Initial-Value Problem for the Non-local Generalization of the Lorentz-Dirac Equation

1976 ◽  
Vol 31 (12) ◽  
pp. 1457-1464
Author(s):  
M. Sorg
Keyword(s):  
Proper Time ◽  
Perturbation Expansion ◽  
Equation Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Dirac Theory ◽  
Initial Value ◽  
Local Character ◽  
Non Local ◽  
Non Locality

AbstractAlthough the equation of motion, recently proposed for the classical radiating electron, is of non-local character in proper time, the Newtonian initial data (position and velocity) are sufficient to guarantee existence and uniqueness of the solutions. The corresponding existence proof is accomplished by the Picard-Lindelöf method of successive approximations. This method indicates the possibility of a perturbation expansion of the exact solution in terms of the non-locality parameter. Such a perturbation expansion does not seem to be possible in the Lorentz-Dirac theory.

Causality Violation and Non-local Generalization of the Lorentz-Dirac Equation

1977 ◽  
Vol 32 (3-4) ◽  
pp. 319-326
Author(s):  
P. Alber ◽  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Dirac Equation ◽  
Equation Of Motion ◽  
Local Theory ◽  
Constant Force ◽  
Dirac Theory ◽  
Local Equation ◽  
Causality Violation ◽  
Finite Duration ◽  
Non Local

AbstractIt is demonstrated by a concrete example (constant force of finite duration) that the recently proposed, non-local equation of motion for the radiating electron does exhibit the effect of causality violation. This phenomenon, which occurs in the non-local theory in form of self-oscillations, is however less severe than in the Lorentz-Dirac theory, if only physically reasonable forces are admitted.

ON SOLVING STOCHASTIC INITIAL-VALUE DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 715-745 ◽  
Author(s):  
IVO BABUŠKA ◽  
KANG-MAN LIU
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Element Solution ◽  
Initial Value ◽  
Random Functions ◽  
Priori Error Estimates

This paper addresses the issues involved in solving systems of linear ODE's with stochastic coefficients and loadings described by the Karhunen–Loeve expansion. The Karhunen–Loeve expansion is used to discretize random functions into a denumerable set of uncorrelated random variables, thus providing us for transforming this problem into an equivalent deterministic one. Perturbation error estimates and a priori error estimates between the exact solution and the finite element solution in the framework of Sobolev space are given. The method of successive approximations for finite element solutions is analyzed.

Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument

2020 ◽  
pp. 31-56
Author(s):  
T.K. Yuldashev ◽  
J.A. Artykova
Keyword(s):  
Differential Equation ◽  
Initial Value Problem ◽  
Higher Order ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Degenerate Kernel ◽  
Hyperbolic Operator ◽  
Initial Value ◽  
Partial Differential ◽  
Integro Differential Equation

In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.

Finite Size of the Electron and Runaway Solutions

1977 ◽  
Vol 32 (5) ◽  
pp. 383-389 ◽  
Author(s):  
J. Petzold ◽  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Dirac Equation ◽  
Free Electron ◽  
Constant Velocity ◽  
Initial Conditions ◽  
Perturbation Expansion ◽  
Finite Size ◽  
Equation Of Motion ◽  
Local Equation ◽  
Causality Violation ◽  
Non Local

Abstract The problem of runaway solutions is studied within the framework of a non-local equation of motion for the classically radiating electron. It is found that the force-free electron oscillates down to a constant velocity under emission of radiation, if certain restrictions on the initial conditions are imposed. Causality violation is not present in this model, but penetrates into the theory as consequence of a false perturbation expansion leading to the notorious Lorentz-Dirac equation of motion.

scholarly journals A Non-local Phase Field Model of Bohm’s Quantum Potential

2021 ◽  
Vol 51 (2) ◽  
Author(s):  
Roberto Mauri
Keyword(s):  
Body Force ◽  
Field Model ◽  
Mass Density ◽  
Phase Field Model ◽  
Equation Of Motion ◽  
Quantum Potential ◽  
The Body ◽  
Local Phase ◽  
Non Local ◽  
Non Locality

AbstractAssuming that the free energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are identically obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics admit a classical interpretation based on non-locality.

scholarly journals Evaluating probabilistic forecasts of football matches: the case against the ranked probability score

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Edward Wheatcroft
Keyword(s):  
Scoring Rules ◽  
Brier Score ◽  
Sporting Events ◽  
Forecast Performance ◽  
Scoring Rule ◽  
Probability Score ◽  
Probabilistic Forecasts ◽  
Non Local ◽  
Evaluating Forecasts ◽  
Non Locality

Abstract A scoring rule is a function of a probabilistic forecast and a corresponding outcome used to evaluate forecast performance. There is some debate as to which scoring rules are most appropriate for evaluating forecasts of sporting events. This paper focuses on forecasts of the outcomes of football matches. The ranked probability score (RPS) is often recommended since it is ‘sensitive to distance’, that is it takes into account the ordering in the outcomes (a home win is ‘closer’ to a draw than it is to an away win). In this paper, this reasoning is disputed on the basis that it adds nothing in terms of the usual aims of using scoring rules. A local scoring rule is one that only takes the probability placed on the outcome into consideration. Two simulation experiments are carried out to compare the performance of the RPS, which is non-local and sensitive to distance, the Brier score, which is non-local and insensitive to distance, and the Ignorance score, which is local and insensitive to distance. The Ignorance score outperforms both the RPS and the Brier score, casting doubt on the value of non-locality and sensitivity to distance as properties of scoring rules in this context.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

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