scholarly journals Field lines of gravity, their curvature and torsion, the Lagrange and the Hamilton equations of the plumbline

1997 ◽  
Vol 40 (5) ◽  
Author(s):  
E. W. Grafarend
Keyword(s):  
Differential Equations ◽  
Three Dimensional ◽  
Physical Geodesy ◽  
Equipotential Surface ◽  
Second Order ◽  
Gravity Potential ◽  
Problem Solver ◽  
Generalized Force ◽  
Field Lines ◽  
Curvature And Torsion

The length of the gravitational field lines/of the orthogonal trajectories of a family of gravity equipotential surfaces/of the plumbline between a terrestrial topographic point and a point on a reference equipotential surface like the geoid í also known as the orthometric height í plays a central role in Satellite Geodesy as well as in Physical Geodesy. As soon as we determine the geometry of the Earth pointwise by means of a satellite GPS (Global Positioning System: «global problem solver») we are left with the problem of converting ellipsoidal heights (geometric heights) into orthometric heights (physical heights). For the computation of the plumbline we derive its three differential equations of first order as well as the three geodesic equations of second order. The three differential equations of second order take the form of a Newton differential equation when we introduce the parameter time via the Marussi gauge on a conformally flat three-dimensional Riemann manifold and the generalized force field, the gradient of the superpotential, namely the modulus of gravity squared and taken half. In particular, we compute curvature and torsion of the plumbline and prove their functional relationship to the second and third derivatives of the gravity potential. For a spherically symmetric gravity field, curvature and torsion of the plumbline are zero, the plumbline is straight. Finally we derive the three Lagrangean as well as the six Hamiltonian differential equations of the plumbline, in particular in their star form with respect to Marussi gauge.

THREE-DIMENSIONAL STAGNATION-POINT FLOW OVER A PERMEABLE STRETCHING/SHRINKING SHEET WITH A SECOND ORDER SLIP FLOW

2021 ◽  
Vol 10 (9) ◽  
pp. 3273-3282
Author(s):  
M.E.H. Hafidzuddin ◽  
R. Nazar ◽  
N.M. Arifin ◽  
I. Pop
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Flow Model ◽  
Nonlinear Partial Differential Equations ◽  
Slip Flow ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Second Order ◽  
Stagnation Point Flow ◽  
Shrinking Sheet

The problem of steady laminar three-dimensional stagnation-point flow on a permeable stretching/shrinking sheet with second order slip flow model is studied numerically. Similarity transformation has been used to reduce the governing system of nonlinear partial differential equations into the system of ordinary (similarity) differential equations. The transformed equations are then solved numerically using the \texttt{bvp4c} function in MATLAB. Multiple solutions are found for a certain range of the governing parameters. The effects of the governing parameters on the skin friction coefficients and the velocity profiles are presented and discussed. It is found that the second order slip flow model is necessary to predict the flow characteristics accurately.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

Pushing the limit for the grid-based treatment of Schrödinger's equation: a sparse Numerov approach for one, two and three dimensional quantum problems

2016 ◽  
Vol 18 (46) ◽  
pp. 31521-31533 ◽  
Author(s):  
Ulrich Kuenzer ◽  
Jan-Andrè Sorarù ◽  
Thomas S. Hofer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Second Order ◽  
Special Focus ◽  
Schrödinger’S Equation ◽  
Numerov Method ◽  
Schrödinger's Equation ◽  
Grid Based

The general Numerov method employed to numerically solve ordinary differential equations of second order was adapted with a special focus on matrix sparsity and applications in higher dimensions.

Symmetry and conservation laws for tuberculosis model

2017 ◽  
Vol 10 (03) ◽  
pp. 1750042 ◽  
Author(s):  
Maba Boniface Matadi
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Three Dimensional ◽  
Symmetry And Conservation Laws ◽  
Second Order ◽  
Dimensional System ◽  
First Order ◽  
Tuberculosis Model ◽  
Nonlinear Lagrangian ◽  
Three Dimensional System

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

Three dimensional MHD stagnation point flow of Al-Cu alloy suspended water based nanofluid with second order slip and convective heating

2017 ◽  
Vol 27 (12) ◽  
pp. 2879-2901
Author(s):  
N. Nithyadevi ◽  
P. Gayathri ◽  
A. Chamkha
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Stagnation Point ◽  
Three Dimensional ◽  
Second Order ◽  
Stagnation Point Flow ◽  
Convective Heating ◽  
Slip Effect ◽  
Content Type ◽  
Water Based

Purpose The paper aims to examine the boundary layers of a three-dimensional stagnation point flow of Al-Cu nanoparticle-suspended water-based nanofluid in an electrically conducting medium. The effect of magnetic field on second-order slip effect and convective heating is also taken into account. Design/methodology/approach The thermophysical properties of alloy nanoparticles such as density, specific heat capacity and thermal conductivity are computed using appropriate formula. The non-linear parabolic partial differential equations are transformed to ordinary differential equations and solved by shooting technique. Findings The influence of compositional variation of alloy nanoparticle, nanoparticle concentration, magnetic effect, slip parameters and Biot number are presented for various flow characteristics. Interesting results on skin friction and Nusselt number are obtained for different composition of aluminium and copper. Originality/value A novel result of the analysis reveals that impact of magnetic field near the boundary is suppressed by the slip effect.

scholarly journals Algorithmic differentiation improves the computational efficiency of OpenSim-based optimal control simulations of movement

2019 ◽  
Author(s):  
Antoine Falisse ◽  
Gil Serrancolí ◽  
Christopher L. Dembia ◽  
Joris Gillis ◽  
Friedl De Groote
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Three Dimensional ◽  
Second Order ◽  
Computational Time ◽  
Code Transformation ◽  
Algorithmic Differentiation ◽  
Implicit Differential Equations ◽  
Direct Collocation ◽  
Source Code Transformation

AbstractAlgorithmic differentiation (AD) is an alternative to finite differences (FD) for evaluating function derivatives. The primarily aim of this study was to demonstrate the computational benefits of using AD instead of FD in OpenSim-based optimal control simulations. The secondary aim was to evaluate computational choices including different AD tools, different linear solvers, and the use of first- or second-order derivatives. First, we enabled the use of AD in OpenSim through a custom source code transformation tool and through the operator overloading tool ADOL-C. Second, we developed an interface between OpenSim and CasADi to perform optimal control simulations. Third, we evaluated computational choices through simulations of perturbed balance, two-dimensional predictive simulations of walking, and three-dimensional tracking simulations of walking. We performed all simulations using direct collocation and implicit differential equations. Using AD through our custom tool was between 1.8 ± 0.1 and 17.8 ± 4.9 times faster than using FD, and between 3.6 ± 0.3 and 12.3 ± 1.3 times faster than using AD through ADOL-C. The linear solver efficiency was problem-dependent and no solver was consistently more efficient. Using second-order derivatives was more efficient for balance simulations but less efficient for walking simulations. The walking simulations were physiologically realistic. These results highlight how the use of AD drastically decreases computational time of optimal control simulations as compared to more common FD. Overall, combining AD with direct collocation and implicit differential equations decreases the computational burden of optimal control simulations, which will facilitate their use for biomechanical applications.

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