scholarly journals Stiff systems of ordinary differential equations. III. Partially stiff systems

1982 ◽  
Vol 23 (3) ◽  
pp. 310-331 ◽  
Author(s):  
J. J. Mahony ◽  
J. J. Shepherd
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Solution Space ◽  
Homogeneous System ◽  
Stiff Systems ◽  
Stiff System ◽  
Specific System ◽  
Small Positive Parameter ◽  
Image Position

AbstractThe partially stiff system of ordinary differential equationsis studied by the methods developed in the earlier papers in this series. Here e is a small positive parameter, x and y are n- and m-vectors respectively, and A is nonsingular. A useful basis for the solution space of the homogeneous system is constructed and the method of variation of parameters is used to obtain useful representations of all solutions. Sufficient conditions are derived under which the formal approximationis close to the actual solution. it is found that purely imaginary eigenvalues for A require more stringent requirements for the formal technique to be valid. A brief discussion of the case when A is singular shows that there are a great number of possibilities requiring consideration for a general theory. it is suggested that local computation of such cases is likely to be the most effective weapon for any specific system.

scholarly journals Stiff systems of ordinary differential equations. Part 1. Completely stiff, homogeneous systems

1981 ◽  
Vol 23 (1) ◽  
pp. 17-51 ◽  
Author(s):  
J. J. Mahony ◽  
J. J. Shepherd
Keyword(s):  
Differential Equations ◽  
Finite Number ◽  
Solution Space ◽  
Homogeneous System ◽  
Positive Parameter ◽  
Stiff Systems ◽  
Linearly Independent ◽  
Small Positive Parameter ◽  
Independent Eigenvector ◽  
Image Position

AbstractFor the completely stiff real homogeneous systemwhere e is a small positive parameter, a method is given for the construction of a basis for the solution space.If A has n linearly independent eigenvector functions, then there exists a choice of these, {si}, with corresponding eigenvalue functions {λi}, such that there is a local basis for solution, that takes the formwhere vi is a vector that tends to zero with e. In general, a basis of this form exists only on an interval in which the distinct eigenvalues have their real parts ordered. A construction is provided for continuing any solution across the boundaries of any such interval. These results are proved for a finite or infinite interval for which there are only a finite number of points at which the ordering of the real parts of eigenvalues changes.

Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations

1981 ◽  
Vol 90 (1-2) ◽  
pp. 25-40 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Kyoko Tanaka
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Solution Space ◽  
Extreme Situation ◽  
Nonoscillatory Solutions ◽  
Linear Ordinary Differential Equations ◽  
Image Position ◽  
Monotone Solutions

SynopsisThe equation to be considered iswhere pi(t), 0≦i≦n, and q(t) are continuous and positive on some half-line [a, ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [a, ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.

scholarly journals Stiff systems of ordinary differential equations

1981 ◽  
Vol 23 (2) ◽  
pp. 136-172 ◽  
Author(s):  
J. J. Mahony ◽  
J. J. Shepherd
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Singular Points ◽  
Operator Norm ◽  
Existence Theory ◽  
Stiff Systems ◽  
Stiff System ◽  
Linear Differential ◽  
Image Position

AbstractSolutions of the stiff system of linear differential equationsare obtained in a form yielding tight estimates of their properties, and conditions are obtained under which the operator norm of the map from r to the solution x does not become exponentially large for small values of ε. When these conditions are satisfied, the solutions are shown to be close to those of Ax + r = 0, save at any singular points of A, and in boundary layers. The behaviour of solutions near admissible singular points is also obtained.The results are used to characterize those boundary-value problems for the above system in which the solution defines maps from the data that are of “moderate” operator norm. This leads to a constructive existence theory for a limited class of boundary-value problems for the nonlinear systemIt is suggested that the treatment of more general classes of boundary-value problems may be simplified using these results. By the use of simple examples, the problems involving large operator norms are shown to be related to the stability properties of the possible branches of the outer solutions close to those of

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals On periodic-type solutions of systems of linear ordinary differential equations

2004 ◽  
Vol 2004 (5) ◽  
pp. 395-406 ◽  
Author(s):  
I. Kiguradze
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Type Solution ◽  
Linear Ordinary Differential Equations

We establish nonimprovable, in a certain sense, sufficient conditions for the existence of a unique periodic-type solution for systems of linear ordinary differential equations.

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