Determinants of Regular Singular Sturm ‐ Liouville Operators

In the paper we deal with the Heun functions --- solutions of the Heun\nequation, which is the most general Fuchsian equation of second order with four\nregular singular points. Despite the increasing interest to the equation and\nnumerous applications of the functions in a wide variety of physical problems,\nit is only Maple amidst known software packages which is able to evaluate the\nHeun functions numerically. But the Maple routine is known to be imperfect:\neven at regular points it may return infinities or end up with no result.\nImproving the situation is difficult because the code is not publicly\navailable. The purpose of the work is to suggest and develop alternative\nalgorithms for numerical evaluation of the Heun functions. A procedure based on\npower series expansions and analytic continuation is suggested which allows us\nto avoid numerical integration of the differential equation and to achieve\nreasonable efficiency and accuracy. Results of numerical tests are given.\n

We study the integrability properties of nonlinearly perturbed Euler equations (linear ordinary differential equations with one regular singular point in the complex plane plus a nonlinear perturbation) near the singular point. We allow for first integrals with essential singularities and give sufficient conditions for the nonintegrability of the equations in the complex domain. We extend normal form theorems for singular equations and argue that equivalence to normal forms captures the spirit of the poly-Painlevé test and is a powerful tool for a rigorous approach to nonintegrability.

Chapter 7 - Ordinary Differential Equations

In this paper, we propose an efficient continuation method for locating multiple power flow solutions. We adopt the holomorphic embedding technique to represent solution curves as holomorphic functions in the complex plane. The holomorphicity, which provides global information of the curve at any regular point, enables large step sizes in the path-following procedure such that non-singular curve segments can be traversed with very few steps. When approaching singular points, we switch to the traditional predictor-corrector routine to pass through them and switch back afterward to the holomorphic embedding routine. We also propose a warm starter when switching to the predictor-corrector routine, i.e. a large initial step size based on the poles of the Pad\'{e} approximation of the derived holomorphic function, since these poles reveal the locations of singularities on the curve. Numerical analysis and experiments on many standard IEEE test cases are presented, along with the comparison to the full predictor-corrector routine, confirming the efficiency of the method.

We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out around the regular singular points $x=\pm1$ of the Legendre differential equations. The solutions $y_{n}(x)$ of the Legendre equation are predicted to be oscillatory in $|x| < 1$ for $n\geq3$ and nonoscillatory outside of that interval for all values of n. In contrast, the solutions $y_{n}^{m}(x)$ of the associated Legendre equation are predicted to be oscillatory for $n\geq3$ and $m\leq n-2$ only in smaller subintervals $|x| < x_{*} < 1$ , the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than $(-1, +1)$ .

In this paper, we develop the eigenfunction expansion theory of a selfadjoint operator generated by a symmetric sixth order differential equation L6(y)=λy. This differential equation has regular singular points at x=1, and is in the limit-5 case at each end. This means that two boundary conditions are needed at ±1 to ensure a well-posed boundary value problem. Not many examples are known of such higher order singular differential equations. The example that we give is interesting because the eigenvalue problem L6(y)=λy has a sequence of polynomial solutions that are orthogonal on [−1,1] with respect to the weight distribution W(x)=1/Aδ(x+1)+1/Bδ(x−1)+C.

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

An effective method for solving boundary value problems for the Laplace and Helmoltz equations (in domains possessing a definite symmetry) is the method of separation of variables. The general idea of this method is to find a set of solutions of the homogeneous partial differential equation in question that satisfy certain boundary conditions. These solutions then serve as “atoms”, from which, based on the linear superposition principle, one constructs the “general” solution. Since each of these “atoms” is a solution of the corresponding homogeneous equation, their linear combination is also a solution of the same equation.KeywordsDirichlet ProblemPoisson EquationArbitrary ConstantElliptic ProblemLaplace EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study in this article the structure and its stability of 2-D divergence-freevector fields with the Dirichlet boundary conditions. First we classify boundarypoints into two new categories: $\partial$−singular points and $\partial$−regular points, and establishan explicit formulation of divergence-free vector fields near the boundary.Second, local orbit structure near the boundary is classified. Then a structural stabilitytheorem for divergence-free vector fields with the Dirichlet boundary conditionsis obtained, providing necessary and sufficient conditions of a divergence-free vectorfields. These structurally stability conditions are extremely easy to verify, and exampleson stability of typical flow patterns are given.The main motivation of this article is to provide an important step for a forthcomingpaper, where, for the first time, we are able to establish precise rigorous criteriaon boundary layer separations of incompressible fluid flows, a long standing problemin fluid mechanics.

The main aim of this chapter will be to use Theorem 4 of Chapter 2 – and specifically both existence and uniqueness of solutions – to develop a theory that will describe the solutions of the homogeneous linear second-order equation where p 0; p 1; p 2 are continuous real-valued functions on [a, b] and p 2(x) &amp;lt; 0 for each x in [a, b]. (‘Homogeneous’ here reflects the zero on the right-hand side of the equation which allows λy to be a solution (for any real constant λ) whenever y is a given solution.) The language of elementary linear algebra will be used and the theory of simultaneous linear equations will be presumed. Central to our discussion will be the Wronskian, or Wronskian determinant: if y 1 : [a, b] → ℝ and y 2 : [a, b] → ℝ are differentiable functions on the closed interval [a, b], the Wronskian of y 1 and y 2, W(y 1; y 2) : [a, b] → ℝ, is defined, for x ∈ [a, b], by If y 1 and y 2 are solutions of (1), it will turn out that either W(y 1, y 2) is identically zero or never zero in [a, b].

Many Engineering structures, including electronic component assemblies, are inherently multi-scale in nature. These structures often experience complex local nonlinear behavior such as plasticity, damage or fracture. These local behaviors eventually lead to the failure at the macro length scale. Connecting the behavior across the length scales to develop an understanding of the failure mechanism is important for developing reliable products.To solve multi-scale problems in which the critical region is much smaller than the entire structure, an iterative solution approach based on domain decomposition techniques is proposed. Two independent models are constructed to model the global and local substructures respectively. The unbalanced force at the interface is iteratively reduced to ensure force equilibrium of the overall structure in the final solution. The approach is non-intrusive since only nodal values on the interface are transferred between the global and local models. Solution acceleration using SR1 and BFGS updates is also demonstrated. Equally importantly, the two updates are applied in a non-intrusive manner, meaning that the technique is implemented without needing access to the codes using which the sub-domains are analyzed. Code- and mesh-agnostic solutions for problems with local nonlinear material behavior or local crack growth are demonstrated. Analysis in which the global and local models are solved using two different commercial codes is also demonstrated.Engineering analysis using numerical models are helpful in providing insight into the connection between the structure, loading history, behavior and failure. Specifically, Isogeometric analysis (IGA) is advantageous for engineering problems with evolving geometry compared to the traditional finite element method (FEM). IGA carries out analysis by building behavioral approximations isoparametrically on the geometrical model (commonly NURBS) and is thus a promising approach to integrating Computer-Aided Design (CAD) with Computer-Aided Engineering (CAE).In enriched isogeometric Analysis (EIGA), the solution is enriched with known behavior on lower dimensional geometrical features such as crack tips or interfaces. In the present research, enriched field approximation techniques are developed for the application of boundary conditions, coupling patches with non-matching discretizations and for modeling singular stresses in the structure.The first problem solution discussed is to apply Dirichlet and Neumann boundary conditions on boundary representation (B-rep) CAD models immersed in an underlying domain of regular grid points. The boundary conditions are applied on the degrees of freedom of the lower dimensional B-rep part directly. The solution approach for the immersed analysis uses signed algebraic level sets constructed from the B-rep surfaces to blend the enrichedfield with the underlying field. The algebraic level sets provide a surrogate for distance, are non-iteratively (or algebraically) computed and allow implicit Boolean compositions.The methodology is also applied to couple solution approximations of decomposed patches by smoothly blending incompatible geometries to an arbitrary degree of smoothness. A parametrically described frame or interface is introduced to “stitch” the adjacent patches. A hierarchical blending procedure is then developed to stitch multiple unstructured patches including those with T-junctions or extraordinary vertices.Finally, using the EIGA technique, a computational method for analyzing general multimaterial sharp corners that enables accurate estimations of the generalized stress intensity factors is proposed. Explicitly modeled geometries of material junctions, crack tips and deboned interfaces are isogeometrically and hierarchically enriched to construct approximations with the known local behavior. specifically, a vertex enrichment is used to approximate the asymptotic field near the re-entrant corner or crack tip, Heaviside function is used to approximate the discontinuous crack face and the parametric smooth stitching technique is used to approximate the behavior across material interface. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is validated through analysis of a bi-material corner, interface crack and growth of an inclined crack in a homogeneous solid. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly coarse discretizations.

An invariant matrix asymptotic formula for the approximation of solutions of second-order linear ordinary differential equations y″=φ(x)y is proposed and elaborated upon. This is utilized to develop scalar versions of asymptotic formulas for two linearly independent solutions and for their derivatives. This formula is shown to be valid in a half neighbourhood of a point x0. The validity holds whether x0 is an ordinary (regular) point for the ODE, whether x0 is a singular regular point for the ODE, if some exceptional case is avoided, whether x0 is a singular irregular point for the ODE, and whether or not x0 is finite. The matrix version of our formulas is shown to be valid also at a turning point. The Liouville-Green approximation is extracted as a particular case of our formula. Examples are given. The formula has additional “globality” properties. Examples are given where the ODE is considered on an infinite interval (0, ∞) and its coefficient φ(x) is singular at x = 0 as well as at x = ∞. A uniformly valid approximation on the entire infinite interval is then provided.

Efficient accelero-summation of holonomic functions