Department of Mathematics
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Item Absolute-convective instability of a mixed forced-free convection boundary layer(Fluid Dynamics Research, 2010-08-18) Mureithi, Eunice; Denier, JamesA spatio-temporal inviscid instability of a mixed forced-free convection boundary layer is investigated. The base flow considered is the self-similar flow with free-stream velocity ue xn. Such a boundary-layer flow presents the unusual behaviour of generating a region of velocity overshoot, in which the streamwise velocity within the boundary layer exceeds the free-stream speed.A linear stability analysis has been carried out. Saddle points have been located and a critical value for the buoyancy parameter, G0c 3.6896, has been determined below which the flow is convectively unstable and above which the flow becomes absolutely unstable. Two spatial modes have been obtained, one mode being convective in nature and the other absolute. The convective-type spatial mode shows mode crossing behaviour at lower frequencies. Thermal buoyancy is shown to be destabilizing to the absolutely unstable spatial mode.Item Absolute-convective instability of mixed forced-free convection boundary layers(2010-08) Mureithi, Eunice; Denier, JamesA spatio-temporal inviscid instability of a mixed forced-free convection boundary layer is investigated. The base flow considered is the self-similar flow with free-stream velocity ue ~ xn. Such a boundary-layer flow presents the unusual behaviour of generating a region of velocity overshoot, in which the streamwise velocity within the boundary layer exceeds the free-stream speed. A linear stability analysis has been carried out. Saddle points have been located and a critical value for the buoyancy parameter, G0c ≈ 3.6896, has been determined below which the flow is convectively unstable and above which the flow becomes absolutely unstable. Two spatial modes have been obtained, one mode being convective in nature and the other absolute. The convective-type spatial mode shows mode crossing behaviour at lower frequencies. Thermal buoyancy is shown to be destabilizing to the absolutely unstable spatial mode.Item Adaptive Stochastic Numerical Scheme in Parallel Random Walk Models for Transport Problems in Shallow Water(Elsevier, 2009) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing. We start by considering a basic stochastic Euler scheme for integration of the diffusion and drift terms of the SDEs, with a strong order 1 in the strong sense. The errors due to this scheme depend on the location of the pollutant; it is dominated by the diffusion term near boundaries, and by the deterministic drift further away from the boundaries. Using a pair of integration schemes, one of strong order 1.5 near the boundary and one of strong order 2.0 elsewhere, we can estimate the error and approximate an optimal step size for a given error tolerance. The resulting algorithm is developed such that it allows for complete flexibility of the step size, while guaranteeing the correct Brownian behaviour. Modelling pollutants by non-interacting particles enables the use of parallel processing in the simulation. We take advantage of this by implementing the algorithm using the MPI library. The inherent asynchronic nature of the particle simulation, in addition to the parallel processing, makes it difficult to get a coherent picture of the results at any given points. However, by inserting internal synchronisation points in the temporal discretisation, the code allows pollution snapshots and particle counts to be made at times specified by the user.Item Item (alpha, beta) -Lp 2 -norm orthogonality and characterizations of 2 - inner product spaces,(Eudoxus Press, LLC of TN, 2008) Vinai, K. Singh; Kumar, Santosh; Singh, A. K.In the present paper we have characterised (alpha, beta, )−Lp orthogonality in a 2- normed linear space. In some way the results proved in this paper generalize some of the similar characterization of generalized Lp- orthogonality derived earlier by Zheng Liu[8].Item Analysis and Dynamically Consistent Numerical Schemes for the SIS Model and Related Reaction Diffusion Equation(2011-10) Lubuma, Jean M.S.; Mureithi, Eunice; Terefe, Yibeltal A.The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease-free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.Item Analysis and dynamically consistent numerical schemes for the SIS model and related reaction diffusion equation(Proceedings of the 3rd International Conference on Applied Mathematics in Technical and Natural Sciences,(AMiTaNS’11), (Albena, Bulgaria), Institute of Physics-AIP Conf. Proc., 2011) Lubuma, Jean; Mureithi, Eunice; Terefe, YibeltalThe classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease-free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.Item Application of Coloured Noise as a Driving Force in the Stochastic Differential Equations(2010) Charles, Wilson M.In this chapter we explore the application of coloured noise as a driving force to a set of stochastic differential equations(SDEs). These stochastic differential equations are sometimes called Random flight models as in A. W. Heemink (1990). They are used for prediction of the dispersion of pollutants in atmosphere or in shallow waters e.g Lake, Rivers etc. Usually the advection and diffusion of pollutants in shallow waters use the well known partial differential equations called Advection diffusion equations(ADEs)R.W.Barber et al. (2005). These are consistent with the stochastic differential equations which are driven by Wiener processes as in P.E. Kloeden et al. (2003). The stochastic differential equations which are driven by Wiener processes are called particle models. When the Kolmogorov’s forward partial differential equations(Fokker-Planck equation) is interpreted as an advection diffusion equation, the associated set of stochastic differential equations called particle model are derived and are exactly consistent with the advection-diffusion equation as in A. W. Heemink (1990); W. M. Charles et al. (2009). Still, neither the advection-diffusion equation nor the related traditional particle model accurately takes into account the short term spreading behaviour of particles. This is due to the fact that the driving forces are Wiener processes and these have independent increments as in A. W. Heemink (1990); H.B. Fischer et al. (1979). To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this chapter. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection-diffusion equation.Item Approximating fixed point of generalized nonexpansive mappings(ASCENT PUBLICATION, 2012) Kumar, SantoshUnder certain conditions the convergence of Ishikawa iterate to a unique xed point is proved for nonexpansive type mappings in a uniformly convex Banach space. In this paper we improve the results given by Pathak and Khan [6].Item Are There Observable Precursors to HPC Platform Failures?(2010) Brandt, James M.; James, MakunguItem Can Climate-Driven Change Influence Silicon Assimilation by Cereals and Hence the Distribution of Lepidopteran Stem Borers in East Africa?(Elsevier, 2016) Calatayud, P. A.; Njuguna, E.; Mwalusepo, Sizah; Gathara, Mary; Okuku, G.; Kibe, A.; Musyoka, B.; Williamson, David; Ong’amo, Georges; Gerald, Juma; Johansson, Tino; Subramanian, Sevgan; Gatebe, E.; Le Ru, Bruno P.In East Africa, lepidopteran stemborers such as Chilo partellus and Busseola fusca are major constraints to production of maize, which is the main staple food crop in the region. Cereals depend on silicon (Si)-based defences to fight off herbivores. Using altitudinal ranges in the East African highlands as ecological surrogates for inferring climate change, it was shown that Si concentrations in soil and maize decreased with altitude. This was attributed, in part, to low temperatures at high altitudes, which negatively affected Si assimilation by maize. Experiments showed that B. fusca was more susceptible to Si than C. partellus. Hence the predominance of B. fusca in the highlands and of C. partellus in the lowlands could be partly explained by altitudinal differences in Si concentrations in maize plants. Therefore, a rise in temperature due to climate change should enhance the plants’ Si assimilation and as a result C. partellus might move into the higher altitudes and increasingly displace B. fusca.Item Characteristics of Soils in Selected Maize Growing Sites along Altitudinal Gradients in East African Highlands(2015) Ong’amo, George; Njuguna, Elijah; Gathara, Mary; Nadir, Stanley; Mwalusepo, Sizah; Juma, Gerald; Kimani, Jackson; Landmann, Tobias; Williamson, DavidMaize is the main staple crop in the East African Mountains. Understanding how the edaphic characteristics change along altitudinal gradients is important for maximizing maize production in East African Highlands, which are the key maize production areas in the region. This study evaluated and compared the levels of some macro and micro-elements (Al, Ca, Fe, K, Mg, Mn, Na and P) and other soil parameters (pH, organic carbon content, soil texture [i.e. % Sand, % Clay and % Silt], cation exchange capacity [CEC], electric conductivity [EC], and water holding capacity [HC]). Soil samples were taken from maize plots along three altitudinal gradients in East African highlands (namely Machakos Hills, Taita Hills and Mount Kilimanjaro) characterized by graded changes in climatic conditions. For all transects, pH, Ca, K and Mg decreased with the increase in altitude. In contrast, % Silt, organic carbon content, Al and water holding capacity (HC) increased with increasing altitude. The research provides information on the status of the physical-chemical characteristics of soils along three altitudinal ranges of East African Highlands and includes data available for further research.Item Coloured Noise for Dispersion of Contaminants in Shallow Waters(Elsevier, 2009) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.Item COMMON FIXED POINT FOR A PAIR OF NON-SELF MAPPINGS IN PARTIAL METRIC SPACES(Busan International Nonlinear Analysis Academy, 2017-06) Rugumisa, Terentius; Kumar, SantoshIn this paper we prove a common fixed point theorem for a pair of non-self mappings in partial metric spaces. We generalize a theorem by Imdad and Kumar which was proved for metric spaces. We also provide an illustrative example.Item COMMON FIXED POINTS FOR FOUR NON-SELF-MAPPINGS(Vijñāna Parishad of India: Jnanabha, 2017-12) Kumar, Santosh; Rugumisa, TerentiusIn this paper, we formulate a quasi-contraction type non-self mapping on Takahashi convex metric spaces and common fixed point theorems that applies to two pairs of mappings. The result generalizes the fixed point theorems of some previous authorsItem COMMON FIXED POINTS FOR FOUR NON-SELF-MAPPINGS IN PARTIAL METRIC SPACES(Mathematica Bohemica, 2018) Terentius, Rugumisa; Kumar, Santosh; Mohammad, ImdadIn this paper, we formulate a common fi xed point theorem for four non-self mappings in convex partial metric spaces. The result extends a fi xed point theorem by Gajic and Rakocevic [Pair of non-self mappings and common fi xed points. Appl. Math. Comp. 187 (2007), 999-1006] proved for two non-self mappings in metric spaces with a Takahashi convex structure. We also provide an illustrative example on the use of the theorem.Item Common Fixed Points for Weakly Compatible Mappings in Partial Metric Spaces(Busan International Nonlinear Analysis Academy, 2018-03-01) Johnson, Kessy; Santosh, Kumar; Grayson, KakikoIn many situations the arising mappings need not be self-mappings. It is well known that metrically convex metric spaces are desirable for the study of fixed points of non-self mappings. Partial metric spaces are generalization of the notion of a metric space that allows non-zero self distance. In this paper, we present a common fixed point theorem for weakly compatible mappings on complete metrically convex partial metric spaces. Our main result generalizes a common metric fixed point theorem due to ´Ciri´c and Ume, which is an extension of a metric fixed point theorem due to Rhoades, to partial metric spaces.Item COMMON FIXED POINTS IN METRICALLY CONVEX PARTIAL METRIC SPACES(Dergi Konuralp Journal of Mathematics, 2017-12) Kumar, Santosh; Rugumisa, Terentius; Imdad, Mohd.In this study, we extend some common xed points theorems for mappings in metrically convex metric spaces into partial metric spaces. We generalize earlier results by Imdad et al. We also provide an illustrative example.Item Common fixed points of a pair of multivalued non-self mappings in partial metric spaces(Malaya Journal of Matematik, 2018, 2018-07-01) Kumar, Santosh; Rugumisa, TerentiusIn this paper, we utilize the concept of the partial Hausdorff metric, first introduced by Aydi et al.[4] for partial metric space, to consider a pair of multivalued mappings which are non-self almost contractions on metrically convex partial metric spaces. We establish the existence of fixed point in such mappings.Item COMMON FIXED POINTS OF HYBRID PAIRS OF NON-SELF MAPPINGS SATISFYING AN IMPLICIT RELATION IN PARTIAL METRIC SPACES(Aligarh Muslim University, Aligarh, India, 2017) Kumar, Santosh; Kessy, JohnsonIn this paper, we define a hybrid-type tangential property in the sense of Ahmed [8] in the setting of partial metric spaces. We obtain some results for coincidence and common fixed points of two hybrid pairs of non-self mappings satisfying an implicit relation due to Popa [19] under the tangential property in a partial metric space. Our results unify and generalize some existing ones in the literature.