Related papers: Riccati-parameter solutions of nonlinear second-or…

A reduced-order model algorithm, based on approximations of Lax pairs, is proposed to solve nonlinear evolution partial differential equations. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the space where…

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution…

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs…

Tensor decompositions, which represent an $N$-order tensor using approximately $N$ factors of much smaller dimensions, can significantly reduce the number of parameters. This is particularly beneficial for high-order tensors, as the number…

The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to…

We use a new approach with a matrix transformation to obtain a new global solvability criterion for matrix Riccati equations. The proven theorem completes an well known result in directions of extension of classes of coefficient of…

In this paper, the fractional projective Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Burgers equation, the space-time…

We systematically analyze the nonlinear partial differential equation that determines the behaviour of a bounded radiating spherical mass in general relativity. Four categories of solution are possible. These are identified in terms of…

Some twenty years ago we introduced a nonstandard matrix Riccati equation to solve the partial stochastic realization problem. In this paper we provide a new derivation of this equation in the context of system identification. This allows…

The low-rank alternating direction implicit (ADI) method is an efficient and effective solver for large-scale standard continuous-time algebraic Riccati equations that admit low-rank solutions. However, the existing low-rank ADI algorithm…

This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…

This paper introduces a generalization of the well-known Riccati recursion for solving the discrete-time equality-constrained linear quadratic optimal control problem. The recursion can be used to compute the solutions as well as optimal…

We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory…

For large-scale discrete-time algebraic Riccati equations (DAREs) with high-rank nonlinear and constant terms, the stabilizing solutions are no longer numerically low-rank, resulting in the obstacle in the computation and storage. However,…

We consider factorizations of the stationary and non-stationary Schroedinger equation in R^n which are based on appropriate Dirac operators. These factorizations lead to a Miura transform which is an analogue of the classical…

A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known…

We address asymptotic formulae for the classical Poincar\'e-Perron problem of linear differential equations with almost constant coefficients in a half line $[t_0,+\infty)$ for high order equation $n\ge 5$ and some $t_0\in\mathbb{R}$. By…

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the…

We study the $T$-periodic solutions of the real Riccati differential equation $x' = x^2 + \gamma(t),$ where $x=x(t)$ and $\gamma$ is a $T$-periodic function. Our goal is to define a real-valued discriminant $\Delta_{\gamma}$ that determines…

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…