Related papers: Numerical solution of Riemann--Hilbert problems: r…
$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\},…
We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr\"{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem…
In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials…
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts…
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…
The processes of simultaneous coagulation and Ostwald ripening of particles in the concluding stage of phase transformation are considered. We solve the integro-differential system of Smoluchowski-type kinetic and mass balance equations…
In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using…
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic…
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility…
When homogenizing elliptic partial differential equations, the so-called corrector problem is pivotal to compute the macroscale effective coefficients from the microscale information. To solve this corrector problem in the periodic setting,…
Explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi-dimensional spaces in a continuous sequence of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also given…
The vector Riemann-Hilbert problem is analyzed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical…