GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical results indicate the usefulness of the proposed method.

A novel clinical model for risk prediction and stratification of new-onset diabetes mellitus after distal pancreatectomy

Background:The incidence of new-onset diabetes mellitus(NODM)after distal pancreatectomy(DP)remains high.Few studies have focused on NODM in patients with pancreatic benign or low-grade malignant lesions(PBLML).This study aimed to develop and validate an effective clinical model for risk prediction and stratification of NODM after DP in patients with PBLML.Methods:A follow-up survey was conducted to investigate NODM in patients without preoperative DM who underwent DP.Four hundred and forty-eight patients from Peking Union Medical College Hospital(PUMCH)and 178 from Guangdong Provincial People’s Hospital(GDPH)met the inclusion criteria.They constituted the training cohort and the validation cohort,respectively.Univariate and multivariate Cox regression,as well as least absolute shrinkage and selection operator(LASSO)analyses,were used to identify the independent risk factors.The nomogram was constructed and verified.Concordance index(C-index),receiver operating characteristic(ROC)curve,calibration curves,and decision curve analysis(DCA)were applied to assess its predictive performance and clinical utility.Accordingly,the optimal cut-off point was determined by maximally selected rank statistics method,and the cumulative risk curves for the high-and low-risk populations were plotted to evaluate the discrimination ability of the nomogram.Results:The median follow-up duration was 42.8 months in the PUMCH cohort and 42.9 months in the GDPH cohort.The postoperative cumulative 5-year incidences of DM were 29.1%and 22.1%,respectively.Age,body mass index(BMI),length of pancreatic resection,intraoperative blood loss,and concomitant splenectomy were significant risk factors.The nomogram demonstrated significant predictive utility for post-pancreatectomy DM.The C-indexes of the nomogram were 0.739 and 0.719 in the training and validation cohorts,respectively.ROC curves demonstrated the predictive accuracy of the nomogram,and the calibration curves revealed that prediction results were in general agreement with the actual results.The considerable clinical applicability of the nomogram was certified by DCA.The optimal cut-off point for Background:The incidence of new-onset diabetes mellitus(NODM)after distal pancreatectomy(DP)remains high.Few studies have focused on NODM in patients with pancreatic benign or low-grade malignant lesions(PBLML).This study aimed to develop and validate an effective clinical model for risk prediction and stratification of NODM after DP in patients with PBLML.Methods:A follow-up survey was conducted to investigate NODM in patients without preoperative DM who underwent DP.Four hundred and forty-eight patients from Peking Union Medical College Hospital(PUMCH)and 178 from Guangdong Provincial People’s Hospital(GDPH)met the inclusion criteria.They constituted the training cohort and the validation cohort,respectively.Univariate and multivariate Cox regression,as well as least absolute shrinkage and selection operator(LASSO)analyses,were used to identify the independent risk factors.The nomogram was constructed and verified.Concordance index(C-index),receiver operating characteristic(ROC)curve,calibration curves,and decision curve analysis(DCA)were applied to assess its predictive performance and clinical utility.Accordingly,the optimal cut-off point was determined by maximally selected rank statistics method,and the cumulative risk curves for the high-and low-risk populations were plotted to evaluate the discrimination ability of the nomogram.Results:The median follow-up duration was 42.8 months in the PUMCH cohort and 42.9 months in the GDPH cohort.The postoperative cumulative 5-year incidences of DM were 29.1%and 22.1%,respectively.Age,body mass index(BMI),length of pancreatic resection,intraoperative blood loss,and concomitant splenectomy were significant risk factors.The nomogram demonstrated significant predictive utility for post-pancreatectomy DM.The C-indexes of the nomogram were 0.739 and 0.719 in the training and validation cohorts,respectively.ROC curves demonstrated the predictive accuracy of the nomogram,and the calibration curves revealed that prediction results were in general agreement with the actual results.The considerable clinical applicability of the nomogram was certified by DCA.The optimal cut-off point for risk prediction value was 2.88, and the cumulative risk curves of each cohort showed significant differences between the high- and low-risk groups. Conclusions: The nomogram could predict and identify the NODM risk population, and provide guidance to physicians in monitoring and controlling blood glucose levels in PBLML patients after DP.

Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

In this paper,the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay.In this collocation method,the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation.Derivative approximation in the sense of integral is constructed in numerical format,and the convergence of the spectral collocation method in the sense of the L¥and L2 norm is proved by the Dirichlet formula.At the same time,the error convergence also meets the effect of spectral accuracy convergence.The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.