Mathematics
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Item Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion(Mathematics, 2019) Perera, U.; Böckmann, C.In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. View Full-TextItem Role of total curvature on rays of non-compact Riemannian 2-manifold(2019-10-25) Malwatta, P.B.It is interesting to study the geometry of total curvature on complete open surfaces. Cohn-Vossen’s inequality states that in every connected noncompact finitely connected complete Riemannian 2-manifold 𝑀 with finite total curvature 𝑐(𝑀) and finite Euler characteristic 𝜒(𝑀), we have 𝑐(𝑀)≤2𝜋𝜒(𝑀). Huber extended this result, if a connected, infinitely connected complete Riemannian 2-manifold 𝑀 without boundary admits a total curvature 𝑐(𝑀), then 𝑐(𝑀)= −∞. The value 2𝜋𝜒(𝑀)−𝑐(𝑀) plays an important role in the study of rays on complete, noncompact Riemannian 2-manifolds. A ray 𝛾:[0,∞]⟶𝑀, on a complete, non-compact Riemannian manifold 𝑀 is by definition a unit speed geodesic every subarc of which is minimizing. Due to the completeness and non-compactness of the Riemannian 2-manifold 𝑀, there exists at least one ray emanating from every point of a manifold. If 𝐴(𝑝) is the collection of all rays emanating from 𝑝∈𝑀 and 𝜇 is the natural measure induced by the Riemannian metric then lim𝑛→∞𝜇𝜊𝐴(𝑝𝑛)⊂𝐴(𝑝) , where {𝑝𝑛} is a sequence of points of 𝑀 converging to 𝑝. Also we have the function 𝜇𝜊𝐴∶𝑀⟶[0,2𝜋] is upper semi-continous and hence Lebesgue integrable. If 𝑀 is connected, finitely connected, complete and non-compact Riemannian 2-manifold, we then investigated the relationship between 𝑐(𝑀) and the function 𝜇𝜊𝐴, proving that if 𝑀 is homeomorphic to 𝑅2 and if Gaussian curvature 𝐺≥0, then 𝜇𝜊𝐴 ≥2𝜋−𝑐(𝑀), and in particular 𝑖𝑛𝑓𝑀𝜇𝜊𝐴=2𝜋−𝑐(𝑀).Item Analytical solutions of the time-fractional non- linear Schrodinger equation with zero and non-zero trapping potential through the sumudu decomposition method(Journal of Science of the University of Kelaniya Volume:10, 2019) Mathanaranjan, K.; Himalini, K.Sumudu decomposition method is used to construct the approximate analytical solutions of time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential. The Sumudu decomposition method is a combined form of the Sumudu transform and the Adomian decomposition method. The fractional derivatives are defined in the Caputo sense. The exact solutions of some nonlinear Schrodinger equations are given as a special case of our approximate analytical solutions. The computations show that the described method is easy to apply, and it needs smaller size of computation as compared to the other existing methods. Further, the solutions are derived in a convergent series form which shows the effectiveness of the method for solving a wide variety of nonlinear fractional differential equations.Item Tidal variation m the west coastal area of Sri Lanka.(International Conference on Computational Modelling and Simulation-2017 (ICCMS 117), Department of Mathematics, Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Munasinghe, J.; Gunasekera, H.D.S.The present study was carried out in an attempt to observe and analyze the tidal height changes due to the motion of the sun, moon and earth, in the west coastal area of Sri Lanka. Tidal height deviations from the Mean Sea Level (MSL) were measured every 15 minutes throughout the year 2015 using the tide pole installed in the sea, 100m away from Colombo Fort, which was built by the Hydrography Survey Unit of the Sri Lanka Navy. Using the obtained data, the behaviour of tidal waves was identified. The main tidal constituents were obtained using the Fast Fourier Transformation (FFT) and the Interpolation method. The mean value of the High Water Level (MHWL) and the mean value of the Low Water Level (MLWL) of the tides were then calculated for each month of the year. These mean values were used to update the Mean Sea Level (MSL). The main tidal constituents for each month were then used to identify the behaviour of tidal waves.Item Centrality Measures to Identify Traffic Congestion on Road Networks: A Case Study of Sri Lanka.(IOSR Journal of Mathematics (IOSR-JM), Department of Mathematics, Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Jayaweera, I.M.L.N.; Perera, K.K.K.R.; Munasinghe, J.This study presents a graph theoretical approach to identify the traffic congestion on a road network. Problem address on a city called Kiribathgoda situated in the western province of Sri Lanka. In the analysis of social networks, centrality measures played a vital role to identify the central nodes in a given network. We look at the applicability of centrality and betweenness measures in order to identify the most important locations which directly affect to the traffic congestion in road networks in Sri Lanka. Using the graph theoretical approach a traffic network for a selected area was constructed and several centrality measures were calculated. According to our simulation results, it was noted that the practically identified locations could be identified from the simulations carried out using the centrality measures.Item Confirmation of earth’s closed loop orbit using tidal waves.(Sri Lanka Association for the Advancement of Science Proceedings of the 73rd Annual Sessions - 2017 , Department of Mathematics, University of Kelaniya,Sri Lanka., 2017) Munasinghe, J; Kekulawala, K.I.S.The word 'tide' is a prevalent term used to define the alternating rise and fall in the sea level wit respect to the land, produced by the gravitational attraction of the moon and the sun. To a bett understanding of the tide,-it is necessary to study each astronomical motion, together with it associated tide producing forces, separately. The present study was carried out with an attempt t confirm the Earth’s closed loop orbit using tidal height changes due to the motion of the Sun, Moo and Earth in the Trincomalee coastal area of Sri Lanka. Tidal heights from the Mean Sea Level (MS were measured every fifteen minutes throughout the year 2015 (365 days) using the tide pole installs in the sea in the Trincomalee coastal area, which was built by the Hydrography Survey Unit of Sri Land Navy. Using the data obtained, the behavior of tidal waves was identified. The main tidal constituen were obtained using Tidal Analysis Tool (TAT). The tidal constituent ’Principal solar semidiurn constituent’ (52), which is the consideration of the tidal effect caused by the sun, was then chosen fro TAT application because the Earth‘s orbit around the Sun is caused only by the forces between t Earth and the Sun. Fast Fourier Transformation (FFT) and Interpolation methods were used to analyz the chosen tidal constituent, Sz, together with the obtained tidal data to confirm the Earth’s close loop orbit around the Sun. Meteorological factors and human errors can occur while collectin data and hence there are eighteen peaks towards the inside of the loop. The following ellipti shaped orbit was obtained at the confidence level of 50% after removing such data. Figure 1: Earth’s closed loop orbitItem On the validity of three body model calculations(Sri Lanka Association for the Advancement of Science, 2013) Dahanayaka, S.D.; Piyadasa, R.A.D.Item The identity of Fermat equation and simple proof of Fermat’s last theorem for n=5(Sri Lanka Association for the Advancement of Science, 2013) Pallewatta, P.G.M.O.; Piyadasa, R.A.D.Item Proof of Fermat’s last theorem for n=5 and many odd primes(Sri Lanka Association for the Advancement of Science, 2013) Ubeynarayana, C.U.; Dharmasiri, K.G.E.U.; Piyadasa, R.A.D.Item The equality of Schrödinger’s Theory and Heisenberg’s S-matrix Theory(Sri Lanka Association for the Advancement of Science, 2011) Silva, H.I.R.U.The main aim of this work is to show that the energy discrete eigen values given by the Schrödinger’s theory and Heisenberg’s theory are the same. To obtain this result, we have used Parabolic co-ordinates to solve the Schrödinger’s equation for the Hydrogen Atom. By using the Hyper Geometric Confluent functions we have expressed the S-matrix element using Gamma functions; ( ) ( ) (l in) l in S k n l G + − G + + = 1 1 where k e n 2 2 h μ = − By the definition of Gamma function, ( ) ( )Õ ¥ = − + + = 1 2 2 1 1 p p in in l e p z p z e z z S n g Then it is apparent that the S-matrix element contains infinite number of poles and zeros. Considering the relevant simple pole, we have derived an equation for the energy eigen values of the form 2 2 4 2 n e En h μ = − This shows that it is the same as the equation we have obtained in Schrödinger’s theory. Therefore Heisenberg’s S-matrix theory and Schrödinger’s wave mechanics give exactly the same eigen values in the cases we have examined.