A second-order nonlinear anisotropic diffusion-based model for Gaussian additive noise removal is proposed. The method is based on a properly constructed edge-stopping function and provides an efficient detail-preserving denoising. It removes additive noise, overcomes blurring effect, reduces the image staircasing and does not generate multiplicative noise, thus preserving boundaries and all the essential image features very well. The corresponding PDE model is solved by a robust finite-difference based iterative scheme consistent with the diffusion model. The method converges very fast to the model solution, the existence and regularity of which is rigorously proved.

A pansharpening method based on the Rudin-Osher-Fatemi model and using a reproducing kernel Hilbert space along with the approximated Heaviside function is developed. The corresponding minimisation problem is solved by the alternating direction method of multipliers. Numerous numerical experiments with Pléiades and IKONOS satellite datasets demonstrate the efficiency of the method in preserving spectral and spatial information and show its superiority to other approaches.

A Crank-Nicolson finite volume scheme for the modeling of the Riesz space distributed-order diffusion equation is proposed. The corresponding linear system has a symmetric positive definite Toeplitz matrix. It can be efficiently stored in  $\mathcal{O}$($NK$) memory. Moreover, for the finite volume scheme, a fast version of conjugate gradient (FCG) method is developed. Compared with the Gaussian elimination method, the computational complexity is reduced from $\mathcal{O}$($MN$³ + $NK$) to $\mathcal{O}$($l$_$A$$MN$log$N$ + $NK$), where $l$_$A$ is the average number of iterations at a time level. Further reduction of the computational cost is achieved due to use of a circulant preconditioner. The preconditioned fast finite volume method is combined with the Levenberg-Marquardt method to identify the free parameters of a distribution function. Numerical experiments show the efficiency of the method.

The convergence of a compact finite difference scheme for one- and two-dimensional time fractional fourth order equations with the first Dirichlet boundary conditions is studied. In one-dimensional case, a Hermite interpolating polynomial is used to transform the boundary conditions into the homogeneous ones. The Stephenson scheme is employed for the spatial derivatives discretisation. The approximate values of the normal derivative are obtained as a by-product of the method. For periodic problems, the stability of the method and its convergence with the accuracy $\mathcal{O}$(τ^2−$α$) + $\mathcal{O}$($h$⁴) are established, with the similar error estimates for two-dimensional problems. The results of numerical experiments are consistent with the theoretical findings.

A multistep collocation method on uniform grids for weakly singular Volterra integral equations is considered. The initial integral equation is modified by a smoothing transformation and then solved by a multistep collocation method on a uniform grid. Convergence and linear stability are also studied. Numerical results demonstrate the efficiency of the method.

The convergence of $H$¹-Galerkin mixed finite element methods for elliptic optimal control problems is studied and postprocessing operators are used to establish the superconvergence for control, state and adjoint state variables. A numerical example confirms the validity of theoretical results.

A general RTMS iteration method for linear complementarity problems is proposed. Choosing various pairs of relaxation parameters, we obtain new two-sweep modulus-based matrix splitting iteration methods and already known iteration procedures such as the MS [1] and TMS [27] iteration methods. If the system matrix is positive definite or an $H_+$-matrix and the relaxation parameters $ω_1$ and $ω_2$ satisfy the inequality 0≤$ω_1$, $ω_2$≤1, sufficient conditions for the uniform convergence of MS, TMS and NTMS iteration methods are established. Numerical results show that with quasi-optimal parameters, RTMS iteration method outperforms MS and TMS iteration methods in terms of computing efficiency.

A predator-prey system with Holling type II functional response and a time lag is described by a delayed differential-algebra system and the local asymptotic stability and Hopf bifurcation of such system is studied. It is shown that if the time lag increases, a sequence of Hopf bifurcations can occur. The stability and direction of the Hopf bifurcations are studied by using center manifold theory for functional differential equations. A numerical example illustrates our theoretical findings.

A variational iteration method involving Adomian polynomials to solve a strongly nonlinear boundary value problem is considered. After its convergence is established, the efficiency and accuracy of the proposed method are tested on problems with exponential nonlinearity.

An $h$-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method with a positivity-preserving technique to simulate classical two-dimensional detonation waves is developed. The KXRCF troubled-cell indicator is used to detect the troubled cells with possible discontinuities or high gradients. At each time-level, an adaptive mesh is generated by refining troubled cells and coarsening others. In order to avoid the situations where detonation front moves too fast and there are not enough cells to describe detonation front before it leaves, a recursive multi-level mesh refinement technique is designed. The numerical results show that for smooth solutions, this $h$-adaptive method does not degrade the optimal convergence order of the nonadaptive method and outperforms it in terms of computational storage for shocked flows.

Linear partial differential equations in (3 + 1)-dimensions consisting of all mixed second-order derivatives are considered, and Maple symbolic computations are made to construct their lump and interaction solutions, including lump-periodic, lump-kink and lump-soliton solutions.

A collocation method based on exponential B-splines for two-dimensional second-order non-linear hyperbolic equations is studied. The initial equation is split into a system of coupled equations, each of which is transformed into a system of ordinary differential equations. The corresponding differential equations are solved by SSP-RK(2,2) method. It is shown that the method under consideration is unconditionally stable. Numerical experiments demonstrate its efficiency and accuracy.

First and second order numerical schemes for the fourth order parabolic equation with Peng-Robinson equation of state, which are based on recently proposed invariant energy quadratisation method are developed. Both schemes are linear, unconditionally energy stable and uniquely solvable. The reduced linear systems are symmetric and positive definite, so that their solutions can be efficiently found. Numerical results demonstrate the good performance of the schemes, consistent with experimental data.