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Research Interests: Mechanical Engineering, Civil Engineering, Ocean Engineering, Aerospace Engineering, Mathematics, and 11 moreApplied Mathematics, Computational Mechanics, Numerical Analysis, Finite element method, Mathematical Analysis, Finite Difference, Smoothed Finite Element Method, Eigenvalue Problems, Civil and Structural Engineering, Finite Difference Methods, and Building and Construction
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The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search... more
The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p 0 to reach ‘by chance’ a site p for the first time? Further important are recurrence and transience features of a random walk: A random walker starting at p 0 will he ever reach site p (ever return to p 0)? How often a site is visited? Here we investigate Markovian random walks generated by fractional (Laplacian) generator matrices L\( \frac{\alpha }{2} \) 2 (0 < \( \alpha \) ≤2) where L stands for ‘simple’ Laplacian matrices. This walk we refer to as ‘Fractional Random Walk’ (FRW). In contrast to classical Polya type walks where only local steps to next neighbor sites are possible, the FRW allows nonlocal long-range moves where a remarkably rich dynamics and new features arise. We analyze recurrence and transience features of the FRW on infinite d-dimensional simple cubic lattices. We deduce by means of lattice Green’s function (probability generating functions) the mean residence times (MRT) of the walker at preselected sites. For the infinite 1D lattice (infinite ring) we obtain for the transient regime (0 < \( \alpha \) < 1) closed form expressions for these characteristics. The lattice Green’s function on infinite lattices existing in the transient regime fulfills Riesz potential asymptotics being a landmark of anomalous diffusion, i.e. random motion (Levy flights) where the step lengths are drawn from a Levy \( \alpha \)-stable distribution.
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Abstract The behavior of 1-dimensional shock waves in deformable dielectric materials with polarization gradients, which are non-conductor of heat, is analyzed in the case of quasi-electrostatics. The differential equation governing the... more
Abstract The behavior of 1-dimensional shock waves in deformable dielectric materials with polarization gradients, which are non-conductor of heat, is analyzed in the case of quasi-electrostatics. The differential equation governing the amplitude of wave and criteria concerning the polarization gradient and temperature changes across the shock are deduced.
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Research Interests: Mathematics, Applied Mathematics, Physics, Statistical Mechanics, Fractional Differential Equations, and 8 moreFractional calculus and its applications, Mathematical Sciences, Mathematical Analysis, Physical sciences, Generalized Functions, Fractional Calculus, Laplace operator, and Limit (Mathematics)
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Emploi de la theorie des ondes non lineaires pour mettre en evidence, ou etudier des influences caracteristiques, des phenomenes dynamiques dans des materiaux a proprietes de couplages electromecaniques. Traitement des ondes... more
Emploi de la theorie des ondes non lineaires pour mettre en evidence, ou etudier des influences caracteristiques, des phenomenes dynamiques dans des materiaux a proprietes de couplages electromecaniques. Traitement des ondes d'acceleration et de choc dans des cristaux ioniques; proposition d'une classification thermodynamique des vitesses de propagation d'une discontinuite faible qui, sous certaines conditions, peut se transformer en une discontinuite forte (ou onde de choc). Etude analytique des ondes de choc dans les materiaux piezoelectriques non lineaires pour mieux cerner les phenomenes de polarisation et depolarisation induite par une discontinuite forte. Analyse de l'influence des pertes dielectriques dans les milieux piezoelectriques dans le cas de la propagation d'ondes de choc, d'impulsions de haute frequence ou d'ondes transitoires dans un massif semi-infini. Determination de la distance de formation d'une onde de choc (ou d'acceleratio...
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ABSTRACT We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each... more
ABSTRACT We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders of the displacement field leading by application of Hamilton&#39;s variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. &quot;Anomalous&quot; dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.
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Two recursive surface impedance methods are described for acoustic wave propagation in multilayered piezoelectric structures. Both methods have the advantage of conceptual simplicity and flexibility brought about by the transfer matrix... more
Two recursive surface impedance methods are described for acoustic wave propagation in multilayered piezoelectric structures. Both methods have the advantage of conceptual simplicity and flexibility brought about by the transfer matrix method. Moreover they do not have a priori computational limitations with respect to the total number of layers of the stratified structure nor with respect to the thickness of individual layers; nor is the computational stability limited by the frequency range. For both methods, numerical simulations were carried out in order to illustrate their performances and robustness when combined with suitable recursive numerical algorithms.
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The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search... more
The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p 0 to reach ‘by chance’ a site p for the first time? Further important are recurrence and transience features of a random walk: A random walker starting at p 0 will he ever reach site p (ever return to p 0)? How often a site is visited? Here we investigate Markovian random walks generated by fractional (Laplacian) generator matrices L\( \frac{\alpha }{2} \) 2 (0 < \( \alpha \) ≤2) where L stands for ‘simple’ Laplacian matrices. This walk we refer to as ‘Fractional Random Walk’ (FRW). In contrast to classical Polya type walks where only local steps to next neighbor sites are possible, the FRW allows nonlocal long-range moves where a remarkably rich dynamics and new features arise. We analyze recurrence and transience features of the FRW on infinite d-dimensional simple cubic lattices. We deduce by means of lattice Green’s function (probability generating functions) the mean residence times (MRT) of the walker at preselected sites. For the infinite 1D lattice (infinite ring) we obtain for the transient regime (0 < \( \alpha \) < 1) closed form expressions for these characteristics. The lattice Green’s function on infinite lattices existing in the transient regime fulfills Riesz potential asymptotics being a landmark of anomalous diffusion, i.e. random motion (Levy flights) where the step lengths are drawn from a Levy \( \alpha \)-stable distribution.
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We introduce positive elastic potentials in the harmonic approximation leading by Hamilton's variational principle to fractional Laplacian matrices having the forms of power law matrix functions of the simple local Bornvon Karman... more
We introduce positive elastic potentials in the harmonic approximation leading by Hamilton's variational principle to fractional Laplacian matrices having the forms of power law matrix functions of the simple local Bornvon Karman Laplacian. The fractional Laplacian matrices are well defined on periodic and infinite lattices in $n=1,2,3,..$ dimensions. The present approach generalizes the central symmetric second differenceoperator (Born von Karman Laplacian) to its fractional central symmetric counterpart (Fractional Laplacian matrix).For non-integer powers of the Born von Karman Laplacian, the fractional Laplacian matrix is nondiagonal with nonzero matrix elements everywhere, corresponding to nonlocal behavior: For large lattices the matrix elements far from the diagonal expose power law asymptotics leading to continuum limit kernels of Riesz fractional derivative type. We present explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear...
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The goal of the present work deals with the analysis the propagation of shear horizontal (SH) and Lamb waves in piezoelectric sandwich structures. For both kinds of wave, the exact solutions are obtained using the formulation of state... more
The goal of the present work deals with the analysis the propagation of shear horizontal (SH) and Lamb waves in piezoelectric sandwich structures. For both kinds of wave, the exact solutions are obtained using the formulation of state vector combined with the method of surface local impedance and the technique of the global matrix. The spectra of dispersion, the asymptotic behaviours as well as the profiles across the thickness of the composite structures are numerically obtained. The surface (Bleustein-Guylaev and piezoelectric Rayleigh) and interface (Maerfeld-Tournois and piezoelectric Stoneley) waves obtained at high frequency are examined and discussed.
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Research Interests: Mathematics, Applied Mathematics, Physics, Statistical Mechanics, Fractional Differential Equations, and 8 moreFractional calculus and its applications, Mathematical Sciences, Mathematical Analysis, Physical sciences, Generalized Functions, Fractional Calculus, Laplace operator, and Limit (Mathematics)
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Research Interests: Engineering, Physics, Applied Physics, Multidisciplinary, Mathematical Sciences, and 12 moreMathematical Analysis, Physical sciences, Potassium, Piezoelectricity, Boundary Condition, Rayleigh Scattering, Boundary Value Problem, Rayleigh Waves, Rayleigh Wave, Electric Field, Surface Wave, and Polynomial
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Abstract In this paper, the behavior of plane shock waves propagating in elastic dielectrics is examined. The differential equation governing the amplitude of wave and formulae concerning the polarization changes across the shock are... more
Abstract In this paper, the behavior of plane shock waves propagating in elastic dielectrics is examined. The differential equation governing the amplitude of wave and formulae concerning the polarization changes across the shock are deduced. Some particular situations are studied in detail.