Darboux's Principle asserts that a power series or Fourier coefficient $a$_$n$ for an analytic function $f$($z$) is approximated as $n$ → $∞$ by a sum of terms, one for each singularity of $f$($z$) in the complex plane. This is crucial to understanding why Fourier series converge rapidly or slowly, and thus crucial to Fourier numerical methods. We partially refute Darboux's Principle by an explicit counterexample constructed by applying the Poisson Summation Theorem to a Fourier Transform pair found explicitly by Ramanujan. The Fourier coefficients show a geometric rate of decay proportional to exp(−$πχ$$n$) multiplied by sin($φ$) where the “phase" is $φ$ = $π$$χ$²$n$² mod (2$π$). We prove that the Fourier series converges everywhere within the largest strip centered on the real axis which is singularity-free, here |$\mathcal{I}$($z$)| < $πχ$. We present strong evidence that the boundaries of the strip of convergence are natural boundaries. Because the function $f$($z$) is singular everywhere on the lines $\mathcal{I}$($z$) = ±$πχ$, there is no simple way to extrapolate the asymptotic form of the Fourier coefficients from knowledge of the singularities, as is possible through Darboux's Theorem when the singularities are isolated poles or branch points.

We studied the influence of the arrays of ceramic balls on the hardness of water and discovered effects, which can possibly explain the correlation between the balls and the growth of calcite crystals of CaCO₃. A model to compute the electric field at the surface of the balls is proposed. It is shown that the number of polarised nuclei contributing to scale prevention is considerably larger than in natural water. Numerical simulations for a two-dimensional macroscopic model show that the effect of ceramic balls can be reproduced in configurations studied experimentally and industrially. Summarising, we note that ceramic balls induce a polarisation in the calcite particles thus decreasing the surface tension energy of crystals in the vicinity of the ceramic balls.

A stochastic three-dimensional Navier-Stokes system with the axisymmetric initial data and white noise is studied. It is shown that if the swirl component of the initial velocity field and the white noise are sufficiently small, then the axisymmetric pathwise solution is global in probability. Moreover, in the absence of the swirl, the pathwise axisymmetric solution is global almost surely.

A family of explicit parametric stochastic Runge-Kutta methods for stochastic Poisson systems is developed. The methods are based on perturbed collocation methods with truncated random variables and are energy-preserving. Under certain conditions, the truncation does not change the convergence order. More exactly, the methods retain the mean-square convergence order of the original stochastic Runge-Kutta method. Numerical examples show the efficiency of the methods constructed.

Fast algorithms for boundary integral equations connected with Robin boundary value problem for the Laplace equation in domains with ellipse or close to ellipse boundaries are developed. It is shown that the coefficient matrices of discretisation systems have a special structure. This fact is used to develop a fast algorithm for matrix vector multiplication and to implement it in the numerical methods used. Such an approach is especially helpful in numerical methods for inverse problems, since many methods of their solution repeatedly use forward solvers. The efficiency of the methods is illustrated by numerical examples.

A new approach to construction of generalised backward differentiation formulae for fractional differential equations is proposed. The consistence and convergence of the method are considered and numerical simulations are presented.

A fast and accurate exponential Runge-Kutta method for a class of time-dependent advection-diffusion-reaction equations is developed. To discretise the convection term, a modified upwind difference scheme is used. This allows to avoid numerical oscillation and achieve second order spatial accuracy. The method demonstrates good stability and numerical examples show the applicability of the method to advection-diffusion-reaction problems with stiff nonlinearities.

Two second-order convolution quadrature methods for fractional nonlinear diffusion-wave equations with Caputo derivative in time and Riesz derivative in space are constructed. To improve the numerical stability, the fractional diffusion-wave equations are firstly transformed into equivalent partial integro-differential equations. Then, a second-order convolution quadrature is applied to approximate the Riemann-Liouville integral. This deduced convolution quadrature method can handle solutions with low regularity in time. In addition, another second-order convolution quadrature method based on a new second-order approximation for discretising the Riemann-Liouville integral at time $t$_$k$−1/2 is constructed. This method reduces computational complexity if Crank-Nicolson technique is used. The stability and convergence of the methods are rigorously proved. Numerical experiments support the theoretical results.

An $H$(div)-conforming finite element method for the Biot's consolidation model is developed, with displacements and fluid velocity approximated by elements from BDM_$k$ space. The use of $H$(div)-conforming elements for flow variables ensures the local mass conservation. In the $H$(div)-conforming approximation of displacement, the tangential components are discretised in the interior penalty discontinuous Galerkin framework, and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully discrete scheme, prove their unique solvability and establish optimal error estimates for each variable.

Numerical experiments are used to determine the motion of singularities in the heat flow of harmonic maps into a unit sphere. These singularities are closely related to point defects in a nematic liquid crystals. The motion of singularities is affected by initial positions and interaction between singularities and boundary conditions. In particular, it is shown that the motion of singularities is the same as the motion of point defects in nematic liquid crystals under Neumann boundary conditions. For Dirichlet boundary conditions, the results do not properly reflect the crystal defect motion due to the shortcoming of the model.

The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

An easily implementable noise strength estimation algorithm to analyse the urban traffic is developed. It improves the accuracy of the road velocity predictions. Using real urban traffic data from Beijing Taxi GPS system, we demonstrate the efficiency of the algorithm. It is also shown that the BV denoising method with the best noise-strength estimates significantly improves the road clustering.