources and weighted nonlocal boundary conditions. The main aim of this paper is to study how the reaction terms, the diffusion terms, and the weight functions in the boundary conditions affect the global and blow-up properties to a porous medium system. The conditions on the global existence and blow-up in finite time for nonnegative solutions are given. Furthermore, the blow-up rate estimates of the blow-up solutions are also established. Copyright © 2011 John Wiley & Sons, Ltd.
- Published
- 2011
45. Estimates for the asymptotic expansion of a viscous fluid satisfying Navier's law on a rugous boundary
- Author
-
Juan Casado-Díaz, Francisco Javier Suárez-Grau, and Manuel Luna-Laynez
- Subjects
Boundary layer ,Amplitude ,General Mathematics ,Law ,Mathematical analysis ,General Engineering ,No-slip condition ,Boundary (topology) ,Viscous liquid ,Stokes flow ,Asymptotic expansion ,Mathematics ,Ansatz - Abstract
In a previous paper, we have studied the asymptotic behavior of a viscous fluid satisfying Navier's law on a periodic rugous boundary of period e and amplitude δe, with δe/e tending to zero. In the critical size, δe∼e3/2, in order to obtain a strong approximation of the velocity and the pressure it is necessary to consider a boundary layer term in the corresponding ansatz. The purpose of this paper is to estimate the approximation given by this ansatz. Copyright © 2011 John Wiley & Sons, Ltd.
- Published
- 2011
46. Numerical iterative method for Volterra equations of the convolution type
- Author
-
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
47. The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity
- Author
-
Wei Yan and Yongsheng Li
- Subjects
Sobolev space ,Conservation law ,Pure mathematics ,Homogeneous ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Engineering ,Bilinear interpolation ,Initial value problem ,Wave equation ,Mathematics - Abstract
This paper is devoted to studying the initial-value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces Xs, b(R2) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I-method as well as L2 conservation law, we show that this fifth-order shallow water wave equation is globally well-posed for the initial data in the Sobolev spaces Hs(R) with . Copyright © 2010 John Wiley & Sons, Ltd.
- Published
- 2010
48. Non-perturbative solution of three-dimensional Navier-Stokes equations for the flow near an infinite rotating disk
- Author
-
Sefa Anıl Sezer and Ahmet Yildirim
- Subjects
Algebraic equation ,Runge–Kutta methods ,Nonlinear system ,Flow (mathematics) ,Transcendental equation ,General Mathematics ,Numerical analysis ,Mathematical analysis ,General Engineering ,Padé approximant ,Navier–Stokes equations ,Mathematics::Numerical Analysis ,Mathematics - Abstract
In this paper, we present Homotopy perturbation method (HPM) and Pade technique, for finding non-perturbative solution of three-dimensional viscous flow near an infinite rotating disk. We compared our solution with the numerical solution (fourth-order Runge–Kutta). The results show that the HPM–Pade technique is an appropriate method in solving the systems of nonlinear equations. The mathematical technique employed in this paper is significant in studying some other problems of engineering. Copyright © 2009 John Wiley & Sons, Ltd.
- Published
- 2009
49. Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids
- Author
-
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
50. Analysis for the identification of an unknown diffusion coefficient via semigroup approach
- Author
-
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
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