Related papers: The Pad\'e interpolation method applied to additiv…
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 show how the newly developed method of Periodic Unfolding on Riemannian manifolds can be applied to PDE problems: We consider the homogenization of an elliptic model problem. In the limit, we obtain a generalization of the well-known…
We review recent results on the connection between Hermite-Pad\'e approximation problem, multiple orthogonal polynomials, and multidimensional Toda equations in continuous and discrete time. In order to motivate interest in the subject we…
In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for…
We construct the elliptic Painlev\'e equation and its higher dimensional analogs as the action of line bundles on 1-dimensional sheaves on noncommutative surfaces.
Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according…
In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin.…
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
Variational formulations of time-dependent PDEs in space and time yield $(d+1)$-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other hand, this approach enables…
Novel hybrid Ermakov-Painlev\'{e} IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. B\"{a}cklund transformations are then employed to generate classes of exact solutions via the…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these…
A method for the resummation of nonalternating divergent perturbation series is described. The procedure constitutes a generalization of the Borel-Pad\'{e} method. Of crucial importance is a special integration contour in the complex plane.…
The problem of having to reconstruct the decay rates and corresponding amplitudes of the single-exponential components of a noisy multi-exponential signal is common in many other areas of physics and engineering besides lattice field…
When approximating a function that depends on a parameter, one encounters many practical examples where linear interpolation or linear approximation with respect to the parameters prove ineffective. This is particularly true for responses…
We apply the Linear Delta Expansion (LDE) to the Lindstedt-Poincare (``distorted time'') method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case…
We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order…
Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed…
For all non-equivalent matrix systems of Painlev\'e-4 type found by authors in arXiv:2107.11680, isomonodromic Lax pairs are presented. Limiting transitions from these systems to matrix Painlev\'e-2 equations are found.
We introduce and solve the non-commutative version of the Hermite-Pad\'{e} type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete…