An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of 1/240h6 was achieved with smooth functions.

Exponential spline method for singularly perturbed third-order boundary value problems

The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.

A Varied Density-based Clustering Approach for Event Detection from Heterogeneous Twitter Data

Extracting the latent knowledge from Twitter by applying spatial clustering on geotagged tweets provides the ability to discover events and their locations. DBSCAN (density-based spatial clustering of applications with noise), which has been widely used to retrieve events from geotagged tweets, cannot efficiently detect clusters when there is significant spatial heterogeneity in the dataset, as it is the case for Twitter data where the distribution of users, as well as the intensity of publishing tweets, varies over the study areas. This study proposes VDCT (Varied Density-based spatial Clustering for Twitter data) algorithm that extracts clusters from geotagged tweets by considering spatial heterogeneity. The algorithm employs exponential spline interpolation to determine different search radiuses for cluster detection. Moreover, in addition to spatial proximity, textual similarities among tweets are also taken into account by the algorithm. In order to examine the efficiency of the algorithm, geotagged tweets collected during a hurricane in the United States were used for event detection. The output clusters of VDCT have been compared to those of DBSCAN. Visual and quantitative comparison of the results proved the feasibility of the proposed method.