An adaptive finite element method is adopted to simulate the steady state coupled Schrödinger equations with a small parameter. We use damped Newton iteration to solve the nonlinear algebraic system. When the solution domain is elliptic, our numerical results with Dirichlet or Neumann boundary conditions are consistent with previous theoretical results. For the dumbbell and circular ring domains with Dirichlet boundary conditions, we obtain some new results that may be compared with future theoretical analysis.

The fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on the L2-1σ formula proposed in [A. Alikhanov, J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t–1–α on the interval [Δt, T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials Nexp needed is of order for T≫1 or for TH1 for fixed accuracy ε. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

The Kohlrausch functions $\exp (- {t}^{\beta } )$, with $\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method.

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$ of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of ${K}_{ir} (x)$ and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$.

The coherent states approximation for one-dimensional multi-phased wave functions is considered in this paper. The wave functions are assumed to oscillate on a characteristic wave length 0(ε) with ε ≪ 1. A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic analysis. This algorithm is then extended to multi-phased wave functions. If cross points or caustics exist, the coherent states approximation algorithm based on the parameter recovery will fail in some local regions. In this case, we resort to the windowed Fourier transform technique, and propose a composite coherent states approximation method. Numerical experiments show that the number of coherent states derived by the proposed method is much less than that by the direct windowed Fourier transform technique.

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.