Geometric Numerical Integrators for Hamilton systems

Abstaract: The dynamics of mechanical systems, such as the motion of a pendulum or the motion of the planets in our solar system, can be modeled by the means of ordinary differential equations. In particular these dynamics are described by the Hamiltonian systems. Since these systems are usually too complex to possess closed form solutions, one must resort to numerical methods to approximate these problems. Geometric numerical integration is relative new field in numerical analysis and popular area of research which studies numerical methods preserving important qualitative properties of the differential equations under discretization (the most well known example being sym- plectic integrators for Hamiltonian systems). However, besides preserving qualitative properties, for certain types of problems, one is also interested in producing very accurate numerical solutions with low computational costs. Brief discussion on some new algorithms for numerical solutions of Hamilton 's equations which have been presented in [1] will be the first part of talk. Main emphasis in the second part of talk will be on the benchmark Python library, namely, Higher Order Methods in Python (HOMsPy) [2, 3] which generate described numerical solver of the underlying Hamiltonian system.

Qualitative criteria for solutions of various kind of integro-differential and impulsive differential equations

Abstaract:In this work, a class of non -linear Volterra integro - differential equations, Volterra integro Caputo fractional differential euations with delay and linear periodic impulsive systems with time delay are considered New criteria are presented on the various qualitative properties of solutions of these equations. The Lyapunov – Krasovskii method, the Razumikhin method and some others methods are used as basic tools. Examples are given to verify the obtained results.

A journey through structure-preserving discretization

Abstaract:

Abstaract:In this talk we present recent advances in the numerical approximation of various evolutionary problems by means of methods preserving the qualitative behaviour of the operator along the discretized dynamics. This approach is applied to both deterministic and stochastic problems, with a rigorous setting matched with a proper experimental one that confirms the effectiveness of the introduced methodologies. As regards deterministic structure-preservation, we mostly deal with Hamiltonian problems, treated by multivalue numerical methods whose long-term properties are highlighted. Concerning stochastic problems, a possible structure-preserving framework is introduced for stochastic Hamiltonian problems (with the aim of retaining the known long-term properties on the expected Hamiltonian) as well as and stochastic oscillators (in order to reproduce the long-term properties of the position and the velocity of the oscillating particle).

Second derivative general linear methods for the numerical solution of ordinary differential equations

Abstaract:Traditional numerical methods for the numerical solution of an autonomous system of ordinary differential equation y′ = f(y(x)); y : R ! Rm; f : Rm ! Rm; generally fall into two main classes: linear multistep (multivalue) and Runge–Kutta (multistage) methods. In 1966, Butcher introduced general linear methods as a unifying framework for the traditional methods to study the properties of consistency, stability and convergence, and to formulate new methods with clear advantages over these classes. In fact, this class of the methods includes all the first derivative multivalue and multistage methods. On the other hand, to construct methods with higher order and extensive stability region, some efficient second derivative methods within the class of linear multistep methods and Runge–Kutta methods have been introduced. In 2005, Butcher and Hojjati extended GLMs to the case in which second derivatives, as well as first derivatives, can be calculated. These methods, called SGLMs, were studied more by Abdi and Hojjati. Starting with a discussion on the intrinsic structure of SGLMs, in this talk we will become familiar with the properties of SGLMs and their efficiency in solving of the stiff and nonstiff initial value problems. This is followed by the introducing of some subclasses of SGLMs which preserve qualitative geometrical properties of the flow of the system. Also, some strategies based on SGLMs are given for the numerical solution of conservative problems. Keywords: Initial value problem, General linear methods, Second derivative methods, Conservative problems, Hamiltonian problems.

Solving Boundary Value Problem For Delay Differential Equation using Multistep Block Method

Abstaract:In this study, we propose a multistep block method for the solution of boundary value problem for second order delay differential equations directly. The proposed block method will approximate the solutions at two points simultaneously and will solve the delay differential equations directly without reducing to the system of first order. The shooting technique by using Newton's method will be implemented to compute the guessing values. Some numerical examples are presented to show that the proposed method is capable for solving boundary value problems for delay differential equations.
Keywords: Delay differential equation, boundary value problem, direct block method, shooting method.

Mathematical Model of Boundary Layer Flow in a Porous Medium Filled by a Nanofluid: Stability Analysis

Abstaract:Several types of mathematical model in a porous medium filled by a nanofluid will be discussed. In the presence of a thermal radiation, the flow is generated due to stretching or shrinking sheet. The problem is formulated for three types of nanoparticles, namely, copper (Cu), alumina (Al2O3) and titania (TiO2). The boundary layer conservation equations are transformed using appropriate similarity variables and the resulting nonlinear boundary value problem is numerically solved using shooting method. It is revealed that the corresponding results possessed two branches of solutions, where we then implemented an analysis of stability on those two non-unique solutions to evaluate the most realizable solution and the features of the respective solutions have been discussed in details.

On fractional derivatives and generalized functions

Abstaract:The fractional calculus is considered as an extension of ordinary derivatives and integrals to arbitrary order possible complex number. The historical development of subject is sufficiently old enough and can go back to the times of Leibnitz and Newton. After the introduction of fractional derivatives idea, almost a three hundred years the fractional calculus was not much popular in science and engineering. However recently become very famous and has been applied broad range of problems in several areas such as engineering, science, finance, as well as bio engineering etc. In fact it was observed that the modelling with fractional order is more natural than the classical calculus. In the literature there are many different types of related definitions due to the properties. In the present study we extend fractional differential calculus to generalized functions also known as distributions by using the infinitely differentiable functions having compact support as test functions. We also define several new distributions by using the fractional derivatives. And provide some examples in applied sciences such as in finance. 2010 AMS Subject Classifications: Primary 46F10; Secondary; 33B30
Keywords: Test functions, Special Functions, Infinitely Differentiable functions, Dirac delta sequences, Distributions, Convolution, Regular distributions; Fractional derivative.

A method for solving nonlinear fractional partial differential equations

Abstaract:The purpose of the presentation is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain, and to get better and more accurate results The operational matrices for the method will be derived, constructed and utilized for the solution of nonlinear fractional partial differential equations. The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. Many Engineers can utilize the presented method for solving their nonlinear fractional models.

Theoretical Study of Thermal and Radial Effects in Liquid Chromatography

Abstaract:Chromatographic models contain convection-dominated systems of partial differential equations coupled with some differential and algebraic equations describing thermodynamic and kinetic phenomena. Thermal effects are discussed widely in the case of gas chromatography. In liquid chromatography, such effects are typically neglected. The main goal of this work is to quantify how temperature gradients can influence conversion and separation in liquid chromatography. The coupling of concentration and thermal fronts are illustrated and key parameters influencing the column performance are identified. In contrast to previous studies, this work also includes the possibility that radial concentration profiles can develop. For that purpose, two-dimensional models of liquid chromatography are derived in cylindrical geometry. The considered radial gradients are typically ignored, which can be problematic, e.g. in the cases of non-perfect injections and larger column dimensions. To derive analytical solutions the assumption of linear adsorption isotherm is used. The Laplace and Hankel transformations are simultaneously applied to derive analytical solutions of linear models. To further analyze the effects of different kinetic parameters on the elution profiles, statistical temporal moments are derived from the Hankel-Laplace domains solutions. In the case of nonlinear isotherms, a high resolution flux limiting finite volume scheme is applied to solve the model equations. Several case studies are carried out to analyze the effects of different parameters on the elution profiles. The developed analytical and numerical solutions could be useful for improving the process and for the estimation of model parameters from results of laboratory-scale experiments.

Numerical Computation of Shock Waves by using simplified ghost point treatment

Abstaract:In this work we discuss simplified ghost point treatment to calculate shock waves over immersed object. The scheme was introduced by the author of this paper to easily compute shock waves. We solve compressible flow over immersed bodies while discussing benchmark problems. We find supersonic and subsonic flow over a wedge, circular arc airfoil. We find good agreement from literature.

The Peristaltic Flow of Newtonian fluid in a Slightly Curved Circular Tube

Abstaract:This paper investigates the peristaltic flow of a Newtonian fluid in a slightly curved circular tube employing a model for the flow in the toroidal coordinate system. Using the assumptions of lubrication theory, an approximate asymptotic solution is obtained by the method of perturbation expansion for the axial velocity and pressure gradient. The influence of curvature and polar angle on various flow quantities is analyzed. It is shown that peristalsis in a straight tube is just the limiting case of this study.

Numerical Study of Reaction Mechanism of Methane

Abstaract:The main purpose of the work we done is to study the reaction mechanism of methane and its properties and get an understanding of combustion processes to come to the point where we can reduced the full mechanism by constraining it to our desired situation. We take one dimensional premixed laminar[1](when a fluid flows in parallel layers, with no disruption between the layers) flames. Many practical combustors, such as internal combustion engines, rely on premixedflame propagation. Laminar flamespeed is often used to characterize the combustion of various fuel-oxidizercombinations and in determining mixture flammability limits. Therefore, the ability tomodel chemical kinetics and transport processes in these flames is critical toflammability studies, interpreting flame experiments, and to understanding thecombustion process itself. Examples of the use of flame modeling to interpret experimental observations and to verify combustion chemistry and pollution formationcan be found in Miller, et al.

Second-order slip effects on oscillating fractionalized Maxwell fluid in porous medium

Abstaract:The objective of this article is to investigate the effects of second order slip on the MHD flow of fractionalized Maxwell fluid through a porous medium due to oscillatory motion of an infinite plate. The governing equations are developed by fractional calculus approach using Caputo-Fabrizio fractional derivative definition. The exact analytical solutions for velocity field and associated shear stress are calculated using Laplace transforms and presented in series form in terms of generalized M-function satisfying all imposed initial and boundary conditions. The flow of fractionalized Maxwell fluid with and without slips, in the presence and absence of magnetic effect, the solutions for ordinary Maxwell and Newtonian fluid performing the similar motion are derived as the limiting cases. The impact of fractional parameter, magnetic, slip coefficients, and porosity parameter over the velocity field and shear stress are discussed and analyzed through graphical illustrations.
Keywords: Second-order slip, Magnetohydrodynamics (MHD), Maxwell fluid, fractional calculus, integral transforms, exact solutions, graphical analysis

Gravitational Decoupled Anisotropic Solutions

Abstaract:The purpose of this paper is to obtain exact solutions for charged anisotropic spherically symmetric matter configuration. For this purpose, we consider a known solution for the isotropic spherical system in the presence of the electromagnetic field and extend it to two types of anisotropic charged solutions through gravitational decoupling approach. We examine the physical characteristics of the resulting models. It is found that only first solution is physically acceptable as it meets all the energy bounds as well as the stability criterion. We conclude that the stability of the first model is enhanced with the increase of charge.

Numerical comparisons of finite element stabilized methods for high Reynolds numbers vortex dynamics simulations

Abstaract:In this talk, I will present up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared in high Reynolds numbers vortex dynamics simulations. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical. Streamline-Upwind Petrov{Galerkin (SUPG) method together with grad-div stabilization, a standard onelevel Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulations of high Reynolds numbers flows with vortical structures on relatively coarse grids are showcased, by focusing on two-dimensional plane mixing-layer flows. Both Inf-Sup Stable (ISS) and Equal Order (EO) FE pairs are explored, using a secondorder semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies, it is concluded that the SUPG method using both ISS and EO FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend SUPG method by the higher order terms of the RB-VMS method.
Key words: Variational multiscale methods; finite element stabilized methods; high Reynolds numbers. incompressible flows; vortex dynamics problems

Trigonometric Hamming similarity operators

Abstaract:"This talk is aimed to a present several new similarity measures based on trigonometric Hamming similarity operators of rough neutrosophic sets and their applications in decision making. Some properties of the proposed similarity measures are established. Also, a numerical example will be given to illustrate the applicability of the proposed similarity measures in decision making.