Blow-up solutions of quadratic differential systems

Baris, J.; Baris, P.; Ruchlewicz, B.

doi:10.1007/s10958-008-0070-8

Cited by 8 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Based on other ideas, some special methods of numerical integration of blowup problems are described, for example, in [1,2,[5][6][7][8][9][10][11][12][13]. In particular, it was suggested in [8,13] to investigate such problems via compactifications, which are point transformations of the special form (whose inverse transformations have singularities).…”

Section: Problems Arising In Numerical Solutions Of Blow-up Problemsmentioning

confidence: 99%

Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints

Polyanin,

Shingareva

2017

Preprint

View full text Add to dashboard Cite

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first-and secondorder are described. Solutions of such problems have singularities whose positions are unknown a priori (for this reason, the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first * This article is a significantly extended and more general version of the article A.D. Polyanin and I.K. Shingareva, The use of differential and non-local transformations for numerical integration of non-linear blow-up problems,

show abstract

Section: Problems Arising In Numerical Solutions Of Blow-up Problemsmentioning

confidence: 99%

Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints

Polyanin,

Shingareva

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…is the switching function, which defines the types of the optimal controls u * (t), v * (t) according to formulas (12), (13).…”

Section: Pontryagin Maximum Principlementioning

confidence: 99%

“…Lemma 2 and formulas (12), (13) show that the optimal controls u * (t), v * (t) are bang-bang functions taking values {u min ; u max }, {v min ; v max }, respectively. Moreover, these controls switch from maximum values to minimum values and vice versa at the same moments of switching.…”

Section: Properties Of the Switching Functionmentioning

confidence: 99%

“…The problem of continuability of solutions of non-autonomous quadratic systems of differential equations was considered in [13,14]. In these studies, as well as in the classic textbook on differential equations [15], it was noted that, when the initial conditions g j (0) = g 0 j , j = 1, 3 were added to system (23), an appropriate solution g(t) = (g 1 (t), g 2 (t), g 3 (t)) will be determined on the interval [0, t max ), which is the maximum possible interval for the existence of such solution, and either t max = +∞ or t max < +∞.…”

Section: Properties Of the Switching Functionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics

Grigorieva

Хайлов

2015

Mathematics

View full text Add to dashboard Cite

A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle's theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided.

show abstract

Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints

Polyanin

Shingareva

2018

Applied Mathematics and Computation

View full text Add to dashboard Cite

Blow-up solutions of quadratic differential systems

Cited by 8 publications

References 6 publications

Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints

Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints

Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics

Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints

Contact Info

Product

Resources

About