Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic ...

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome th...

We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noise. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random ...

The aim of this paper is to carry out a rigorous error analysis for the Strang splitting Laguerre–Hermite/Hermite collocation methods for the time-dependent Gross–Pitaevskii equation (GPE). We derive error estimates for full discretizations of the three-dimensional GPE with cylindrical symmetry by the Strang splitting Laguerre–Hermite collocation method, and for the d-dimensional GPE by the Str...

In this paper we discuss the collocation method for a large class of Fredholm linear integro-differential equations. It will be shown that, when a certain higher order interpolation operation is added to the collocation solution of this equation, the new approximations will, under suitable assumptions, admit a multiterm error expansion in even powers of the step-size h. Based on this expansion,...

In this article we use discrete collocation method for solving Fredholm–Volterra integro– differential equations, because these kinds of integral equations are used in applied sciences and engineering such as models of epidemic diffusion, population dynamics, reaction–diffusion in small cells. Also the above integral equations with convolution kernel will be solved by discrete collocation metho...

Although the semantic analysis and the grammatical distribution are treated as separate issues in linguistic theories, there is a close interconnection between the two in that the differences in the lexical meanings are often realized at the levels both of the grammatical function and of the lexical collocation. This is the basic assumption utilized when we design and develop Knowledge Base of ...

Languages are inherently ambiguous. Four out of five words in English have more than one meaning. Nowadays there is a growing number of small proprietary thesauri used for knowledge management for different applications. In order to enable the usage of these thesauri for automatic text annotations, we introduce a robust method for discriminating word senses using hypernyms. The method uses coll...

This paper describes how to automatically classify the functional relations from the Factotum knowledge base via a statistical machine learning algorithm. This incorporates a method for inferring prepositional relation indicators from corpus data. It also uses lexical collocations (i.e., word associations) and class-based collocations based on the WordNet hypernym relations (i.e., is-subset-of)...