scholarly journals Analytic tadpole coefficients of one-loop integrals

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Bo Feng ◽  
Tingfei Li ◽  
Xiaodi Li
Keyword(s):  
Differential Equations ◽  
Recurrence Relations ◽  
Differential Operators ◽  
Tensor Structure ◽  
Loop Integral ◽  
Cut Method ◽  
Analytic Expressions

Abstract One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

scholarly journals Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space

1994 ◽  
Vol 135 ◽  
pp. 165-196 ◽  
Author(s):  
Masatake Miyake ◽  
Masafumi Yoshino
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Index Theorem ◽  
Fredholm Property ◽  
Goursat Problem ◽  
Point Of View ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Gevrey Spaces

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

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