Terms Of Confluent Hypergeometric Functions Research Articles

Transient Analysis of Slip Flow and Heat Transfer in Microchannels

Hybrid analytical-numerical solutions for transient flow and transient convective heat transfer within microchannels are presented. Analytical solutions for flow transients in microchannels are obtained by making use of the integral transform approach. The proposed model involves the transient fully developed flow equation for laminar regime and incompressible flow with slip at the walls in simple channel geometries. The solution is constructed so as to account for any general functional form of the time variation of the pressure gradient along the duct. Then, the transient-state convection heat transfer is solved for laminar slip flow inside microchannels formed by parallel-plates, making use of the generalized integral transform technique (GITT) and the exact analytical solution of the corresponding eigenvalue problem in terms of confluent hypergeometric functions so as to eliminate the transversal coordinate. The resulting system of transformed partial differential equations in the longitudinal coordinate is numerically solved by the Method of Lines as implemented in the routine NDSolve of the Mathematica system. Mixed symbolic-numerical algorithms are developed under the Mathematica platform.

Relativistic and nonrelativistic solutions for diatomic molecules in the presence of double ring-shaped Kratzer potential

The authors investigate solutions of the three dimensional Klein-Gordon and Schrodinger equations in the presence of a new exactly solvable potential of V(r,theta)=-2De(re/r-(1/2)(re2/r2))+b/r2 sin2 theta+a/r2 cos2 theta type, the so-called double ring-shaped Kratzer potential. For a diatomic molecule system in double ring-shaped Kratzer potential, the exact bound state energy eigenvalues and corresponding wave functions have been determined within the framework of the asymptotic iteration method. Bound state eigenfunction solutions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric function and Jacobi polynomial. This new formulation is tested by calculating the energies of rovibrational states of a number of diatomic molecules. Also, the author-prove that in the nonrelativistic limit c-->infinity, where c is the speed of light, solutions of the Klein-Gordon system converge to those of the Schrodinger system.

EXACT ANALYTICAL SOLUTION OF THE KLEIN–GORDON EQUATION FOR THE PIONIC ATOM BY ASYMPTOTIC ITERATION METHOD

Within the framework of the asymptotic iteration method, we investigate the exact analytical solution for pionic atom in the Coulomb field of a nucleus. Exact bound state energy eigenvalues and corresponding eigenfunctions are determined for the case of angular momentum l≠0, for which the Coulomb potential is exactly solvable. Bound state eigenfunctions solutions, which have been extremely used in applications related with molecular spectroscopy, are obtained in terms of confluent hypergeometric functions.

Effective transport properties of arrays of multicoated or graded spheres with spherically transversely isotropic constituents

This work is concerned with the determination of the effective conductivity and potential fields of a periodic array of spherically transversely isotropic spheres in an isotropic matrix. We generalize Rayleigh’s method to account for the periodic arrangements of the inclusions. The inclusions considered in the formulation could be multicoated, generally graded, or exponentially graded. For the multicoated spheres, we derive a recurrence procedure valid for any number of coatings. We show that a (2×2) array alone can mathematically represent the effect of the multiple coatings. For a graded inclusion, the method of Frobenius is adopted to obtain series solutions for the potential fields. For an exponentially graded sphere, we show that the admissible potential field in the inclusion admits a closed-form expression in terms of confluent hypergeometric functions. All these types of inclusions can be characterized by simple scalar coefficients Tl in the estimate of effective conductivities. Simple orthorhombic, body-centered orthorhombic, and face-centered orthorhombic lattice structures are considered in the formulation. Numerical results are presented for selected systems with sufficient accuracy. We demonstrate that the anisotropy of the spheres can strongly influence the potential fields inside the inclusions. The effects of spherical anisotropy, multiple coatings, and the grading factor are also studied.

Transmission of optical communication signals by distributed parametric amplification

For the first time, to the authors' knowledge, distributed parametric amplification (DPA), i.e., the use of a transmission fiber itself for parametric amplification of communication signals is proposed and demonstrated. To account for the inevitable fiber loss, solutions were derived for the distributed amplifier, with either one or two pumps: They are obtained in terms of confluent hypergeometric functions. Low-penalty DPA of a 10-Gb/s nonreturn-to-zero (NRZ) signal over a 75-km dispersion-shifted fiber (DSF), is demonstrated by using only 66.5 mW of pump power. Three adjacent channels have been simultaneously transmitted, with little penalty due to nonlinear crosstalk. It is experimentally verified that DPA requires less pump power than distributed Raman amplification (DRA), for similar power penalties.

Energy levels and states of parabolic cylindrical lens shaped quantum dots

The parabolic cylindrical lens shaped quantum dot is investigated theoretically. The Schr?dinger equation for an electron confined in this structure is solved in the parabolic cylindrical coordinate system. The wavefunctions for the electron are presented in terms of confluent hypergeometric functions and the electron energy spectra are also obtained.

Exact Solutions for a Fluid-Saturated Porous Medium with Heat and Mass Transfer

AbstractAn analysis is performed to the study of the momentum, heat and mass transfer of a viscous fluid-saturated porous medium past an impermeable, non-isothermal stretching sheet with internal heat generation or absorption and chemical reaction. The governing partial differential equations are converted into ordinary differential equations by means of a similarity transformation. Exact solutions of velocity components together with the pressure distribution, which can not be found in the boundary layer theory, are obtained analytically; in addition, the temperature and concentration functions are given in terms of confluent hypergeometric functions. The velocity, temperature (concentration) profiles and thermal characteristics at the sheet for relevant parameters are plotted, tabulated and discussed.

Mapping of the position-dependent mass Schrödinger equation under point canonical transformation

In this paper, the three-dimensional radial position-dependent mass Schrodinger equation is exactly solved through mapping this wave equation into the constant mass Schrodinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.

Mixed symbolic–numerical computation of convective heat transfer with slip flow in microchannels

Steady-state convection heat transfer is analytically solved for laminar flow inside microchannels formed by parallel-plates, making use of the integral transform method and the exact analytical solution of the corresponding eigenvalue problem in terms of confluent hypergeometric functions. A mixed symbolic–numerical algorithm is developed under the Mathematica platform. The paper was also prepared in the Mathematica notebook format, available upon request, allowing for the immediate reproduction of the results and comprehension of the symbolic and computational rules developed.

137 粘弾性体のフラクタル構造と分数階微分(粘弾性体の減衰のモデル化, OS-11 ダンピング(2))

The dynamical behavior of an viscoelastic body is known to be well described by fractional derivatives of the amount of deformation. The physical reason of this fact is the underlying fractal nature of the molecular structure of such materials. By using the Schiessel-Blumen ladder of mechanical system, it has been shown that the fractal structure would produce the asymptotic power law decay of the deformation, which is the most distinguished characteristics of fractional derivatives. So far, this has been shown by asymptotic behavior of the Laplace transform of the impulse response of the relaxation. Here, we generalize this result, and show that the exact solution of the impulse response function is obtained for a large class of parameter values in terms of confluent hypergeometric functions. The asymptotic expression is then readily obtained which shows the power law decay.

Fluid queues driven by a discouraged arrivals queue

We consider a fluid queue driven by a discouraged arrivals queue and obtain explicit expressions for the stationary distribution function of the buffer content in terms of confluent hypergeometric functions. We compare it with a fluid queue driven by an infinite server queue. Numerical results are presented to compare the behaviour of the buffer content distributions for both these models.

Laminar flow in a porous channel with large wall suction and a weakly oscillatory pressure

The laminar oscillatory flow inside a rectangular channel with wall suction is considered here. The scope is limited to large suction imposed uniformly along the permeable walls. Inside the channel, the onset of small amplitude pressure disturbances produces an oscillatory field that we wish to investigate. Based on the normalized pressure-wave amplitude, the conservation equations are linearized and split into leading-order (steady) and first-order (time-dependent) equations. The first-order set is subdivided into an acoustic, pressure-driven, wave equation, and a vortical, boundary-driven, viscous equation. For longitudinal pressure oscillations, both equations are written to the order of the wall suction Mach number. The resulting equations are then solved in an exact fashion. The novelty lies in the vortical response that reduces to a Weber equation following a Liouville–Green transformation. The emerging rotational solution is expressible in terms of confluent hypergeometric functions of the suction Reynolds number, Strouhal number, and spatial coordinates. The total solution is then constructed and found to coincide with the numerical solution of the linearized momentum equation. The oscillatory velocity exhibits similar characteristics to the exact Stokes profile for oscillations inside a long channel with hard walls. In particular, a thin rotational layer is observed in addition to the small velocity overshoot near the wall. Both depth and overshoot are nowhere near their values obtained by switching from mass extraction to mass addition. In contrast to former studies involving injection, the so-called acoustic boundary layer is found to depreciate when suction is increased or when viscosity is reduced. This response is similar to that of the Stokes layer over hard walls. Overall, the effect of increasing frequency is that of compressing the rotational layer near the wall.

On functionally graded balls and cones

The heat equation, for both steady and unsteady situations, is considered when the material parameters are spherically symmetric functions of position. Explicit separated solutions are derived when the material parameters are exponential functions; the radial part of these solutions is given in terms of confluent hypergeometric functions or Whittaker functions. In the steady case, explicit solutions are found when the conductivity k(r)= exp(−βrq), where β and q are parameters with q>0. The behaviour near the tip of a spherically-graded cone is also investigated.

Asymptotic expansion of 2F1(a,b;c;x) in terms of confluent hypergeometric functions

We have developed a new asymptotic formula for 2F1(a,b;c;x) in which a and c are finite, |b| tends to infinity and the argument x tends to zero. We make use of this result to obtain asymptotic expansions of the electric dipole (E1) quantal differential excitation function for the adiabaticity parameter ξ=ηf−ηi and the Sommerfeld parameter ηf tending to infinity simultaneously, while the parameter ηi remains finite.

Exact solutions for modeling sound propagation through a combustion zone.

Exact analytical solutions for one-dimensional sound propagation through a combustion zone, taking the effects of mean temperature gradient and oscillatory heat release into account, are presented in this paper. The wave equation is derived starting from the momentum and energy equations. Using appropriate transformations, solutions are derived for the case of an exponential mean temperature gradient in terms of Bessel functions. For the case of a linear mean temperature profile, solutions are derived in terms of confluent hypergeometric functions. Example calculations show that the accuracy in modeling combustion-acoustics interactions can be significantly increased by the use of these solutions.

On hitting times for compound Poisson dams with exponential jumps and linear release rate

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

On hitting times for compound Poisson dams with exponential jumps and linear release rate

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).

Influence of two-photon absorption on modulational instability

The instability of a plane wave in an optical medium with two-photon absorption is studied. The analysis is based on the modified nonlinear Schrödinger equation. The linearized equation for the modulation is shown to have an exact solution in terms of confluent hypergeometric functions. It is found that the gain spectrum varies with position. This may result in a change of the wave dynamics and in a decrease of the repetition rate of the pulse train developed from the plane wave. The application of the results to the optical pulse propagation in semiconductor gratings and fiber gratings is discussed.

Integrated Brownian Motion, Conditioned to be Positive

We study the two-dimensional process of integrated Brownian motion and Brownian motion, where integrated Brownian motion is conditioned to be positive. The transition density of this process is derived from the asymptotic behavior of hitting times of the unconditioned process. Explicit expressions for the transition density in terms of confluent hypergeometric functions are derived, and it is shown how our results on the hitting time distributions imply previous results of Isozaki-Watanabe and Goldman. The conditioned process is characterized by a system of stochastic differential equations (SDEs) for which we prove an existence and unicity result. Some sample path properties are derived from the SDEs and it is shown that $t \to t^{9/10}$ is a “critical curve” for the conditioned process in the sense that the expected time that the integral part of the conditioned process spends below any curve $t \to t^{\alpha}$ is finite for $\alpha < 9 /10$ and infinite for $\alpha \geq 9/10$.

Kronig-Penney problem for a biparabolic potential: Quasifree particle motion

The Bloch states of a particle in a one-dimensional, periodic, biparabolic field are determined. The solutions are expressed in terms of confluent hypergeometric functions. It is shown that the first term of the Tricomi expansion for confluent hypergeometric functions represents the quasifree above-barrier motion of a particle and is identical to the exact Kronig-Penney solution for a square potential.