Bell Polynomials and Nonlinear Inverse Relations

By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).

The affine connection

The connection and the covariant derivative are treated. Connection coefficients are introduced in their role of expressing the change in the coordinate basis vectors between neighbouring points. The covariant derivative of a vector is then defined. Next we relate the connection to the metric, and obtain the Levi-Civita connection. The logic concerning what is defined and what is derived is explained carefuly. The notion of a derivative along a curve is defined. The emphasis through is on clarity and avoiding confusions arising from the plethora of concepts and symbols.

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters $$u=(u_1,\ldots ,u_n)$$ u = ( u 1 , … , u n ) , which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters $$u_1,\ldots ,u_n$$ u 1 , … , u n . The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials

Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coefficients for the series of orthogonal polynomials, we find the possibility to extend the Tricomi method to more general series expansions. Some examples showing the effectiveness of the considered procedure are shown.

Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

The principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

Wavelet Solution for Nonlinear Reaction-Diffusion Equations

In this paper, we consider Fisher’s equation to find the approximate solution to overcome the difficulty to handle its nonlinearity. For solving this nonlinear PDE, we propose a method based on Legendre wavelets with lesser number of connection coefficients. We also study the theoretical analysis and error bound for the proposed technique. Two examples are tested with the proposed method to show the applicability and efficiency. The outcomes show that this approach fulfils the error bound conditions.

Curvilinear coordinates

This chapter presents a discussion on curvilinear coordinates in line with the introduction on Cartesian coordinates in Chapter 1. First, the chapter introduces a new system C of curvilinear coordinates xⁱ = xⁱ(Xj) (also sometimes referred to as Gaussian coordinates), which are nonlinearly related to Cartesian coordinates. It then introduces the components of the covariant derivative, before considering parallel transport in a system of curvilinear coordinates. Next, the chapter shows how connection coefficients of the covariant derivative as well as the Euclidean metric can be related to each other. Finally, this chapter turns to the kinematics of a point particle as well as the divergence and Laplacian of a vector and the Levi-Civita symbol and the volume element.