Optimal $H^1$-error estimates for a Crank-Nicolson finite difference scheme for 2D-Gross-Pitaevskii equation with angular momentum rotation term are derived. The analysis is based on classical energy estimate method and on the lifting technique. With no constraint on the grid ratio, we show that the convergence rate of approximate solutions is equivalent to $O$($τ^2$+$h^2$), consistent with numerical results of the existing studies.

The Feynman-Kac formulas are used to develop new second-order numerical schemes for the forward-backward stochastic differential equations (FBSDEs) of the first and second order. The methods are simple and allow an easy implementation. Numerous numerical tests for FBSDEs, fully nonlinear second-order parabolic partial differential equations and the Hamilton-Jacobi-Bellman equations show the stability and a high accuracy of the methods.

A nonlinear innovation diffusion model which incorporates the evaluation stage (time delay) is proposed to describe the dynamics of three population classes for non-adopter and adopter densities. The local stability of the various equilibrium points is investigated. It is observed that the system is locally asymptotically stable for a delay limit and produces periodic orbits via a Hopf bifurcation when evaluation period crosses a critical value. Applying normal form theory and center manifold theorem, we study the properties of the bifurcating periodic solutions. The model shows that the adopter population density achieves its maturity stage faster if the cumulative density of external influences increases. Several numerical examples confirm our theoretical results.

A new image denoising-decomposition model is proposed. It is based on the TV minimisation and fractional derivatives of various order. It is shown that the texture details are better described by fractional derivatives of order greater than 1, whereas the noise part by fractional derivative of order smaller than 1, and this effect is used in image denoising. The model is able to eliminate staircase effect and keeps the edge and texture information of the image. Various experiments confirm the efficiency of the model.

A simple and efficient algorithm for solving a mean curvature based model is proposed. It uses linearisation technique and allows to find closed form solutions of all the subproblems involved. The experimental results show that the method is more efficient in terms of CPU time than the augmented Lagrangian methods considered earlier. Numerical examples demonstrate the convergence of the method.

The Hirota bilinear method is applied to a generalised (3 + 1)-dimensional nonlinear evolution equation. Using the Riemann theta function, we construct periodic wave solutions of the Eq. (1.1) and discuss their properties. Graphic examples show the propagation of the corresponding waves for different sets of parameters. We also study the asymptotic of periodic waves and their relation with solution solutions.

The alternating direction method of multipliers is applied to three-block separable quadratic programming problems whose objective function is the sum of three functions without coupled variables. Necessary and sufficient conditions for the unique solvability of this problem are established. The convergence of the method is considered from the viewpoints of matrix computation and numerical optimisation.

Considering a reduced (3 + 1)-dimensional shallow water equation, we use Hirota formulation and symbolic calculation to derive positive lump solitons rationally localised in all directions of the $(x, y)$-plane. The interaction of the lump and one stripe solitons is studied. Numerical experiments show that the collision of such solutions is completely inelastic and the lump soliton is swallowed by the stripe one. Exploring the interaction of the lump and a couple of resonance stripe solitons, we note that the lump soliton transforms into a ghost soliton. Most of the time it remains hidden in stripe solitons, but appears at a certain time and fades after that.

Modulus-based multisplitting iterative methods for large sparse nonlinear complementarity problems are developed. The approach is based on a reformulation of nonlinear complimentarily problems as implicit fixed-point equations and includes Jacobi, Gauss-Seidel and SOR iteration methods. For systems with positive definite matrices the convergence of the methods is proved. The methods are suitable for implementation on multiprocessor systems and numerical experiments confirm their high efficiency.

An integrable coherently coupled nonlinear Schrödinger system describing the propagation of polarised optical waves in an isotropic medium with a generalized 4 × 4 matrix Ablowitz-Kaup-Newell-Segur-type Lax pair is studied. The corresponding initial-boundary value problem is reduced to a matrix Riemann-Hilbert problem in the complex plane. Moreover, it is shown that the associated spectral functions depend on each other and satisfy a global relationship.

A two-grid based finite element method for nonlinear Sobolev equations is studied. It consists in solving small nonlinear systems related to coarse-grids, following the solution of linear systems in fine-grid spaces. The method has the same accuracy as the standard finite element method but reduces workload and saves CPU time. The $H^1$-error estimates show that the two-grid methods have optimal convergence if the coarse $H$ and fine $h$ mesh sizes satisfy the condition $h=\mathscr{O}(H^2)$. Numerical examples confirm the theoretical findings.

The sparse reconstruction of functions via a transformed $ℓ_1$ (TL1) minimisation is studied and theoretical results concerning recoverability and accuracy of such reconstruction from undersampled measurements are obtained. To identify the coefficients of sparse orthogonal polynomial expansions in uncertainty quantification, the method is combined with the stochastic collocation approach. The DCA-TL1 algorithm [37] is used in implementing the TL1 minimisation. Various numerical examples demonstrate the recoverability and efficiency of this method.

A variational $ℓ_q$-seminorm model to reduce the impulse noise is proposed. For $0<q<1$, it captures sparsity better than the $ℓ_1$-norm model. Numerical experiments show that for small $q$ this model is more efficient than TV$ℓ_1$ model if the noise level is low. If the noise level grows, the best possible parameter $q$ in the model approaches 1.

A projection gradient method for computing the ground state of the spin-orbit-coupled spin-1 Bose-Einstein condensate at extremely low temperatures is proposed. The continuous gradient flows are discretised by a second-order finite difference method in space and the Crank-Nicolson method in time. Our discretisation preserves the total mass conservation and the energy diminishing property. Numerical results show the efficiency of the method.