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We present a novel deep learning method for computing eigenvalues of the fractional Schr\"odinger operator. Our approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior…
This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure,…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
A novel overlapping domain decomposition splitting algorithm based on a Crank-Nisolson method is developed for the stochastic nonlinear Schroedinger equation driven by a multiplicative noise with non-periodic boundary conditions. The…
This research deals with the numerical solution of non-linear fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article aims to present a new formula for the…
We construct a new class of higher-dimensional column-vector loop algebras. Based on it, a method for generating higher-dimensional isospectral-nonisospectral integrable hierarchies is proposed. As an application, we derive a generalized…
This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…
In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and…
This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…
This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
The method of moments in the context of Nonlinear Schrodinger Equations relies on defining a set of integral quantities, which characterize the solution of this partial differential equation and whose evolution can be obtained from a set of…
In this paper we propose a method to solve the Kadomtsev--Petviashvili equation based on splitting the linear part of the equation from the nonlinear part. The linear part is treated using FFTs, while the nonlinear part is approximated…
This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential $V$. The main concern is treating nonlinearities whose degree is low…
We investigate discretizations of the integrable discrete nonlinear Schr\"odinger dynamical system and related symplectic structures. We develop an effective scheme of invariant reducing the corresponding infinite system of ordinary…
Operator-splitting methods are widely used to solve differential equations, especially those that arise from multi-scale or multi-physics models, because a monolithic (single-method) approach may be inefficient or even infeasible. The most…
This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in…