A family of orthogonal generalised Müntz-Jacobi functions is introduced and used for solving singularly perturbed fractional differential equations. Such basis functions can provide much better approximation for boundary layers or endpoint singularities than usual polynomial bases. The fractional integrals and derivatives of generalised Müntz-Jacobi functions are accurately calculated. The corresponding Petrov-Galerkin and Galerkin methods are very efficient. Numerical examples demonstrate a significant improvement in the accuracy of the methods.

The artificial boundary method is used to reformulate the time fractional Schrödinger equation on the real line as a bounded problem with exact artificial boundary conditions. The problem appeared is solved by a numerical method employing the L1-formula for the Caputo derivative and finite differences for spatial derivatives. The convergence of the method studied and optimal error estimates in a special metric are obtained. The technique developed here can be also applied to study the convergence of approximation methods for standard Schrödinger equation.

A fast algorithm with almost optimal memory for the computation of Caputo’s fractional derivative is developed. It is based on a nonuniform splitting of the time interval [0, $t_n$] and a polynomial approximation of the kernel function $(1−τ)^{−α}$. Both the storage requirements and the computational cost are reduced from $\mathscr{O}(n)$ to $(K +1)\mathscr{O}(log~n)$ with $K$ being the degree of the approximated polynomial. The algorithm is applied to linear and nonlinear fractional diffusion equations. Numerical results show that this scheme and the corresponding direct methods have the same order of convergence but the method proposed performs better in terms of computational time.

A framework for solving space fractional partial differential equations by reduced finite element methods is proposed. In particular, we construct reduced bases, study their properties and use them in numerical schemes. The stability and convergence of these methods are investigated. Two numerical examples show that such an approach has a high efficiency and a low computational cost.

Nodal-type Newton-Cotes rules for fractional hypersingular integrals based on the piecewise k-th order Newton interpolations are proposed. A general error estimate is first derived on quasi-uniform meshes and then we show that the even-order rules exhibit the superconvergence phenomenon — i.e. if the singular point is far away from the endpoints then the accuracy of the method is one order higher than the general estimate. Numerical experiments confirm the theoretical results.

A conservative finite difference scheme for nonlinear space fractional Klein-Gordon-Schrödinger systems with high-degree Yukawa interaction is studied. We show that the arising difference equations are uniquely solvable and approximate solutions converge to the exact solution at the rate O ($τ^2+h^2$). Moreover, we prove that the scheme can be decoupled and preserves the mass and energy conservation laws. Numerous examples confirm theoretical results and demonstrate the efficiency of the scheme. They also show the influence of the fractional order and the high-degree term coefficient on the shape and the propagation velocity of solitary waves.

The convergence of Parareal-Euler and -LIIIC2 algorithms using the backward Euler method as a $\mathscr{G}$-propagator for the linear problem $U^′ (t)+α(t)A^ηU(t)=f(t)$ with a non-constant coefficient $α$ is studied. We propose to employ the propagator $G$ to a constant model $U^′(t)+βA^ ηU(t)=f(t)$ with a special coefficient β instead of applying both propagators $\mathscr{G}$ and $\mathscr{F}$ to the same target model. We established a simple formula to find an optimal parameter $β_{opt}$, minimising the convergence factor for all mesh ratios. Numerical results confirm the proximity of theoretical optimal $β_{opt}$ to the optimal numerical parameter.

A finite difference method for a class of time-space fractional diffusion equations is considered. The trapezoidal formula and a fourth-order fractional compact difference scheme are, respectively, used in temporal and spatial discretisations and the method stability is studied. Theoretical estimates of the convergence in the $L_2$ -norm are shown to be $\mathscr{O}(τ^2+h^4)$, where $τ$ and $h$ are time and space mesh sizes. Numerical examples confirm theoretical results.

Iterative Laplace transform methods for fractional partial differential equations and fractional partial integro-differential equations arising in European option pricing with the Lévy α-stable processes and regime-switching or state-dependent jump rates are studied and numerical contour integral methods to inverse the Laplace transform are developed. It is shown that the methods under consideration have the second-order convergence rate in space and spectral-order convergence for Laplace transform inversion.

A numerical method for the generalised second grade fluid through porous media with anomalous diffusion is considered. The method is based on a combination of finite differences in time and a spectral method in space directions. The convergence of the method is rigorously proved and theoretical error estimates are established. The numerical scheme is unconditionally stable and provides a high accuracy if the solution is smooth enough. The results of numerical simulations are consistent with theoretical findings.

A sharp estimate for the L1 formula on graded meshes, which approximates the Caputo derivatives of functions with a weak singularity at t = 0 is obtained. Combining such approximations with the sum-of-exponential approximations of the kernel, we develop fast difference schemes for one- and two-dimensional fractional diffusion equations, the solutions of which have a weak singularity at the starting time. The proof of the stability and convergence is based on the maximum principle. Numerical examples confirm theoretical estimates.