Department of Mathematics
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Item Quasi-Monte Carlo Simulation of Diffusion in a Spatially Nonhomogeneous Medium(Springer, 2009-11) Venkiteswaran, G.We propose and test a quasi-Monte Carlo (QMC) method for solving the diffusion equation in the spatially nonhomogeneous case. For a constant diffusion coefficient, the Monte Carlo (MC) method is a valuable tool for simulating the equation: the solution is approximated by using particles and in every time step the displacement of each particle is drawn from a Gaussian distribution with constant variance. But for a spatially dependent diffusion coefficient, the straightforward extension using a spatially variable variance leads to biased results. A correction to the Gaussian steplength was recently proposed and provides satisfactory results. In the present work, we devise a QMC variant of this corrected MC scheme. We present the results of some numerical experiments showing that our QMC algorithm converges better than the corresponding MC method for the same number of particles.Item Deterministic Particle Methods for High Dimensional Fokker-Planck Equations(Springer, 2006) Venkiteswaran, G.We consider a mathematical model for polymeric liquids which requires the solution of high-dimensional Fokker-Planck equations related to stochastic differential equations. While Monte-Carlo (MC) methods are classically used to construct approximate solutions in this context, we consider an approach based on Quasi- Monte-Carlo (QMC) approximations. Although QMC has proved to be superior to MC in certain integration problems, the advantages are not as pronounced when dealing with stochastic differential equations. In this article, we illustrate the basic difficulty which is related to the construction of QMC product measures.Item A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids(Elsevier, 2005) Venkiteswaran, G.A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker–Planck equation on . In this article, we present QMC methods for the approximate solution of the Fokker–Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23–41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(IDEAS is a RePEc, 2005) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is N−1/2 accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve N−1 convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.Item Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics(Springer, 2009-10) Rana, AnirudhFour basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer’s problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier–Stokes–Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.Item Disintegration process of surface stabilized sol-gel TiO2 nanoparticles by population balances(Elsevier, 2009-12) Kumar, RajeshTitanium oxide () is one of the most useful oxide materials, because of its widespread applications in photocatalysis, solar energy conversion, sensors and optoelectronics. The control of particle size and monodispersity of nanoparticles is a challenging task in processing. The control and prediction of these dynamics are based on the process conditions and the nature of chemicals. In this work, we investigate the effect on the surface stabilization with different surfactants and temperature. The steric stabilization of the polymer and various functional groups of dispersants are also considered. Narrow distributed spherical titania particles in the size range 10–100 nm are produced in a sol–gel synthesis from titanium tetra-isopropoxide. The influence of various precursor concentrations and different surfactants on the particle size distribution is investigated. The population balance model for disintegration leads to a system of integro-partial differential equations which is numerically solved by the cell average technique. The experimental results are also compared with the simulation using two different disintegration kernels.Item On the solution of a partial differential equation representing irrotational flow in bispherical polar coordinates(Elsevier, 2008-11) Tiwari, AshishThe present paper concerns the exact solution of the partial differential equation E2ψ = 0 in bispherical polar coordinates, arising in the study of irrotational fluid motion. It is evident from the solution that the stream function ψ appearing in the equation depends upon Legendre, trigonometric, hyperbolic functions, similar to the solution reported by Khuri and Wazwaz [S.A. Khuri, A.M. Wazwaz, The solution of a partial differential equation arising in fluid flow theory, Appl. Math. Comput. 77 (1996) 295–300] in toroidal coordinates. In addition to this, we have presented a simplified form of solution of E2ψ = 0 in toroidal coordinates, which was previously reported by Khuri and Wazwaz [1].Item A Theorem Relating Multidimensional Generalized Weyl Fractional Integral, Laplace and Varma Transforms with Applications(Tamsui Oxford Journal of Mathematical Sciences, 2002-07) Mathur, TrilokThe main aim of this paper is to establish a theorem which asserts an interesting relationship between the multidimensional Laplace transform, the multidimensional Varma transform and the generalized Weyl fractional integral involving product of a general class of multivariable polynomials and a generalized polynomial set. By specializing the various parameters involved, this general theorem would readily yield several (known and new) results involving simpler integral operators. Further, the theorem is applied to evaluate the generalized Weyl fractional integrals of Fox’s H-function and the (Srivastava – Panda) H-function of several complex variablesItem Bicomplex Gamma and Beta Function(JOURNAL OF RAJASTHAN ACADEMY OF PHYSICAL SCIENCES, 2006) Mathur, TrilokThe aim of this paper is to define bicomplex Gamma and Beta functions. We also discuss conditions of validity and T-holomorphicity for these functions. Various properties including Legendre duplication formula, Gauss multiplication theorem and Binomial theorem are established. These functions which are believed to be new will provide a fundamental tool to develop the theory of bicomplex special functionsItem Total Factor Productivity Growth, Technical Progress and Efficiency Change in Sugar Industry of Uttar Pradesh(Sage, 2006-07) Agarwal, ShiviThis paper examines the total factor productivity (TFP) growth and its components (technical efficiency change and technology change) in the sugar industry of Uttar Pradesh. The TFP growth is estimated applying SBM-DEA-based Malmquist Productivity Index (MPI) on the panel data of 36 sugar mills for the period 1996-97 to 2002-03. The study finds that the average TFP in the industry grew at a moderate rate of 1.6 per cent per annum during the entire period. The decomposition of TFP growth into technical efficiency change (catch-up effect) and technical change (frontier shift) reveals that the TFP growth is primarily contributed by technical change rather than by technical efficiency change. Sector-wise estimation of the TFP testifies that the private sector has the highest growth, followed by the cooperative sector. Regional pattern of the growth demonstrates that the sugar mills in the western region achieved relatively better TFP growth than their counterparts in other two regions. The study also finds that the mills with bigger plant size attained higher productivity growth than the smaller ones. Further, relatively higher TFP growth achieved during the later part of the study period provides some indication that the policy-induced factors, such as de-licensing and partial decontrol of sugar sector have made some positive impact on the TFP growth