Related papers: The Pad\'e interpolation method applied to $q$-Pai…
We establish interpolation problems related to all the $q$-Painlev\'e equations of types from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$. By solving those problems, we can derive the evolution equations, the scalar Lax pairs and the determinant…
We study Pad\'e interpolation problems on an additive grid, related to additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By choosing suitable Pad\'e problems, we can derive time…
We give a simple form of the evolution equation and a scalar Lax pair for the $q$-Garnier system. Some reductions to the $q$-Painlev\'e equations and the autonomous case as a generalized QRT system are discussed. Using two kinds of Pad\'e…
The $q$-Painlev\'e equation of type $E^{(1)}_6$ is obtained by Pad\'e method. Special solutions in determinant formula to the $q$-Painlev\'e equation is presented. A relation between Pad\'e method and B\"acklund transformation of type…
An interpolation problem related to the elliptic Painlev\'e equation is formulated and solved. A simple form of the elliptic Painlev\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also…
We propose a class of Pad\'e interpolation problems whose solutions are expressible in terms of determinants of hypergeometric series.
A novel application of the Pade approximation is proposed in which the Pade approximant is used as an interpolation for the small and large coupling behaviors of a physical system, resulting in a prediction of the behavior of the system at…
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Pad\'e approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix…
If the small and large coupling behavior of a physical system can be computed perturbatively and expressed respectively as power series in a coupling parameter $g$ and $1/g$, a Pad\'{e} approximant embracing the two series can interpolate…
Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad\'e approximants for solving nonlinear partial differential equations…
We investigate the symmetry of the linear q-difference equations which are associated with some q-Painlev\'e equations. We apply it for adjustment of the expression of the time evolution on the q-Painlev\'e equations in terms of the Weyl…
Parameterized quantum circuits (PQCs) are ubiquitous in the design of hybrid quantum-classical algorithms. In this work, we propose an interpolation-based coordinate descent (ICD) method to address the parameter optimization problem in…
We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…
A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.
It is well known that the Painlev\'e equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou…
To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations,…
Pade approximations appear to be a powerful tool to extend the validity range of expansions around certain kinematical limits and to combine expansions of different limits to a single interpolating function. After a brief outline of the…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to…