Related papers: ADI splitting schemes for a fourth-order nonlinear…
We propose a variational splitting technique for the generalized-$\alpha$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows…
A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the…
An optical flow variational model is proposed for a sequence of images defined on a domain in $\mathbb{R}^2$. We introduce a regularization term given by the $L^1$ norm of a fractional differential operator. To solve the minimization…
In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference…
Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a…
Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric…
We propose a hierarchical splitting approach to differential equations that provides a design principle for constructing splitting methods for $N$-split systems by iteratively applying splitting methods for two-split systems. We analyze the…
We present a method for constructing numerical schemes with up to 3rd strong convergence order for solution of a class of stochastic differential equations, including equations of the Langevin type. The construction proceeds in two stages.…
In several recent works \cite{Causley2013a}, \cite{Causley2013}, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating…
This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical result on…
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the,…
We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately…
The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the $k$-step BDF convolution quadrature to discretize the time-fractional derivative with order…
High order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but…
We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters.…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
We present an algorithm for constructing numerical solutions to one--dimensional nonlinear, variable coefficient boundary value problems. This scheme is based upon applying the Homotopy Analysis Method (HAM) to decompose a nonlinear…
In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
The Adomian decomposition method is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The aim of this paper is to apply Adomian decomposition method to obtain approximate solutions of nonlinear…