Boundary‐Value Problems for the Linearized and Weakly Nonlinear Boltzmann Equation

1967 ◽  
Vol 8 (9) ◽  
pp. 1893-1898 ◽  
Author(s):  
Young‐ping Pao
Keyword(s):  
Boltzmann Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlinear Boltzmann Equation ◽  
Weakly Nonlinear

Nonlinear Boltzmann Equation Solutions and Applications to Fluid Dynamics

1995 ◽  
Vol 48 (12) ◽  
pp. 777-794 ◽  
Author(s):  
N. Bellomo ◽  
P. LeTallec ◽  
B. Perthame
Keyword(s):  
Fluid Dynamics ◽  
Boltzmann Equation ◽  
Qualitative Analysis ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Mathematical Formulation ◽  
Nonlinear Boltzmann Equation ◽  
Computational Aspects ◽  
Formulation Of Problems

This paper provides a review of the mathematical results on the solution of the nonlinear Boltzmann equation. The survey deals both with analytical and computational aspects: Mathematical formulation of problems, initial and/or boundary value problems, a survey of the qualitative analysis of solutions, and computational treatment of fluid dynamical problems. A discussion of some problems deserving further study concludes this work.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

scholarly journals Boundary-value problems for weakly nonlinear ordinary differential equations

1976 ◽  
Vol 15 (3) ◽  
pp. 321-328 ◽  
Author(s):  
E.N. Dancer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Nonlinear Ordinary Differential Equations ◽  
Weakly Nonlinear ◽  
Periodic Boundary Value Problems ◽  
Degenerate Cases

In this paper, we announce results on the Dirichlet and periodic boundary-value problems for the equation -x″(t) = g(x(t)) – f(t) on [0, π]. We consider degenerate cases not covered by the author's earlier work.

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