scholarly journals Spherical harmonic d-tensors

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941002 ◽  
Author(s):  
Manuel Hohmann
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Spherical Harmonic ◽  
Spherical Symmetry ◽  
Finsler Geometry ◽  
Rotation Group ◽  
Mathematical Tool

Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group [Formula: see text]. The aim of this work is to make use of this tool also in the setting of Finsler geometry, or more general geometries on the tangent bundle, where the objects of relevance are d-tensors on the tangent bundle, or tensors in a pullback bundle, instead of ordinary tensors. For this purpose, we construct a set of d-tensor harmonics for spherical symmetry and show how these can be used for calculations in Finsler geometry.

Spherically symmetric differential equations, the rotation group, and tensor spherical functions

1969 ◽  
Vol 65 (1) ◽  
pp. 157-175 ◽  
Author(s):  
R. Burridge
Keyword(s):  
Quantum Theory ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Spherical Symmetry ◽  
Symmetric Tensor ◽  
Scalar Fields ◽  
Rotation Group ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Covariant Differentiation

AbstractIn this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group. These fields may be expanded as series of functions, where,mis fixed and the matricesTl(g) form a 21+ 1 dimensional irreducible representation of.Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with givenl, nare transformed into others with the same values ofl, n. That this must be so follows from Schur's Lemma and the fact that for eachmandlthe functionsform a basis for an invariant subspace of functions onof dimension 2l+ 1 in which an irreducible representation ofacts. Explicit formulae for the results of such operations are presented.The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.

scholarly journals Application of He’s Variational Iteration Method to Nonlinear Integro-Differential Equations

2010 ◽  
Vol 65 (5) ◽  
pp. 418-430 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Variational Iteration ◽  
He’S Variational Iteration Method ◽  
Powerful Mathematical Tool

In this paper, an application of He’s variational iteration method is applied to solve nonlinear integro-differential equations. Some examples are given to illustrate the effectiveness of the method. The results show that the method provides a straightforward and powerful mathematical tool for solving various nonlinear integro-differential equations

scholarly journals Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
F. M. Alharbi ◽  
A. M. Zidan ◽  
Muhammad Naeem ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Suggested Technique ◽  
Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

Towards sample path estimates for fast–slow stochastic partial differential equations

2018 ◽  
Vol 30 (5) ◽  
pp. 1004-1024 ◽  
Author(s):  
MANUEL V. GNANN ◽  
CHRISTIAN KUEHN ◽  
ANNE PEIN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Sample Path ◽  
Slow Manifold ◽  
Mathematical Tool ◽  
Partial Differential ◽  
Sample Paths ◽  
Time Period ◽  
Short Time

Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.

scholarly journals Framework for generalized polytropes with complexity factor

2019 ◽  
Vol 79 (12) ◽  
Author(s):  
Shiraz Khan ◽  
S. A. Mardan ◽  
M. A. Rehman
Keyword(s):  
Equation Of State ◽  
Energy Density ◽  
Differential Equations ◽  
Spherical Symmetry ◽  
Mass Density ◽  
Fluid Distribution ◽  
System Of Differential Equations ◽  
Systems Of Differential Equations ◽  
Polytropic Equation

AbstractA framework is developed for generalized polytropes with the help of complexity factor introduced by Herrera (Phy Rev D 97:044010, 2018), by using the spherical symmetry with anisotropic inner fluid distribution. For this purpose generalized polytropic equation of state will be used, having two cases (i) for mass density $$(\mu _{o})$$(μo), (ii) for energy density $$(\mu )$$(μ), each case leads to a system of differential equations. These systems of differential equations involve two equations with three unknowns and they will be made consistent by using the complexity factor. The analysis of the solutions of these systems will be carried out graphically by using different parametric values involved in the systems.

scholarly journals k-Jet field approximations to geodesic deviation equations

2018 ◽  
Vol 15 (12) ◽  
pp. 1850199
Author(s):  
Ricardo Gallego Torromé ◽  
Jonathan Gratus
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Jacobi Equation ◽  
Smooth Manifold ◽  
Specific Class ◽  
Coordinate Transformations ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Geodesic Deviation Equation ◽  
Deviation Functions

Let [Formula: see text] be a smooth manifold and [Formula: see text] a semi-spray defined on a sub-bundle [Formula: see text] of the tangent bundle [Formula: see text]. In this work, it is proved that the only non-trivial [Formula: see text]-jet approximation to the exact geodesic deviation equation of [Formula: see text], linear on the deviation functions and invariant under an specific class of local coordinate transformations, is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit [Formula: see text]-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher-order geodesic deviation equations, we study the first- and second-order geodesic deviation equations for a Finsler spray.

scholarly journals Metric-Affine Geometries with Spherical Symmetry

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 453 ◽  
Author(s):  
Manuel Hohmann
Keyword(s):  
Spherical Symmetry ◽  
Rotation Group ◽  
Spin Connection ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Comprehensive Overview ◽  
Pure Rotation ◽  
Affine Geometries ◽  
Spherically Symmetric Metric

We provide a comprehensive overview of metric-affine geometries with spherical symmetry, which may be used in order to solve the field equations for generic gravity theories which employ these geometries as their field variables. We discuss the most general class of such geometries, which we display both in the metric-Palatini formulation and in the tetrad/spin connection formulation, and show its characteristic properties: torsion, curvature and nonmetricity. We then use these properties to derive a classification of all possible subclasses of spherically symmetric metric-affine geometries, depending on which of the aforementioned quantities are vanishing or non-vanishing. We discuss both the cases of the pure rotation group SO ( 3 ) , which has been previously studied in the literature, and extend these previous results to the full orthogonal group O ( 3 ) , which also includes reflections. As an example for a potential physical application of the results we present here, we study circular orbits arising from autoparallel motion. Finally, we mention how these results can be extended to cosmological symmetry.

scholarly journals A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

2019 ◽  
Vol 22 (1) ◽  
pp. 27-59 ◽  
Author(s):  
HongGuang Sun ◽  
Ailian Chang ◽  
Yong Zhang ◽  
Wen Chen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Relevant Literature ◽  
Physical Models ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Differential ◽  
Mathematical Foundations ◽  
Selection Of

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

scholarly journals Pin(d, d) covariance of pure spinor equations for supersymmetric vacua and non-Abelian T-duality

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Aybike Çatal-Özer ◽  
Emine Diriöz
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Isometry Group ◽  
Structure Group ◽  
Type Ii ◽  
Pure Spinor ◽  
Four Dimensions ◽  
First Order ◽  
Topological Condition ◽  
First Order Differential Equations

Abstract In a supersymmetric compactification of Type II supergravity, preservation of $$ \mathcal{N} $$ N = 1 supersymmetry in four dimensions requires that the structure group of the generalized tangent bundle TM ⨁ T∗M of the six dimensional internal manifold M is reduced from SO(6) to SU(3) × SU(3). This topological condition on the internal manifold implies existence of two globally defined compatible pure spinors Φ1 and Φ2 of non-vanishing norm. Furthermore, these pure spinors should satisfy certain first order differential equations. In this paper, we show that non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations. We first show that the pure spinor equations are covariant under Pin(d, d) transformations. Then, we use the fact NATD is generated by a coordinate dependent Pin(d, d) transformation. The key point is that the flux produced by this transformation is the same as the geometric flux associated with the isometry group, with respect to which one implements NATD. We demonstrate our method by studying NATD of certain solutions of Type IIB supergravity with SU(2) isometry and SU(3) structure.

Sign in / Sign up

Export Citation Format

Share Document