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In a previous paper, we presented 3 + 2M term recurrence relations with variable dependent coefficients for M-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we present (conjectures of) the recurrence relations with constant coefficients for these multi-indexed orthogonal polynomials. The simplest recurrence relations have 3 + 2l terms, where l (= M) is the degree of the lowest member of the multi-indexed orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present an integrable semi-discretization of the modified Korteweg-deVries (mKdV) equation. We discretize the “time” variable of the mKdV equation and get an integrable differential-difference system. Under a standard limit, the differential-difference system converges to the continuous mKdV equation. By Hirota’s bilinear method, we find some explicit solutions including solitons and breather solutions. From the semi-discrete system, we design a numerical scheme to the mKdV equation and carry out several numerical experiments with the 3-soliton solution and breather solution. [ABSTRACT FROM AUTHOR]

In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their "physical" counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories. [ABSTRACT FROM AUTHOR]

In this paper, we consider the compressible quantum Navier-Stokes-Poisson equations with a linear density-dependent viscosity. By the use of a singular pressure close to vacuum, we prove the global-in-time existence of weak solutions in a three-dimensional torus for large data in the sense of distribution. [ABSTRACT FROM AUTHOR]

In previous papers derivations of the Green function have been given for 5D off-shell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter τ). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the conventional fundamental solutions of linear 5D wave equations published in the mathematical literature. We give physical arguments for the choice of the Green function retarded in the fifth variable τ. [ABSTRACT FROM AUTHOR]

We consider the MHD-α model known as double viscous simplified Bardina MHD (SBMHD) on a 3D periodic box (MHD stands for magnetohydrodynamic). This system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the MHD model neither analytically nor via direct numerical simulation. In a previous paper (joint work with Secchi), the global existence of strong solutions to the SBMHD has been proved as well as the existence of a global attractor of finite fractal dimension. Upper bounds for such dimension are provided both in terms of the modified Grashof number and the dissipation length associated to the mean rate of energy dissipation. In this paper, we commute the same bound in an estimate in terms of the modified Reynolds number R. This result is useful because the classical Kolmogorov theory of turbulence is expressed using the Reynolds number. The global attractor estimate in terms of R is consistent with scaling theories of turbulence and can be interpreted in terms of degrees of freedom of the flow in Landau's sense. However, it does not take into account the effect of strong dissipation-range intermittency, where significant energy lies in wavenumbers larger than the inverse Kolmogorov length. Hence, we prove that there exists a series of time-averaged inverse square length-scales whose members are estimated in terms of R. [ABSTRACT FROM AUTHOR]

Nonisotropic spatiotemporal chaotic vibrations can happen for a linear wave equation with a mixing transport term on the unit interval and with a nonlinear van der Pol boundary condition at one end when the parameter(s) enters some regime [Chen et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 535 (2002)]. In this paper, on the one hand, the equation with general nonlinear boundary conditions is considered. On the other hand, parameter range for the route to chaos is more precisely classified. The results obtained here generalize and sharpen those both in the above paper and the earlier work of [Li, L. L. and Huang, Y., J. Math. Anal. Appl. 361, 69 (2010)]. Two examples and their corresponding numerical simulations of chaotic space-time profiles are also illustrated. [ABSTRACT FROM AUTHOR]

The subject of this paper is solutions of an autoresonance equation.We look for a connection between the parameters of the solution bounded as t ?-8, and the parameters of two two-parameter families of solutions as t ?8. One family consists of the solutions which are not captured into resonance, and another of those increasing solutions which are captured into resonance. In thiswaywe describe the transition through the separatrix for equations with slowly varying parameters and get an estimate for parameters before the resonance of those solutions which may be captured into autoresonance. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the phase portraits features of a barothropic relaxing medium under pressure perturbations. In the starting point, we show within a third-order of accuracy that the previous system is modeled by a "dissipative" cubic nonlinear evolution equation. Paying particular attention to high-frequency perturbations of the system, we solve the initial value problem of the system both analytically and numerically while unveiling the existence of localized multivalued waveguide channels. Accordingly, we find that the "dissipative" term with a "dissipative" parameter less than some limit value does not destroy the ambiguous solutions. We address some physical implications of the results obtained previously. [ABSTRACT FROM AUTHOR]

In this paper, we construct a reciprocal transformation for the Geng-Xue equation and show that, with help of this transformation, we relate the first negative flow of the modified Boussinesq hierarchy to the Geng-Xue equation. Furthermore, we analyze the construction of conserved quantities and present new ones. [ABSTRACT FROM AUTHOR]

A magnetic field is defined by the property that its divergence is zero in a threedimensional oriented Riemannian manifold. Each magnetic field generates amagnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a three-dimensional oriented Riemann manifold, (M³, g). And then, we investigate the trajectories of the magnetic fields called as N-magnetic and Bmagnetic curves. [ABSTRACT FROM AUTHOR]

In the paper, fractional differential inclusions with the Jumarie derivative are studied. We discuss the existence and uniqueness of a solution to such problems. Our study relies on standard variational methods. [ABSTRACT FROM AUTHOR]

In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Liénard type equation, x+f(x)x2+g(x)=0, where f(x) and g(x) are arbitrary functions of x. The symmetry analysis gets divided into two cases, (i) the maximal (eight parameter) symmetry group and (ii) non-maximal (three, two, and one parameter) symmetry groups. We identify the most general form of the quadratic Liénard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the non-maximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel method of reconstructing acoustic obstacles in R2, which follows a similar spirit of the linear sampling method originated by Colton and Kirsch. The reconstruction scheme makes use of the near-field measurements encoded into the boundary Dirichlet-to-Neumann map or the Neumann-to-Dirichlet map. Both the plane waves and cylindrical waves are shown to meet the reconstruction purpose. Rigorous mathematical justification of the reconstruction scheme is established. The mapping properties of the newly introduced function operators involved in the reconstruction scheme are established. These results are of significant mathematical interests for their own sake. Moreover, due to the distinct properties of the function operators, the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems. Numerical experiments are presented to illustrate the effectiveness of the proposed reconstruction scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is threefold. The first and also main purpose is to provide numerical evidence for the conjecture proposed by Bizon et al. ['Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation,' J. Math. Phys. 52, 103703 (2011)] that the blowup evolution of spherically symmetric semilinear Klein-Gordon equations is similar to the evolution of spherically symmetric semilinear wave equations, i.e., the mass term can be neglected when the amplitude of a solution grows. The second aim is to describe the relationship between different types of blowup for energy critical semilinear wave equations. The third goal is to present numerical evidence for the fact that the special class of self-similar profiles of semilinear wave equations found by Kycia ['On self-similar solutions of semilinear wave equations in higher space dimensions,' Appl. Math Comput. 217, 9451-9466 (2011)] play the same role in the evolution of semilinear wave and Klein-Gordon equations as the previously known ordinary profiles. All the results are presented in spherical symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we analyze qualitative properties of the first multi-symplectic scheme for two-dimensional Maxwell's equations. We prove that the scheme is unconditionally stable and convergent, non-dissipative, and divergence-free. The numerical dispersion relation of the scheme is shown to converge to the exact dispersion relation of the Maxwell equations. We also present some numerical results to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4 in 3 × 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis. The product vectors in the kernels of the states belong to a non-zero measure subset of all general unextendible product bases, nevertheless, they can always be locally transformed to the orthogonal form. This fully confirms the surprising numerical results recently reported by Leinaas et al. Parts of the paper rely heavily on the use of Bezout's theorem from algebraic geometry. [ABSTRACT FROM AUTHOR]

In this paper, we propose an effective numerical method for solving nonlinear Volterra partial integro-differential equations. These equations include the partial differentiations of an unknown function and the integral term containing the unknown function as the 'memory' of system. Radial basis functions and finite difference method as the main techniques play the important role to reduce a nonlinear partial integro-differential equation to a linear system of equations. Some examples are demonstrated to describe the method. Numerical results confirm the validity and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonuniform dependence on initial data for the general b-equation. We prove that the solution map of the Cauchy problem of the b-equation is not uniformly continuous in Hs(R), s > 3/2. [ABSTRACT FROM AUTHOR]

A Lyapunov operator is a self-adjoint quantum observable whose expectation value varies monotonically as time increases and may serve as a marker for the flow of time in a quantum system. In this paper it is shown that the existence of a certain type of Lyapunov operator leads to representations of the quantum dynamics, termed transition representations, in which an evolving quantum state ψ(t) is decomposed into a sum ψ(t) = ψb(t) + ψf(t) of a backward asymptotic component and a forward asymptotic component such that the evolution process is represented as a transition from ψb(t) to ψf(t). When applied to the evolution of scattering resonances, such transition representations separate the process of decay of a scattering resonance from the evolution of outgoing waves corresponding to the probability 'released' by the resonance and carried away to spatial infinity. This separation property clearly exhibits the spatial probability distribution profile of a resonance. Moreover, it leads to the definition of exact resonance states as elements of the physical Hilbert space corresponding to the scattering problem. These resonance states evolve naturally according to a semigroup law of evolution. [ABSTRACT FROM AUTHOR]

Motivated by the reservoir engineering concept of the well productivity index (PI) we study a time dependent functional for general nonlinear Forchheimer equation. The PI of the well characterizes the well capacity with respect to drainage area of the well. Unlike the linear case for which this concept is well developed, there are only a few recent publications dedicated to the PI for nonlinear case. In this paper the PI is comprehensively studied both theoretically and numerically. The impact of the nonlinearity of the flow filtration on the value of the PI is analyzed. Exact formula for the so called 'skin factor' in radial case is derived depending on the rate of the flow, the order of nonlinearity and the geometric parameters. Dynamics of the PI for the class of boundary conditions is studied and its convergence to the specific value of steady state PI was justified. Developed framework is applied to obtain nonlinear analog of Peaceman formula for the well-block pressure in unstructured grid. Numerical simulations sustain theoretical results. [ABSTRACT FROM AUTHOR]

The wavelet transform method originated by Wei et al. [Phys. Rev. Lett. 89, 284103.4 (2002)] was proved [Juang and Li, J. Math. Phys. 47, 072704.16 (2006); Juang et al., J. Math. Phys. 47, 122702.11 (2006); Shieh et al., J. Math. Phys. 47, 082701.10 (2006)] to be an effective tool to reduce the order of coupling strength for coupled chaotic systems to acquire the synchrony regardless the size of oscillators. In Juang et al., [IEEE Trans. Circuits Syst., I: Regul. Pap. 56, 840 (2009)] such method was applied to coupled map lattices (CMLs). It was demonstrated that by adjusting the wavelet constant of the method can greatly increase the applicable range of coupling strengths, the parameters, range of the individual oscillator, and the number of nodes for local synchronization of CMLs. No analytical proof is given there. In this paper, the optimal or near optimal wavelet constant can be explicitly identified. As a result, the above described scenario can be rigorously verified. [ABSTRACT FROM AUTHOR]

Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean. [ABSTRACT FROM AUTHOR]

We introduce in this paper the contractions Gc of n-Lie (or Filippov) algebras G and show that they have a semidirect structure as their n = 2 Lie algebra counterparts. As an example, we compute the nontrivial contractions of the simple An+1 Filippov algebras. By using the İnönü-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the G=An+1 simple case) the Lie algebras Lie Gc (the Lie algebra of inner endomorphisms of Gc) with certain contractions (Lie G)IW and (Lie G)W-W of the Lie algebra Lie G associated with G. [ABSTRACT FROM AUTHOR]

In this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times. [ABSTRACT FROM AUTHOR]

In this paper, two multisymplectic integrators, an eight-point Preissman box scheme and a narrow box scheme, are considered for numerical integration of the complex modified Korteweg-de Vries equation. Energy and momentum preservation of both schemes and their dispersive properties are investigated. The performance of both methods is demonstrated through numerical tests on several solitary wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the modified Camassa–Holm equation is Painlevé integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation. [ABSTRACT FROM AUTHOR]

In this paper, we systematically develop the “ghost” symmetry of the Kadomtsev-Petviashvili sub-hierarchy of B type (BKP) through its actions on the Lax operator L, the eigenfunctions, and the τ function. In this process, the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the BKP hierarchy. Moreover, the bilinear identity of the constrained BKP hierarchy and Adler–Shiota–van Moerbeke formula of the BKP hierarchy are rederived compactly by means of the spectral representation and “ghost” symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimal control problem for the viscous nonlinear dispersive wave equation. We first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Furthermore, the existence of an optimal solution to the viscous nonlinear dispersive wave equation is proved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of weak solutions of the coupled time-dependent Ginzburg-Landau equations and establish the global existence of weak solutions to the equations by Galerkin method and compactness theory. [ABSTRACT FROM AUTHOR]

The Heisenberg supermagnet model is an integrable supersymmetric system and has a close relationship with the strong electron correlated Hubbard model. In this paper, we investigate the integrable higher order deformations of Heisenberg supermagnet models with two different constraints: (i) S2=3S-2I for S∈USPL(2/1)/S(U(2)×U(1)) and (ii) S2=S for S∈USPL(2/1)/S(L(1/1)×U(1)). In terms of the gauge transformation, their corresponding gauge equivalent counterparts are derived. [ABSTRACT FROM AUTHOR]

In this paper, we investigate general decay rates of solutions for a quasilinear viscoelastic equation with nonlinear weak damping. [ABSTRACT FROM AUTHOR]

This paper studies low-regularity solutions of the periodic Fornberg–Whitham equation with initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic Fornberg–Whitham equation. [ABSTRACT FROM AUTHOR]

In this paper, Grammian solutions of the Ishimori-I (Ish-I) equation are first obtained by Hirota’s direct method. Utilizing the source generation procedure, this equation with self-consistent sources is then presented and the corresponding Grammian solutions are derived. Finally, as a simple case, the (1,1) dromion solution is examined. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new ten-point multisymplectic scheme for the regularized long-wave equation from its Bridges’ multisymplectic form. The new scheme is an explicit scheme in the sense that it does not need iteration. We discuss some properties of the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long-wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well. [ABSTRACT FROM AUTHOR]

The paper gives the analytical solution for postbuckling of an initially circular inextensible elastic ring under uniform external pressure. [ABSTRACT FROM AUTHOR]

In the paper, we adapt our previous formalism for a mathematical treatment of branes to include processes, specifically the Kondo flow for Wess-Zumino-Witten (WZW) branes. In this framework, we give the precise mathematical definitions and formulate a mathematical conjecture relating WZW branes to nonequivariant twisted K theory in the case of the group SU(n). We also discuss regularization of the Kondo flow, thereby giving a first step toward proving our conjecture. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of the Cauchy problem associated with an n-dimensional generalized Benney-Luke equation, utt-mΔu-Δutt+Δ2u+α(2∇u·∇ut+utΔu)+β∇(|∇u|p∇u)=0, where n=1,2,3,4. We prove the existence and the uniqueness of the global solution of the Cauchy problem for the β≼0 case by using energy conservation law and give the existence and the nonexistence of the global solution of the Cauchy problem for the β>0 case by constructing the stable set and the unstable set. [ABSTRACT FROM AUTHOR]

In this paper we consider the role that numerical computations—in particular Galerkin approximations—can play in problems modeled by the three-dimensional (3D) Navier–Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem. We then consider Galerkin approximations, and show that if a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier–Stokes equations are regular one can justify this particular numerical method rigorously. Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time. We thus introduce the possibility of rigorous computations of the solutions of the 3D Navier–Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples. [ABSTRACT FROM AUTHOR]

The inverse problem in the calculus of variations for a given set of second-order ordinary differential equations consists of deciding whether their solutions are those of Euler–Lagrange equations and whether the Lagrangian, if it exists, is unique. This paper discusses the exterior differential systems approach to this problem. In particular, it proposes an algorithmic procedure towards the construction of a certain differential ideal. The emphasis is not so much on obtaining a complete set of integrability conditions for the problem, but rather on producing a minimal set to expedite the differential ideal process. [ABSTRACT FROM AUTHOR]

In this paper, we review various methods for the numerical approximations of the Ginzburg–Landau models of superconductivity. Particular attention is given to the different treatment of gauge invariance in both the finite element, finite difference, and finite volume settings. Representative theoretical results, typical numerical simulations, and computational challenges are presented. Generalizations to other relevant models are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we give a condition on the bicharacteristic which guarantees the global existence of the mild solution to the Boltzmann equation with an external force for the hard-sphere model and potentials with angular cutoff in infinite vacuum. This generalizes the previous results to the case when the force can have arbitrary strength. The constructive condition on the bicharacteristic is used to obtain the pointwise estimates on the collision operator so that the global existence comes from the contraction mapping theorem. [ABSTRACT FROM AUTHOR]

The generalized Stieltjes transform (GST) is an integral transform that depends on a parameter ρ>0. In previous work a convenient form of the inverse transformation was derived for the case ρ=3/2. This paper generalizes that result to all ρ>0. It is a well-known fact that the GST can be formulated as an iterated Laplace transform, and that therefore its inverse can be expressed as an iterated inverse Laplace transform. The form of the inverse transform derived here is a one-dimensional integral that is considerably simpler. [ABSTRACT FROM AUTHOR]

The Poincaré equation, a second-order partial differential equation describing wave motions in a rotating spheroid of arbitrary eccentricity satisfying a certain set of the boundary condition, is studied. A new polynomial as the general solution of the Poincaré equation in spheroidal geometry is found for the first time. The paper focuses on some unusual and intriguing mathematical properties of the new Poincaré polynomial. The possible completeness of the set of eigenfunctions of the Poincaré equation in the form of the new polynomial is also discussed. The new Poincaré polynomial would provide a powerful basis for the mathematical analysis in many important geophysical and astrophysical problems. [ABSTRACT FROM AUTHOR]

A new characterization for global hyperbolicity is given. The concept of cosmological time function and its regularity was considered by Anderson et al. ['The cosmological time function,' Class. Quantum Grav. 15, 309 (1998)] and it was proved that if the cosmological time function of (M, g) is regular then it is globally hyperbolic. In this paper it is proved that if (M, g) is globally hyperbolic then there is a smooth function Ω > 0 such that the cosmological time function of (M, Ωg) is regular. It is also proved that the cosmological time function of Friedman-Robertson-Walker spacetime ((a, b) × f H, -dt2 + f h), a, b < ∞, is regular and in addition the regularity of cosmological time function for this kind of spacetimes is stable in Lor(M). [ABSTRACT FROM AUTHOR]

We study an infinite system of particles chaotically distributed over a Euclidean space R d . Particles are characterized by their positions x ∈ R d and an internal parameter (spin) σ x ∈ R m and interact via position-position and (position dependent) spin-spin pair potentials. Equilibrium states of such system are described by Gibbs measures on a marked configuration space. Due to the presence of unbounded spins, the model does not fit the classical (super-) stability theory of Ruelle. The main result of the paper is the derivation of sufficient conditions of the existence and uniqueness of the corresponding Gibbs measures. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional Schrödinger operator with a δ-interaction of strength α > 0 supported on an unbounded surface parametrized by the mapping R 2 ∋ x ↦ ( x , β f ( x ) ) , where β ∈ 0 , ∞ and f : R 2 → R , f ≢ 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrödinger operator coincides with − 1 4 α 2 , + ∞. We prove that for all sufficiently small β > 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit β → 0+. In particular, this eigenvalue tends to − 1 4 α 2 exponentially fast as β → 0+. [ABSTRACT FROM AUTHOR]

In this paper, after a brief review of the general theory of dispersive waves in dissipative media, we present a complete discussion of the dispersion relations for both the ordinary and the time-fractional Cattaneo-Maxwell heat equations. Consequently, we provide a complete characterization of the group and phase velocities for these two cases, together with some non-trivial remarks on the nature of wave dispersion in fractional models. [ABSTRACT FROM AUTHOR]

In a remarkable note of Chadan [Il Nuovo Cimento 39, 697–703 (1965)], the author expanded both the regular wave function and the Jost function of the quantum scattering problem using an interpolation theorem of Valiron [Bull. Sci. Math. 49, 181–192 (1925)]. These expansions have a very slow rate of convergence, and applying them to compute the zeros of the Jost function, which lead to the important bound states, gives poor convergence rates. It is our objective in this paper to introduce several efficient interpolation techniques to compute the regular wave solution as well as the Jost function and its zeros approximately. This work continues and improves the results of Chadan and other related studies remarkably. Several worked examples are given with illustrations and comparisons with existing methods. [ABSTRACT FROM AUTHOR]