Numerical Solution for Riesz Fractional Diffusion Equation via Fractional Centered Difference Scheme

In this paper, a mixed matrix transform method with fractional centered difference scheme for solving fractional diffusion equation with Riesz fractional derivative was examined. It was obtained that the numerical scheme was unconditionally stable and feasible using the matrix analysis method. Numerical experiments were, then, carried out to support the theoretical predictions.

The time-splitting Fourier spectral method for Riesz fractional coupled Schrödinger–KdV equations in plasma physics

In this paper, Riesz fractional coupled Schrödinger–KdV (SK) equations have been solved by implementing a new approach viz. time-splitting spectral method. In order to verify the results, it has also been solved by an implicit finite difference method by using fractional centered difference approximation for Riesz fractional derivative. The obtained results manifest that the proposed time-splitting spectral method is very effective and simple for obtaining approximate solutions of Riesz fractional coupled SK equations. In order to show the reliability and efficiency of the proposed methods, numerical solutions obtained by these methods have been presented graphically.

On Preconditioners Based on HSS for the Space Fractional CNLS Equations

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

Effects of Atrium on the Electrical Activity of Sino-Atrial Node

As the electrical source of hearts, sino-atrial node (SAN) plays a significant role in rhythmic firing. The aim of this study is to determine effects of atrium on the electrical activity of SAN. Based on a mathematical model considering central and peripheral SAN cells as well as an atrial model, one-dimensional fiber with a gradual heterogeneity of SANC was developed. The forth-order Runge-Kutta and three-point centered difference methods were used to integrate the conduction model. The results demonstrated that as a load of SAN, atrium decreased the spontaneous sinus rate casing a shift of leading pacemaker site from the periphery to the center. If the ectopic beat produced by a premature stimulation was late after the end of the atrial refractory, the pause following it was compensatory. Otherwise, the pause was non-compensatory. In this case, the peripheral SANC excited earlier than the central SANC. During multiple premature beats, at short intervals the propagation could block in SAN due to a long refractory period of the action potential close to the central SANC. Leading pacemaker site shift was observed during the following several spontaneous SANC excitation. Therefore, the existence of atrium is suggested affecting not only sinus rates, but also the mode of electrical propagation and the site of the earliest excitation.

Counterfactual conditionals in argumentative legal language in Dutch

Legal argumentation is intended to resolve a difference of opinion between two or more legal parties by determining what are the facts in a case and finding an appropriate legal interpretation for these facts. Some of the discussion moves in legal argumentation take the shape of counterfactual conditionals (CTFs). CTFs are conditionals with an antecedent that is implicated to be false, not corresponding to the facts, and they occur in a number of argumentative contexts and argumentation techniques. This paper gives a structured overview of how such non-fact-based CTFs can contribute to resolving a legal and fact- centered difference of opinion. It does so by presenting a bottom-up corpus-based typology of CTFs in lawyers’ conclusions and in judgments in civil cases heard by Dutch-speaking Belgian courts of law. This typology is based on linguistic and pragmatic factors, such as the status of the facts that are referred to in the antecedent, the nature of the relation between antecedent and consequent, and the relation the CTF bears to the argumentative, situational and legal context.