Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics

The object of this article is to present the solution of a fractional generalization of the Schrödinger equation of quantum mechanics in one dimension. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is obtained in a compact and computational form in terms of the H-function. A result given earlier by Debnath for the solution of a generalized Schrödinger equation is obtained in an explicit form in terms of the H-function, as a special case of our findings.     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

New approximations to the principal real-valued branch of the Lambert <Emphasis Type="Italic">W</Emphasis>-function

The Lambert W-function is the solution to the transcendental equation W(x)e ^W(x) = x. It has two real branches, one of which, for x ∈ [?1/e, ∞], is usually denoted as the principal branch. On this branch, the function grows from ? 1 to infinity, logarithmically at large x. The present work is devoted to the construction of accurate approximations for the principal branch of the W-function. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximum relative error smaller than 5 × 10^?3. Starting from it, machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme, here examined for the first time, which is more efficient than standard Newton’s iteration at large x. Analytic bounds for W are also constructed, for x > e, which are much tighter than those currently available. It is noted that the exponential of the upper bounding function yields an upper bound for the prime counting function π(n) that is better than the well-known Chebyshev’s estimates at large n. Finally, the construction of accurate approximations to W based on Chebyshev spectral theory is discussed; the difficulties involved are highlighted, and methods to overcome them are presented.     

Development of the parametric tolerance modeling and optimization schemes and cost-effective solutions

Most of previous research on tolerance optimization seeks the optimal tolerance allocation with process parameters such as fixed process mean and variance. This research, however, differs from the previous studies in two ways. First, an integrated optimization scheme is proposed to determine both the optimal settings of those process parameters and the optimal tolerance simultaneously which is called a parametric tolerance optimization problem in this paper. Second, most tolerance optimization models require rigorous optimization processes using numerical methods, since closed-form solutions are rarely found. This paper shows how the Lambert W function, which is often used in physics, can be applied efficiently to this parametric tolerance optimization problem. By using the Lambert W function, one can express the optimal solutions to the parametric tolerance optimization problem in a closed-form without resorting to numerical methods. For verification purposes, numerical examples for three cases are conducted and sensitivity analyses are performed.     

Convolution integral equations involving a general class of polynomials and the multivariableH-function

In this paper we first solve a convolution integral equation involving product of the general class of polynomials and theH-function of several variables. Due to general nature of the general class of polynomials and theH-function of several variables which occur as kernels in our main convolution integral equation, we can obtain from it solutions of a large number of convolution integral equations involving products of several useful polynomials and special functions as its special cases. We record here only one such special case which involves the product of general class of polynomials and Appell's functionF ₃. We also give exact references of two results recently obtained by Srivastavaet al [10] and Rashmi Jain [3] which follow as special cases of our main result.     

Exact solutions of a novel integrable partial differential equation

We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P_II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P_II with its shift in parameter.We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P_II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel’s equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.     

Application of Wiener-Hopf technique to linear nonhomogeneous integral equations for a new representation of Chandrasekhar's H-function in radiative transfer, its existence and uniqueness

In this paper, the linear nonhomogeneous integral equation of H-functions is considered to find a new form of H-function as its solution. The Wiener-Hopf technique is used to express a known function into two functions with different zones of analyticity. The linear nonhomogeneous integral equation is thereafter expressed into two different sets of functions having the different zones of regularity. The modified form of Liouville's theorem is thereafter used, Cauchy's integral formulae are used to determine functional representation over the cut region in a complex plane. The new form of H-function is derived both for conservative and nonconservative cases. The existence of solution of linear nonhomogeneous integral equations and its uniqueness are shown. For numerical calculation of this new H-function, a set of useful formulae are derived both for conservative and nonconservative cases.     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

Conformal vector fields on complete Finsler spaces of constant Ricci curvature

In this work, it is proved that if a complete Finsler manifold of positive constant Ricci curvature admits a solution to a certain ODE, then it is homeomorphic to the n-sphere. Next, a geometric meaning is obtained for solutions of this ODE, which is applicable to Einstein–Randers spaces. Moreover, some results on Finsler spaces admitting a special conformal vector field are obtained.     

On Kendall-Ressel and related distributions

We delineate a connection of Kendall-Ressel and related laws with the lower real branch of Lambert W function. A characterization of the canonical member of Kendall-Ressel class is found. The Letac-Mora interpretation of the reciprocity of two specific NEFs is extended by considering two related reproductive EDMs. A local limit theorem on gamma convergence for the reproductive back-shifted Kendall-Ressel EDM is derived. Each member of this EDM is self-decomposable and unimodal, but not strongly unimodal. The coefficient of variation, skewness and kurtosis of each representative of this EDM are higher than the corresponding measures for the members of gamma and inverse Gaussian EDMs. An integral representation for the lower real branch of Lambert W function is given.     

Large time behavior of differential equations with drifted periodic coefficients modeling carbon storage in soil

This paper is concerned with the linear ODE in the form y′(t) = λρ(t)y(t) + b(t), λ < 0 which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function ρ(t), a linear drift in the coefficient b(t) involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.     

Adequate predimension inequalities in differential fields

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterovi? and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate.     

Tau-functions,Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.     

Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields

For the Cauchy problem of 1-D first order quasilinear hyperbolic linearly degenerate systems, a new mechanism of singularity formation is given to show that all the W1,p(1<p?+∞) norms of the C¹ solution should blow up simultaneously. It gives a way to verify the property of ODE singularity by directly using the energy method in the framework of C¹ solution.     

First boundary-value problem in the half-strip for a parabolic-type equation with bessel operator and Riemann–Liouville derivative

The first boundary-value problem in the half-strip for a parabolic-type equation with Bessel operator and Riemann–Liouville derivative is studied. In the case of the zero initial condition, the representation of the solution in terms of the Fox H-function is obtained. The uniqueness of the solution for a class of functions vanishing at infinity is proved. It is shown that when the equation under consideration coincides with the Fourier equation, the obtained representation of the solution becomes the known representation of the solution of the corresponding problem.     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

On limiting values of stochastic differential equations with small noise intensity tending to zero

When the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity ε to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case.     

Self-approximation of Dirichlet L-functions

Let d be a real number, let s be in a fixed compact set of the strip 1/2<σ<1, and let L(s,χ) be the Dirichlet L-function. The hypothesis is that for any real number d there exist ‘many’ real numbers τ such that the shifts L(s+iτ,χ) and L(s+idτ,χ) are ‘near’ each other. If d is an algebraic irrational number then this was obtained by T. Nakamura. ?. Pańkowski solved the case then d is a transcendental number. We prove the case then d≠0 is a rational number. If d=0 then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet L-function. We also consider a more general version of the above problem.     

Convolution of Rayleigh functions with respect to the Bessel index

A convolution of Rayleigh functions with respect to the Bessel index can be treated as a special function in its own right. It appears in constructing global-in-time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity. An explicit representation for it is derived which involves the special function ψ(x) (the logarithmic derivative of the Γ-function). The properties of the convolution in question are established. Asymptotic expansions for small and large values of the argument are obtained and the graph is presented.     

Some analytic approximations for neutral stochastic functional differential equations

We consider an analytic iterative method to approximate the solution of a neutral stochastic functional differential equation. More precisely, we define a sequence of approximate equations and we give sufficient conditions under which the approximate solutions converge with probability one and in pth moment sense, p ? 2, to the solution of the initial equation. We introduce the notion of the Z-algorithm for this iterative method and present some examples to illustrate the theory. Especially, we point out that the well-known Picard method of iterations is a special Z-algorithm.