http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/130 Wed, 23 Sep 2026 01:19:07 GMT 2026-09-23T01:19:07Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/13045 Computing harmonic-measure distribution functions of some multiply connected unbounded planar domains Arunmaran, M. A Brownian particle released from a point z0in a two-dimensional region Ωwill move around randomly until it eventually hits the boundary of Ω. We are interested in the probability that it hits the boundary somewhere within distance rof the starting point z0, for each value of r. Putting together these probabilities for all values of rgives a function h(r)called the harmonic-measure distribution function or h-function of Ωwith respect to z0. This h-function encodes information about the shape of the boundary of Ω. In this paper, we compute the h-functions for some multiply connected planar regions whose boundary consists of collinear unequal slits or dissimilar discs. The key tool we use for computing these h-functions is a special function, called the Schottky-Kleinprime function. Furthermore, we have validated our results by simulating the random motion of Brownian particles in the regions mentioned above. Wed, 01 Jan 2025 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/13045 2025-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/13044 Harmonic-measure distribution functions of multiply connected domains with various geometries Green, C.C.; Arunmaran, M.; Ward, L.A. The harmonic-measure distribution functions, or h-functions, associated with several classes of planar multiply connected domains Ω ⊂C and basepoint z0 ∈Ω locations are the principal objects of consideration in this paper. The h-function with respect to Ω and z0 encodes the probability that a particle undergoing Brownian motion in Ω first collides with the boundary ∂Ω within a certain distance from the basepoint z0 where it was initially released. Recently, Green et al. (Green et al. 2022 Proc. R. Soc. A 478, 20210832. (doi:10.1098/rspa.2021.0832)) derived the first explicit formulae for the h-functions of multiply connected symmetrical rectilinear slit domains. In this paper, we generalize and extend the h-function calculations in Green et al. by considering various types of planar domains—those whose boundaries consist of either rectilinear slits or circles—as well as different locations of the basepoint. Throughout, we make judicious use of the Schottky- Klein prime function and its associated theory to derive analytical formulae for the h-functions. Our examples yield solutions to instances of a variant of the conformal Skorokhod embedding problem. Wed, 01 Jan 2025 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/13044 2025-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/9559 On the convergence of the accelerated Riccati iteration method Prasanthan, R.; Jianhong Xu In this paper, we establish results fully addressing two open problems proposed recently by I. Ivanov, see Nonlinear Analysis 69 (2008) 4012–4024, with respect to the convergence of the accelerated Riccati iteration methodfor solving the continuous coupled algebraic Riccati equation, or CCAREfor short. These results confirm several desirable features of that method, including the monotonicity and boundedness of the sequences it produces, its capability of determining whether the CCARE has a solution, the extremal solutions it computes under certain circumstances, and its faster convergence than the regular Riccati iteration method Wed, 01 Jan 2020 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/9559 2020-01-01T00:00:00Z http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/9543 Analysis of the Convergence of More General Linear Iteration Scheme on the Implementation of Implicit Runge-Kutta Methods to Stiff Differential Equations Vigneswaran, R.; Kajanthan, S. A modified Newton scheme is typically used to solve large sets of non-linear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to this scheme, iteration schemes, which sacrifice superlinear convergence for reduced linear algebra costs, have been proposed. A more general linear iterative scheme of this type proposed by Cooper and Butcher in 1983 for implicit Runge-Kutta methods, and he has applied the successive over relaxation technique to improve the convergence rate. In this paper, we establish the convergence result of this scheme by proving some theoretical results suitable for stiff problems. Also these convergence results are verified by two and three stage Gauss method and Radue IIA method. Wed, 01 Jan 2020 00:00:00 GMT http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/9543 2020-01-01T00:00:00Z