Frobenius Series Solution Research Articles

Tandem Recurrence Relations for Coefficients of Logarithmic Frobenius Series Solutions about Regular Singular Points

We enhance Frobenius’ method for solving linear ordinary differential equations about regular singular points. Key to Frobenius’ approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel’s equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.

Fractional Frobenius Series Solutions of Confluent α-Hypergeometric Differential Equation

Fractional Frobenius Series Solutions of Confluent α-Hypergeometric Differential Equation

On the fragility of Alfvén waves in a stratified atmosphere

ABSTRACTComplete asymptotic expansions are developed for slow, Alfvén, and fast magnetohydrodynamic waves at the base of an isothermal 3D plane stratified atmosphere. Together with existing convergent Frobenius series solutions about z = ∞, matchings are numerically calculated that illuminate the fates of slow and Alfvén waves injected from below. An Alfvén wave in a two-dimensional model is 2.5D in the sense that the wave propagates in the plane of the magnetic field but its polarization is normal to it in an ignorable horizontal direction, and the wave remains an Alfvén wave throughout. The rotation of the plane of wave propagation away from the vertical plane of the magnetic field pushes the plasma displacement vector away from horizontal, thereby coupling it to stratification. It is shown that potent slow–Alfvén coupling occurs in such 3D models. It is found that about 50 per cent of direction-averaged Alfvén wave flux generated in the low atmosphere at frequencies comparable to or greater than the acoustic cut-off can reach the top as Alfvén flux for small magnetic field inclinations θ, and this increases to 80 per cent or more with increasing θ. On the other hand, direction-averaged slow waves can be 40 per cent effective in converting to Alfvén waves at small inclination, but this reduces sharply with increasing θ and wave frequency. Together with previously explored fast–slow and fast–Alfvén couplings, this provides valuable insights into which injected transverse waves can reach the upper atmosphere as Alfvén waves, with implications for solar and stellar coronal heating and solar/stellar wind acceleration.

Effects of viscous dissipation on heat convection of viscoelastic flow inside isothermal channels and tubes

In the present paper, analytical solutions for thermal convection of the Finitely Extensible Nonlinear Elastic model with Peterlin’s closure (FENE-P) in isothermal slits and tubes are presented by considering the viscous dissipation effects for the first time. Temperature distributions are derived in closed form, namely the HeunT function, and Frobenius series solutions for the slit and the tube flows, respectively. The effects of fluid elasticity, the Brinkman number and the extensibility parameter on the thermal convection characteristics of FENE-P fluid flows are investigated in detail. Generally, the Brinkman number causes a fall in the Nusselt number, but a rise in the centerline temperature. The results reveal that during the cooling process the Nusselt number experiences a decrease with the Brinkman number, while the centerline temperature rises. However, during the heating process the former increases and the latter decreases. As the main innovation of this study, the results show a strong relation between the Nusselt number and the Brinkman number and also between the centerline temperature and the Brinkman number.

Frobenius Series Solutions of the Schrodinger Equation with Various Types of Symmetric Hyperbolic Potentials in One Dimension

The Schrodinger equation (SE) for a certain class of symmetric hyperbolic potentials is solved with the aid of the Frobenius method (FM). The bound state energies are given as zeros of a calculable function. The calculated bound state energies are successively substituted into the recurrence relations for the expanding coefficients of the Frobenius series representing even and odd solutions in order to obtain wave functions associated with even and odd bound states. As illustrative examples, we consider the hyperbolic Poschl-Teller potential (HPTP) which is an exactly solvable potential, the Manning potential (MP) and a model of the Gaussian potential well (GPW). In each example, the bound state energies obtained by means of the FM are presented and compared with the exact results or the literature ones. In the case of the HPTP, we also make a comparison between exact bound state wave functions and the eigenfunctions obtained by means of the present approach. We find that our results are in good agreement with those given by other methods considered in this work, and that our class of potentials can be a perfect candidate to model the GPW.

Quantum modes of atomic waveguides by series techniques

Atom waveguides are used to manipulate cold atoms in atom interferometers. The creation of atom interferometers using cold atoms in miniature magnetic waveguides is one of many goals of current atom chip research. To achieve a complete understanding of atom propagation in a complicated device such as a guided atom interferometer, a detailed understanding of the ground state and other nearby states is needed. The Frobenius series solutions for the bounded transverse modes of an atomic waveguide are presented here and arbitrary precision arithmetic is used to evaluate the series solutions without roundoff errors. The waveguide potential considered is an infinitely long quadrupole magnetic potential as used in various atom chip waveguides. The simplest case of a guided spin-12 particle is presented here. However, the basic series techniques can be extended to both higher order multipole potentials and higher spin particles, including atoms with hyperfine splitting. The low-field and the high-field seeking states together form the spectrum of the waveguide Hamiltonian. In the limit where the transverse dimension of the guide tends to infinity, the spectrum of the guide changes from a discrete set of low- and high-field seeking states to a continuum of high-field seeking states embedded with a discrete set of low-field seeking states. Although the low-field seeking states are not truly bound, the system is an approximate example of bound states in a continuum first discussed by von Neumann and Wigner. Depending on boundary conditions, the solutions form either a discrete set or a continuum of orthogonal waveguide modes. These are useful for further analysis of ideal waveguide behavior as well as the detailed perturbation studies necessary for analysis of atomic waveguide interferometers.

Series solutions to linear integral equations

Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous equation Kf=f generally has Frobenius series solutions when the kernel is singular, whereas Kf=0 generally has such solutions regardless of singularity: the proviso in each case is that associated “indicial equation” has solutions.

Frobenius series solution of Fuchs second-order ordinary differential equations via complex integration

A method is presented (with standard examples) based on an elementary complex integral expression, for developing Frobenius series solutions for second-order linear homogeneous ordinary Fuchs differential equations. The method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method is straightforward to apply as an algorithm, and eliminates the manipulation of power series, so characteristic of the usual approach [14]. The method is a generalization of a procedure developed by Herrera [4] for finding Maclaurin series solutions for nonlinear differential equations.

Pricing interest rate derivatives under stochastic volatility

PurposeThe purpose of this paper is to derive semi‐closed‐form solutions to a wide variety of interest rate derivatives prices under stochastic volatility in affine‐term structure models.Design/methodology/approachThe paper first derives the Frobenius series solution to the cross‐moment generating function, and then inverts the related characteristic function using the Gauss‐Laguerre quadrature rule for the corresponding cumulative probabilities.FindingsThis paper values options on discount bonds, coupon bond options, swaptions, interest rate caps, floors, and collars, etc. The valuation approach suggested in this paper is found to be both accurate and fast and the approach compares favorably with some alternative methods in the literature.Research limitations/implicationsFuture research could extend the approach adopted in this paper to some non‐affine‐term structure models such as quadratic models.Practical implicationsThe valuation approach in this study can be used to price mortgage‐backed securities, asset‐backed securities and credit default swaps. The approach can also be used to value derivatives on other assets such as commodities. Finally, the approach in this paper is useful for the risk management of fixed‐income portfolios.Originality/valueThis paper utilizes a new approach to value many of the most commonly traded interest rate derivatives in a stochastic volatility framework.

Stress concentration around a hole in a radially inhomogeneous plate

The stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniaxial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress concentration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.

Wave Impedance Matrices for Cylindrically Anisotropic Radially Inhomogeneous Elastic Solids

Impedance matrices are obtained for radially inhomogeneous structures using the Stroh-like system of six first order differential equations for the time harmonic displacement-traction 6-vector. Particular attention is paid to the newly identified solid-cylinder impedance matrix ${\mathbf Z} (r)$ appropriate to cylinders with material at $r=0$, and its limiting value at that point, the solid-cylinder impedance matrix ${\mathbf Z}_0$. We show that ${\mathbf Z}_0$ is a fundamental material property depending only on the elastic moduli and the azimuthal order $n$, that ${\mathbf Z} (r)$ is Hermitian and ${\mathbf Z}_0$ is negative semi-definite. Explicit solutions for ${\mathbf Z}_0$ are presented for monoclinic and higher material symmetry, and the special cases of $n=0$ and 1 are treated in detail. Two methods are proposed for finding ${\mathbf Z} (r)$, one based on the Frobenius series solution and the other using a differential Riccati equation with ${\mathbf Z}_0$ as initial value. %in a consistent manner as the solution of an algebraic Riccati equation. The radiation impedance matrix is defined and shown to be non-Hermitian. These impedance matrices enable concise and efficient formulations of dispersion equations for wave guides, and solutions of scattering and related wave problems in cylinders.

Differential equations with linear algebra

1. First-Order Differential Equations. Differential Equations and Mathematical Model. Integrals as General and Particular Solutions. Direction Fields and Solution Curves. Separable Equations and Applications. Linear First-Order Equations. Substitution Methods and Exact Equations. 2. Mathematical Models and Numerical Methods. Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method. The Runge-Kutta Method. 3. Linear Systems and Matrices. Introduction to Linear Systems. Matrices and Gaussian Elimination. Reduced Row-Echelon Matrices. Matrix Operations. Inverses of Matrices. Determinants. Linear Equations and Curve Fitting. 4. Vector Spaces. The Vector Space R 3. The Vector Space Rn and Subspaces. Linear Combinations and Independence of Vectors. Bases and Dimension for Vector Spaces. General Vector Spaces. 5. Linear Equations of Higher Order. Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Undetermined Coefficients and Variation of Parameters. Forced Oscillations and Resonance. 6. Eigenvalues and Eigenvectors. Introduction to Eigenvalues. Diagonalization of Matrices. Applications Involving Powers of Matrices. 7. Linear Systems of Differential Equations. First-Order Systems and Applications. Matrices and Linear Systems. The Eigenvalue Method for Linear Systems. Second-Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Numerical Methods for Systems. 8. Matrix Exponential Methods. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems. Spectral Decomposition Methods. 9. Nonlinear Systems and Phenomena. Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. 10. Laplace Transform Methods. Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. 11. Power Series Methods. Introduction and Review of Power Series. Power Series Solutions. Frobenius Series Solutions. Bessel's Equation. References for Further Study. Appendix A: The Existence and Uniqueness of Solutions. Appendix B: Theory of Determinants. Answers to Selected Problems. Index.

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.

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.

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.

Validity of the field line resonance expansion

The field line resonance expansion is studied with a magnetohydrodynamic (MHD) model applicable to Earth’s magnetotail. Using a Frobenius series solution, this expansion is found to be inconsistent. The inconsistency is seen by evaluating the terms in the model equations that the expansion neglects as small. Their effect is to provide a solution more singular than that determined by the expansion procedure. A tentative alternative expansion procedure, retaining these neglected terms approximately, yields a nonsingular solution, suggesting that resonant mode conversion rather than field line resonance is the appropriate phenomenon.

Thick E ¼ Stewartson layers in a rapidly rotating gas

The effects of circumferential curvature and strong density variation on E¼ Stewartson layers are investigated. The solution of the third-order ordinary differential equation found to govern the flow is obtained by numerical integration for layers extending in the positive radial direction and in terms of a Frobenius-series solution for layers extending in the opposite direction. Due to the variation of the basic density field, the E¼ layer is compressed in the positive radial direction. E1/4 layers extending in the negative radial direction are likely to extend fully to the axis of symmetry because of the density variation and, consequently, a distinction in terms of a geostrophic flow and an E¼ layer flow cannot be made. Curvature effects are found to play a significant role in this case. A simple case of driving by a differential rotation of part of the horizontal boundaries is examined.

An eigenvalue problem arising in a patch formation model

A simple numerical scheme has been developed for the solution of the eigenvalue problem arising in a patch formation model given by Del Grosso et al. [1]. The scheme is based on finding bounds which separate the eigenvalues. The exact eigenvalues are obtained by solving an algebraic equation given by the corresponding regular Frobenius series solution. At the same time eigenfunctions may also be obtained from this series solution.

Buckling of Continuous Circular Plates

The general theoretical development given in this paper considers the material to be orthotropic. Single Frobenius series solutions and double series solutions are obtained for orthotropic continuous plates. These solutions reduce to exact known solutions in the case of isotropic plates. For the purpose of numerical calculations, two cases of isotropic continuous plates—one over an intermediate rigid ring support and the other over an intermediate elastic ring support—are considered. For the case of a free plate over a rigid intermediate support, buckling loads increase as the support moves towards the center. In the case of an elastically supported fixed plate, the support elasticity considerably influences the buckling values, when the coefficient of support elasticity is more than 100. For lower modes a value of K 1 equal to 10,000 gives an almost fixed support. When the support is at distance of 0.355 times the radius of the plate the buckling resistance is found to be a maximum.