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In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to…
We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical…
Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the constraint functions are deterministic,…
Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefor, it usually pays off to compute local approximations of the used dynamical systems in…
The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it…
We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand,…
Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in…
In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations…
In this paper, we present a numerical homogenization scheme for indefinite, time-harmonic Maxwell's equations involving potentially rough (rapidly oscillating) coefficients. The method involves an $\mathbf{H}(\mathrm{curl})$-stable,…
Inspired by path-integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method,…
Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic…
In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for…
In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.
We introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of…
This paper focuses on the strong convergence of the truncated $\theta$-Milstein method for a class of nonautonomous stochastic differential delay equations whose drift and diffusion coefficients can grow polynomially. The convergence rate,…
Two-time-scale stochastic approximation, a generalized version of the popular stochastic approximation, has found broad applications in many areas including stochastic control, optimization, and machine learning. Despite its popularity,…
Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear…