Abstract. The Cable equation has been one of the most fundamental equations for modeling neuronal dynamics. In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order O( t 2−max{α,β} + t −1 N −m ), where t, N and m are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. α and β are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims.
In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel (x − s) −µ , 0 < µ < 1. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both L ∞ -and weighted L 2 -norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change x → x 1/λ for a suitable real number λ. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.
Background CenteringPregnancy Care is a promising group prenatal care innovation that combines assessment, education, and peer support. In China, it is not clear how best to integrate the CenteringPregnancy Care into existing maternal health care models. This qualitative study aimed to explore Chinese pregnant women’s experience in the Internet-based CenteringPregnancy management model. Methods The Internet-based CenteringPregnancy was applied in a tertiary hospital between 2018 and 2019 in Wuhan, Hubei Province. Through purposive sampling, a total of 9 pregnant women who had experienced Internet-based CenteringPregnancy were recruited. A semi-structured interview was used to collect qualitative data, and Colaizzi’s 7-step method of phenomenological data analysis was used to analyze the collected data. Results Three themes were extracted from the participants’ interviews, including: 1) empowerment; 2) psychological and social support; 3) challenges of the Internet-based CenteringPregnancy. The Internet-based CenteringPregnancy management model retained advantages of CenteringPregnancy, emphasizing the pregnant woman as the subject of health care and promoting them to participate in health care. Participants believed that they could exchange pregnancy knowledge, help each other, and improve mood both timely and efficiently from the new model. However, it was found that there were challenges in seminar time arrangement, topic selection, and discussion management. Conclusion The Internet-based CenteringPregnancy management model positively affected pregnant women’s empowerment, psychological, and social support. It is recommended to improve the seminar’s design in future studies.
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