Related papers: Asymptotic Approximations to Truncation Errors of …
Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary…
Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…
The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear…
This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This…
In this paper, we discuss the application of the author's $\tilde{d}^{(m)}$ transformation to accelerate the convergence of infinite series $\sum^\infty_{n=1}a_n$ when the terms $a_n$ have asymptotic expansions that can be expressed in the…
We introduce a deficiency-based representation and approximation framework for values of the Riemann zeta function. The method is based on comparing two nonlinear accumulation mechanisms: global transformation of a base partial sum and…
In this paper, we construct an asymptotic-preserving neural networks (APNNs) [21] for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss…
Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…
We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…
A new method, called the method of self-similar approximants, and its recent developments are described. The method is based on the ideas of renormalization group theory and optimal control theory. It allows for the effective extrapolation…
Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products…
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a…
In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved…
This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force. When the magnetic field is large, the so-called drift-fluid approximation is obtained. In this…
We introduce templates for exponential asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly…
We study asymptotic behaviour of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function.…
We derive an asymptotic error formula for Gauss--Legendre quadrature applied to functions with limited regularity, using the contour-integral representation of the remainder term. To address the absence of uniformly valid approximations of…