scholarly journals Resummation methods for Master Integrals

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Dhimiter D. Canko ◽  
Nikolaos Syrrakos
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Canonical Basis ◽  
Proof Of Concept

Abstract We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the x → 1 limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters {0, 1} of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.

scholarly journals VI. The waves on a rotating liquid spheroid of finite ellipticity

1889 ◽  
Vol 180 ◽  
pp. 187-219 ◽  
Keyword(s):  
Differential Equation ◽  
Angular Velocity ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Independent Solution ◽  
Ellipsoidal Harmonics ◽  
Rotating Liquid ◽  
The Waves ◽  
Hydrodynamical Problem

1. The hydrodynamical problem of finding the waves or oscillations on a gravitating mass of liquid which, when undisturbed, is rotating as if rigid with finite angular velocity, in the form of an ellipsoid or spheroid, was first successfully attacked by M. Poincaré in 1885. In his important memoir, “Sur l’Équilibre d’une Masse Fluide animée dun Mouvement de Rotation,” Poincare has (§ 13) obtained the differential equations for the oscillations of rotating liquid, and shown that, by a transformation of projection, the determination of the oscillations of any particular period is reducible to finding a suitable solution of Laplace’s equation. He then applies Lamé’s functions to the case of the ellipsoid, showing that the differential equations are satisfied by a series of Lamé’s functions referred to a certain auxiliary ellipsoid, the boundary-conditions, however, involving ellipsoidal harmonics, referred to both the auxiliary and actual fluid ellipsoid. At the same time, Poincare’s analysis does not appear to admit of any definite conclusions being formed as to the nature and frequencies of the various periodic free waves. The present paper contains an application of Poincaré’s methods to the simpler case when the fluid ellipsoid is one of revolution (Maclaurin’s spheroid). The solution is effected by the use of the ordinary tesseral or zonal harmonics applicable to the fluid spheroid and to the auxiliary spheroid required in solving the differential equation. The problem is thus freed from the difficulties attending the use of Lamé’s functions, and is further simplified by the fact that each independent solution contains harmonics of only one particular degree and rank.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Determination of Optimum Profile of One-Dimensional Cooling Fins

1983 ◽  
Vol 105 (3) ◽  
pp. 317-320 ◽  
Author(s):  
S. K. Hati ◽  
S. S. Rao
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Minimum Principle ◽  
Problem Formulation ◽  
One Dimensional ◽  
A New Technique ◽  
The One ◽  
Optimum Profile

The optimum design of an one-dimensional cooling fin is considered by including all modes of heat transfer in the problem formulation. The minimum principle of Pontryagin is applied to determine the optimum profile. A new technique is used to solve the reduced differential equations with split boundary conditions. The optimum profile found is compared with the one obtained by considering only conduction and convection.

scholarly journals Survey of studies on degenerate boundary conditions and finite spectrum

2019 ◽  
Vol 14 (3) ◽  
pp. 184-201
Author(s):  
A.M. Akhtyamov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Complex Plane ◽  
Boundary Value ◽  
Finite Spectrum ◽  
Diffusion Operator ◽  
Odd Order ◽  
Entire Complex

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

scholarly journals CONCEPTION QUASIDERIVATIVES IN THE PROBLEMS OF MATHEMATICAL MODELING OF HEAT TRANSFER

2020 ◽  
Vol 21 ◽  
pp. 79-85
Author(s):  
R. Tatsii ◽  
M. Stasiyk ◽  
O. Pazen ◽  
L. Shypot
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Closed Form ◽  
Critical Radius ◽  
Internal Heat ◽  
Heat Sources ◽  
Internal Heat Sources

In this paper, in closed form, the problems of determining stationary temperature fields in multilayer (flat, cylindrical and spherical) structures in the presence of discrete-continuous internal and point heat sources are solved. The one-dimensional differential equation of thermal conductivity in different coordinate systems is given through one parametric family of quasi-differential equations. It is assumed that the coefficients of the differential equation of thermal conductivity are piecewise constant functions. A system of two linearly independent boundary conditions is added to the equation, which in the general case are nonlocal. The solutions of such problems are constructive and are expressed exclusively through their initial data. The basic provisions of the concept of quasi-derivatives, the provisions of the theory of heat transfer, the theory of generalized systems of linear differential equations, elements of the theory of generalized functions are used. For the mathematical model of stationary thermal conductivity, the practical use of the concept of quasi-derivatives is illustrated, for the efficient construction, in a closed form, of solutions of boundary value problems with the most general boundary conditions. As an example, the problem of finding the critical radii of thermal insulation of multilayer hollow cylinders and spheres, taking into account the internal heat sources in the layers. Boundary conditions of the first and third kind. It is established that the value of the critical radius does not depend on the number of layers and the intensity of internal heat sources, but only on the thermal conductivity of the outer layer of the structure and the heat transfer coefficient between the structure and the environment. The formula for determining the critical radius of thermal insulation for a multilayer cylindrical and spherical structure is derived. The methods developed in this work have the prospect of further development and can be used in engineering calculations.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

scholarly journals Monotone iterative method for fractional p-Laplacian differential equations with four-point boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoping Li ◽  
Minyuan He
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Positive Solutions ◽  
Monotone Iterative Method ◽  
Point Boundary ◽  
Abstract Result ◽  
Monotone Iterative

AbstractA four-point boundary problem for a fractional p-Laplacian differential equation is studied. The existence of two positive solutions is established by means of the monotone iterative method. An example supporting the abstract result is given.

scholarly journals A Pretreatment Method of Volterra the External Boundary Value Problem of Integral Differential Equations

2021 ◽  
Vol 15 ◽  
pp. 1252-1259
Author(s):  
Xiaojuan Chen ◽  
Xiaoxiao Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
External Boundary ◽  
Sinc Function ◽  
Subsequent Calculation ◽  
Integral Differential Equations ◽  
High Level

In the process of traditional methods, the error rate of external boundary value problem is always at a high level, which seriously affects the subsequent calculation and cannot meet the requirements of current Volterra products. To solve this problem, Volterra's preprocessing method for the external boundary value problem of Integro differential equations is studied in this paper. The Sinc function is used to deal with the external value problem of Volterra Integro differential equation, which reduces the error of the external value problem and reduces the error of the external value problem. In order to prove the existence of the solution of the differential equation, when the existence of the solution can be proved, the differential equation is transformed into a Volterra integral equation, the Taylor expansion equation is used, the symplectic function is used to deal with the external value problem of homogeneous boundary conditions, and the uniform effective numerical solution of the external value problem of the equation is obtained by homogeneous transformation according to the non-homogeneous boundary conditions.

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