Exact traveling wave solutions of the fifth order space-time fractional Sawada-Kotera equation are derived by generalised exp(−$Φ(ξ)$)-expansion and an improved fractional sub-equation method. Among the solutions obtained there are hyperbolic, trigonometric, exponential and rational ones. The methods are simple, efficient and can be applied to other nonlinear problems.

Applying Maple symbolic computations, we derive eight sets of mixed lump-soliton solutions to the (2 + 1)-dimensional BKP equation. The solutions are analytic and allow the separation of lumps and line solitons.

A multistep Chebyshev-Gauss-Lobatto spectral collocation method for nonlinear Volterra integral equations with vanishing delays is developed. The convergence of the hp-version of the method in supremum norm is proved. Numerical experiments show the efficiency of the method for equations with highly oscillating, steep gradient and non-smooth solutions.

A new parallel algorithm for convection-dominated diffusion equation is proposed. To derive an approximate solution of the problem at the same time level, we use two domain decomposition methods. The average of the solutions obtained is set to be new numerical solution. The algorithm demonstrates a high accuracy and is suitable for parallel computing. Approximate solution of two-dimensional convection-dominated diffusion equations is also considered. Numerical examples show the efficiency and reliability of the algorithm.

We consider a Galerkin method based on Legendre and Laguerre polynomials and apply it to the Euler-Bernoulli beam equation. The matrices of the method are well structured, which results in substantial reduction of computational cost. Numerical examples demonstrate the efficiency and a high accuracy of the algorithm proposed.

We introduce a bilateral intensity fitting active contour model, which takes into consideration both the space distance and local intensity variation. It demonstrates flexible contour initialisation in handing intensity inhomogeneity, efficient segmentation, robustness to high noise, and ability to work with weak boundaries and complex background. Experiments with synthetic and real images show high efficiency of the model.

An iterative method for complex symmetric systems of linear equations is proposed. Estimates for the spectral radius of the method are obtained and sufficient convergence conditions are established. Numerical experiments show the efficiency of the method for linear systems with a dominant real part.

We reduce a nonlinear fourth order equation of Kirchhoff type to an operator equation for a nonlinear term and establish sufficient conditions for the unique solvability of the original problem. Approximate solutions of the problem are derived by a fast converging iterative method. Numerical examples confirm theoretical results and demonstrate the efficiency of the approach used.

We consider two parameter iteration methods for the coupled Sylvester matrix equations $$AX+YB=E$$ $$CX+YD=F.$$ Having established the convergence of the algorithms, we discuss the choice of optimal parameters. Numerical examples show the efficiency of the methods.

The convergence of modulus-based synchronous multisplitting accelerated overrelaxation iteration methods for linear complementarity problems is studied using the new technique by Zhang, Zhang and Ren. We show that this technique is particularly convenient in the a priori and a posteriori error analysis.

We develop a reliable residual-based a posteriori error estimator for a non-conforming method with non-matching meshes for a harmonic elastodynamics equation and show that the approximation method converges with an optimal order to the exact solution. Moreover, we propose an adaptive strategy to reduce computational cost and derive better approximations for problems with singularities and with large approximating systems. Numerical experiments confirm theoretical conclusions.