scholarly journals Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

1968 ◽  
Vol 5 (4) ◽  
pp. 680-716 ◽  
Author(s):  
Joel N. Franklin ◽  
Eugene R. Rodemich
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Analysis ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

scholarly journals USING GENERALIZED STOCHASTIC METHOD TO EVALUATE PROBABILITY OF CONFLICT IN CONTROLLED AIR TRAFFIC

Aviation ◽  
2007 ◽  
Vol 11 (2) ◽  
pp. 31-36 ◽  
Author(s):  
Vitaly Babak ◽  
Volodymyr Kharchenko ◽  
Volodymyr Vasylyev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Traffic Safety ◽  
Evaluation Method ◽  
Air Traffic ◽  
Collision Probability ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Potential Conflict ◽  
Infinitesimal Operator

The introduction of the new concepts of air traffic management (ATM) and transition from centralized to decentralized air traffic control (ATC) with the change of traditional ATM to Cooperative ATM sets new tasks and opens new capabilities for air traffic safety systems. This paper is devoted to the problem of evaluating the probability of aircraft collision under the condition of Cooperative ATM, when the necessary information is available to the subjects involved in the decision‐making process. The generalized stochastic conflict probability evaluation method is developed. This method is based on the generalized conflict probability equation for evaluation of potential conflict probability and aircraft collision probability that is derived by taking into account stochastic nature and time correlation of deviation from planned flight trajectory in controlled air traffic. This equation is described as a multi‐dimensional parabolic partial differential equation using a differential (infinitesimal) operator of the multi‐dimensional stochastic process of relative aircraft movement. The common procedure for the prediction of conflict probability is given, and the practical application of the generalized method presented is shown. All equational coefficients of a differential operator for a practical solution of a parabolic partial differential equation are derived. For some conditions, the numerical solution of the conflict probability equation is obtained and illustrated graphically.

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