Frobenius Series Research Articles

ON THE VALUES OF CONTINUED FRACTIONS: q-SERIES II

Let polynomials S(t), T(t) be given, then the convergence of the q-continued fraction [Formula: see text] will be studied using the Poincaré–Perron Theorem and Frobenius series solutions of the corresponding q-difference equation S(t)H(q2t) = -T(t)H(qt) + H(t). Our applications include a generalization of a q-continued fraction identity of Ramanujan and certain q-fractions, which arise in the theory of q-orthogonal polynomials.

The scattering of electro-elastic waves by a spherical piezoelectric particle in a polymer matrix

This paper is devoted to the study of scattering of plane harmonic waves by a piezoelectric sphere with spherical isotropy embedded in an unbounded isotropic polymer matrix. The scattered displacement field and the electric potential in the matrix are expressed in terms of spherical vector wave functions and spherical harmonic functions, respectively. For the field points inside the inhomogeneity, new displacement functions are introduced. Expansion of the new displacement functions and the electric potential in terms of spherical harmonic functions, the equations of motion and electrostatic lead to four second order ordinary differential equations (odes), where three of them are coupled. The coupled system of odes is solved by the generalized Frobenius series. This approach is readily used to handle low and high frequencies. Three different types of piezoelectric inhomogeneities, PZT-4, PZT-5H, and BaTiO 3 are considered and the associated piezoelectric effects on the electro-mechanical fields, differential and total scattering cross-sections are addressed.

The analytical and experimental studies of the parallel-plate concurrent dialysis system coupled with ultrafiltration

The mass transfer in a parallel-plate concurrent dialysis system coupled with ultrafiltration operation is investigated theoretically and experimentally in this study. The separation efficiency improvement in mass transfer and flow stream composition variation along the parallel-plate channel were investigated analytically using the Frobenius series method. The mass transfer efficiency enhancement results by introducing a transmembrane pressure in the device are represented graphically and compared with those in a conventional dialysis device considering only the concentration difference as the driving force. The analytical results show that a dialysis system coupled with ultrafiltration can effectively enhance the separation efficiency compared with operation without transmembrane pressure. Moreover, the theoretical predictions are in qualitative agreement with the experimental results.

Power Laws, Scale Invariance and the Generalized Frobenius Series: Applications to Newtonian and TOV Stars Near Criticality

We present a self-contained formalism for calculating the background solution, the linearized solutions and a class of generalized Frobenius-like solutions to a system of scale-invariant differential equations. We first cast the scale-invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fixed points, we find a second linearized solution, which provides a distinct collection of power laws characterizing the deviations from the fixed point. We prove that generically there will be a region surrounding the fixed point in which the complete general solution can be represented as a generalized Frobenius-like power series with exponents that are integer multiples of the exponents arising in the linearized problem. While discussions of the linearized system are common, and one can often find a discussion of power-series with integer exponents, power series with irrational (indeed complex) exponents are much rarer in the extant literature. The Frobenius-like series we encounter can be viewed as a variant of the rarely-discussed Liapunov expansion theorem (not to be confused with the more commonly encountered Liapunov functions and Liapunov exponents). As specific examples we apply these ideas to Newtonian and relativistic isothermal stars and construct two separate power series with the overlapping radius of convergence. The second of these power series solutions represents an expansion around "spatial infinity," and in realistic models it is this second power series that gives information about the stellar core, and the damped oscillations in core mass and core radius as the central pressure goes to infinity. The power-series solutions we obtain extend classical results; as exemplified for instance by the work of Lane, Emden, and Chandrasekhar in the Newtonian case, and that of Harrison, Thorne, Wakano, and Wheeler in the relativistic case. We also indicate how to extend these ideas to situations where fixed points may not exist — either due to "monotone" flow or due to the presence of limit cycles. Monotone flow generically leads to logarithmic deviations from scaling, while limit cycles generally lead to discrete self-similar solutions.

The Frobenius Power Series Solution for Cylindrically Anisotropic Radially Inhomogeneous Elastic Materials

Solution in the form of Frobenius power series to the system of equations of three-dimensional elasticity, describing propagation of cylindrical waves, is constructed and studied. A general case of radially inhomogeneous materials with arbitrary cylindrical anisotropy is considered. The governing ordinary differential system is taken in the form of six first-order equations. The intrinsic algebraic symmetry of the matrix of coefficients reveals basic properties of the indicial equation and ensuing features of the fundamental solution. On this basis, complemented by energy considerations, it is shown that partial solutions in the form of the Frobenius series satisfy specific orthogonality. Its physical meaning is that the corresponding wave modes do not carry energy flux across cylindrical surfaces. The limiting behaviour of the partial solutions near the cylinder axis (r → 0) is analysed. It is proved that they always partition into two triplets, one with converging displacement amplitudes tending to zero or to a rigid-body translation at r → 0, and the other with displacements diverging at r → 0. Analogies with elasticity of rectilinearly anisotropic media are discussed.

Torsion of uniform bars with polygon cross-section

The solution of the torsion problem of uniform bars with polygon cross-section by means of the Trefftz integral for the complex torsion function is presented. The governing equations consist of the first-order integral of the Schwarz-Christoffel mapping, and another third-order integral, known as the Trefftz integral. The known series solution of the Schwarz-Christoffel integral is used. The solution of Trefftz's integral is carried out in series form according to the method of von Koppenfels and Stallmann, by the separation into regular and singular corner functions. The result is in the form of explicit corner functions, Frobenius and Taylor series. The series solutions are computed with convolution number algebra. An additional Fourier analysis allows one to compute the shear in a polygon cross-section with a round hole. Torsion stiffness is computed by integrals along the boundary. An estimate of the effect of rounding of corners is given.

Phase shift and attenuation characteristics of acoustic waves in a flowing gas confined by cylindrical walls

Wave propagation in flowing ideal gases confined by cylindrical waveguides is described in the low-frequency range using an iterative Frobenius series expansion method. The primary concern is to present a mathematical model enabling any radial-dependent flow profile to be analyzed. In contrast to previous analytical results, the present model is applicable in the general case where cubic and higher order terms in the axial acoustic velocity become important and to examine the influence of a non-vanishing radial velocity term. As a numerical test case, it is found that a gas flow velocity w( r)—for simplifying reasons assumed to be a linear combination of a flat flow profile and a parabolic flow profile corresponding to a mean flow equal to w ̄ —is well approximated by a flat flow profile of the same mean flow value w ̄ at low shear wavenumbers and at higher shear wavenumbers (calculations were done for shear wavenumbers up to 8). In actual fact, the error introduced by making this mean flow approximation is smaller than the error introduced by neglecting the radial velocity term.

Elastic properties of particulate composites with imperfect interface

An effective analytical approach is developed for the problem of particulate composites containing spherical inclusion with imperfect interface between the matrix and spherical inclusions. In this paper, a general interface model for a variety of interfacial defects has been presented, in which both displacement discontinuity across the interface and the elastic moduli varing with radius outside of the inclusion are considered. The imperfect interface conditions are appropriate in the case of thin coatings on the inclusion. Furthermore, in the case of thin elastic interphase, the displacement field and the stress field in the inclusion and matrix are exactly solved for the boundary problem of hydrostatic compression of an infinite spherical symmetrical body by Frobenius series, and the expression of the normal interface parameter, Dr, is derived. In addition, it has been proved that two previous results derived in some literatures by considering the interface to be a thin interphase with displacement jump or with some variance in its moduli can be reverted from the present formula, respectively. Numerical results are given to demonstrate the significance of the general imperfect interface effects.

Error analysis in the series evaluation of the exponential type integral e zE 1( z)

The method of convolution algebra is used to compute values of the exponential type integral e zE 1( z), ▪ by expansion of the integrand in a string of Taylor series' along the real s-axis for any complex parameter z, accurate within ±1 of the last digit of seven-digit computation. Accuracy is verified by comparison with existing tables of E 1 and related integrals. This method is used to assess the accuracy of the error estimates of all subsequent computations. Three errors of a Taylor series are identified. These consist of a Taylor series truncation error, a digital truncation error, and a stability error. Methods are developed to estimate the error. By iteration a numerical radius of convergence for a given accuracy is determined. The z-plane is divided into three regions in which three different types of series are used to expand the function f( z) directly in z. Around the center the well-known Frobenius series is used. In the outer region the well-known asymptotic approximation is used. Their accuracy boundaries are determined. In the near-annular region in between, a set of Taylor series is introduced. As the result, the function f( z) can be computed fast with the appropriate series for any complex argument z, to an accuracy within less than relative error of 5 × 10 −7.

Pedagogical implications in the thermal analysis of uniform annular fins: Alternative analytic solutions by series

The dimensionless temperature along annular fins of uniform thickness and constant physical properties is governed by an ordinary differential equation of second order with variable coefficients, called the modified Bessel equation of zero order. This educational article addresses two alternative computational procedures of approximate nature for solving this modified Bessel equation: (a) the Frobenius method, and (b) the power series method coupled with a curve fit. Approximate temperature distributions of good quality have been obtained with available symbolic algebra software such as Maple or Mathematica on a personal computer. The primary specification when designing annular fins of uniform thickness is the amount of heat transfer from a tube to the surrounding fluid. These calculations are traditionally done with the help of a fin efficiency diagram. The fin efficiencies can be computed approximately by integration or by differentiation via the Frobenius and power series methods as well. Students of heat transfer courses can benefit from these two alternative computational procedures that seek to circumvent the use and operations with Bessel functions and still produce approximate analytic results of good caliber. A course on ordinary differential equations is the only mathematical background that students need to have to implement these computational techniques. © 1998 John Wiley & Sons, Inc. Comput Appl Eng Educ 6: 31–40, 1998

Calculating connection coefficients for Meromorphic differential equations

We consider systems of linear differential equations near an irregular singularity of rank one with a leading coefficient matrix having all distinct eigenvalues. The main results express connection coefficients (Stokes' multipliers and connection factors) as sums of convergent series or as limits of sequences, where the terms may be calculated in various ways. One way is to form certain explicit weighted sums of the coefficients in the formal or Frobenius series solutions. Another way is to recursively solve for the coefficients of a solution of a certain singular integral equation similar to the Frobenius method for differential equations near a regular singular point. In case the original differential equation has rational coefficients, then associated linear differential equations with rational coefficients can be constructed whose Frobenius solution determines die terms in the series or limits; hence all transformations are done within the class of “holonomic” functions.

Elliptic integrals and the Schwarz-Christoffel transformation

The real elliptic integrals of the first and second kind in Jacobi's normal form are computed efficiently, using the convolution number in conjunction with the method of Frobenius. For this purpose certain treatments of the Laurent series are included. Different regions of convergence on the real axis are determined, and for each one a different series is developed. The real elliptic integral of the third kind is solved within a limited parameter plane by the same method. The integral of the Schwarz-Christoffel transformation is solved in the complex variable by complex convolution number algebra, using the unit disk as mapping region. Different regions of convergence of Frobenius, Laurent, and Taylor series are determined to cover the whole disk. The complex evaluation of the elliptic integral of the third kind is included. A Schwarz-Christoffel formula for an infinite periodic mapping is given. The solutions for exterior, interior, periodic, and cyclic polygons are separately treated. Examples of several polygon mappings are presented graphically, and compared with previous numerically integrated solutions. The parameter problem is solved by the Newton-Raphson method, using a quotient matrix as approximation for the Jacobian matrix. The coordinate relations are simplified by using an overdetermined system. An exact analytical Jacobian matrix is computed, solving Leibniz' derivative of the Schwarz-Christoffel integral, and results are compared with the approximate quotient matrix method.

Effect of the Interphase Zone on the Bulk Modulus of a Particulate Composite

An exact solution is found for the problem of hydrostatic compression of an infinite body containing a spherical inclusion, with the elastic moduli varying with radius outside of the inclusion. This may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. Both the shear and the bulk moduli are assumed to be equal to a constant term plus a power-law term that decays away from the inclusion. The method of Frobenius series is used to generate an exact solution for the displacements and stresses. The solution is then used to estimate the effective bulk modulus of a material containing a random dispersion of these inclusions. The results demonstrate the manner in which a localized interphase zone around an inclusion may markedly affect both the stress concentrations at the interface, and the overall bulk modulus of the material.

THERMAL STRESSES AND EFFECTIVE THERMAL EXPANSION COEFFICIENT OF A FUNCTIONALLY GRADIENT SPHERE

We present the exact solution to the problem of uniform heating of a spherical body whose elastic moduli and thermal expansion coefficient vary linearly with the radius. The Frobenius series method is used to find exact expressions for the displacements and stresses. Both the radial and hoop stresses are largest in magnitude at the center of the sphere. The radial stress decays to zero at the outer edge, whereas the hoop stresses always change sign at some intermediate value of the radius. We also find an exact expression for the effective thermal expansion coefficient. For the special case where the thermal expansion coefficient varies with the radius but the elastic moduli are uniform, the effective thermal expansion coefficient of the sphere is equal to the volumetric average of the local thermal expansion coefficient. In the more general case, where the moduli also vary, the moduli variations have very little influence on the effective thermal expansion coefficient.

Effects of Poiseuille flow on peristaltic transport of a particulate suspension

The problem of peristaltic transport induced by sinusoidal waves of a particle-fluid mixture in the presence of a Poiseuille flow, is analysed. The governing equations of motion resulting from the Navier-Stokes equations for both the fluid and particle phases are solved and closed form solutions are obtained for limiting values of Reynolds number, wave number and the Poiseuille flow parameter while the method of Frobenius series solution is used for the general case. It is found that the mean flow is strongly dependent on the Poiseuille flow parameter. The effects of particle concentration in the fluid is well discharged throughout the analysis and the results are compared with the other studies in the literature.

Wave motion in a three-layered, orthotropic-isotropic-orthotropic, composite shell

A classification of the possible free wave motions in a three-layered composite thick cylinder is presented. The governing equations of motion for free wave motion for isotropic and orthotropic elastic media are given and a combined solution is developed for a thick cylinder made of three different materials, such as orthotropic-isotropic-orthotropic layers. The use of Bessel and special Frobenius series is required to obtain a correct, closed form solution for the propagating waves. Numerical results are given in the form of dispersion curves and these are discussed. A correct approach to the calculation of the cut-off frequencies is presented. It is shown that the wave speeds of decoupled longitudinal-shear motion in orthotropic shells is dependent upon the ratio of the different shear moduli,[formula]. The influence of a thin, soft rubbery material at the centre of a sandwich configuration is thoroughly analyzed. The effect of the isotropic core properties (its stiffness and thickness) on the dynamics of the wave motion is investigated.

Solutions to equations of vibrations of spherical and cylindrical shells

The governing equations of the free vibrations of spherical and cylindrical shells with a regular singularity, are solved by Frobenius Series Method in the form of matrix. Considering the relationship of the roots of the indicial equation, we get some various expressions of solutions according to different cases. This work lays a foundation of solving certain elastic problems by analytical method.

Dispersion properties of three-layered orthotropic shells

This paper studies the harmonic wave propagation in thick, cylindrical, three-layered shells of infinite length. Both the outer layers and the core are composites made of short strand fiberglass and polyester resin. The randomly oriented fibers were approximately in parallel planes to constitute a transversely isotropic fibre composite. The planes of isotropy in the outer layers are orthogonal to the plane of isotropy at the core. A closed form solution of the exact linear equations of elasticity in sought in terms of a Frobenius series. The influence of the core thickness on the dynamics of the wave motion is estimated from numerically computed dispersion curves. Asymmetric wave motion is given prime consideration and the different types of waves which can occur are identified over a wide range of wave numbers.

Analytical treatment of Alfvén resonances and singularities in nonuniform magnetoplasmas

The linear resonant excitation of Alfvén waves in a cold plasma permeated by a nonuniform magnetic field is considered. The equilibrium magnetic field is irrotational and possesses an invariant coordinate perpendicular to the direction of the field. By solving for the coefficients in a Generalized Frobenius Series the regular and singular solutions may be generated. The singular solution is logarithmic and produces a net absorption of energy at the resonant field line. The efficiency of coupling between the fast mode and the resonant Alfvén mode is determined by the following overlap integral along the resonant field line, ∫(ξβrbγ0/hβ)dl; ξβr is the resonant eigenfunction, bγ0 is the compressional/parallel magnetic field perturbation, and hβ is proportional to the separation of background lines of force in the invariant direction. The amplitude of the singular solution is proportional to this integral, while the rate of energy absorption at the resonance is proportional to its square. It is also shown how the analytical solution at the resonance may be used to avoid problems encountered in numerical solutions.

Resonant Alfvén wave excitation in two‐dimensional systems: Singularities in partial differential equations

The resonant excitation of Alfvén waves is considered in a cold plasma embedded in a uniform magnetic field . All wave fields are assumed to vary as exp[i(λy ‐ ωt)], and the background medium is invariant in y. The background density distribution ρ0(x, z) is otherwise completely arbitrary. Regular and singular solutions for the waves are derived systematically in the vicinity of a resonance by considering a generalized Frobenius series, and we are able to recover many results found in earlier studies. Some new features of our work include a generalization of the overlap integral determining the efficiency with which any resonance may be excited, and the possibility that ξy ∝ 1/xN (N = 1,2,3 …) at the resonance. Hitherto only the solution with N = 1 has been considered.