Parametrically defined nonlinear differential equations, differential–algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

Polyanin, Andrei D.; Zhurov, Alexei I.

doi:10.1016/j.aml.2016.08.006

Cited by 9 publications

(4 citation statements)

References 5 publications

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“…where k = const > 0 (and the limiting case k = ∞ is also allowed); otherwise the function g can be chosen rather arbitrarily. From (10) and the second condition (12) it follows that x ′ ξ → 0 as ξ → ∞. The Cauchy problem (11) can be integrated numerically applying the Runge-Kutta method or other standard numerical methods.…”

Section: Solution Methods Based On Non-local Transformationsmentioning

confidence: 99%

“…Remark 1. Systems of equations ( 2) and ( 16) are particular cases of parametrically defined nonlinear differential equations, which are considered in [9,10]. In [10], the general solutions of several parametrically defined ODEs were obtained via differential transformations, based on introducing a new independent variable t = y ′ x .…”

Section: Solution Methods Based On a Differential Transformationmentioning

confidence: 99%

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New numerical methods for blow-up problems

Polyanin,

Shingareva

2017

Preprint

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Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable. The second method is based on introducing a new non-local variable that reduces ODE to a system of coupled ODEs. Both methods lead to problems whose solutions do not have blowing-up singular points; therefore the standard numerical methods can be applied. The efficiency of the proposed methods is illustrated with several test problems.

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Section: Solution Methods Based On Non-local Transformationsmentioning

confidence: 99%

Section: Solution Methods Based On a Differential Transformationmentioning

confidence: 99%

New numerical methods for blow-up problems

Polyanin,

Shingareva

2017

Preprint

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“…Remark 10. Systems of differential-algebraic equations ( 12) and (81) are particular cases of parametrically defined nonlinear differential equations, which are considered in [34,35]. In [35], the general solutions of several parametrically defined ODEs were constructed via differential transformations based on introducing a new differential independent variable t = y ′ x .…”

Section: Solution Methods Based On Introducing a Differential Variablementioning

confidence: 99%

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

Polyanin,

Shingareva

2017

Preprint

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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,

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“…Basic concepts and definitions of the theory of differential equations with deviating arguments, as well as existence theorems and approximate methods of their solution, are covered in [1][2][3][4][5]. One of the latest significant works, which provides an overview of the most common mathematical models with delay used in population theory, biology, medicine, and other applications, is the work of a group of authors [6]. A fairly complete list of sources related to problems for differential equations with delay is also given here.…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

On a problem for a delay differential equation

Iskakova

Temesheva

Uteshova

2023

Math Methods in App Sciences

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The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem.

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Parametrically defined nonlinear differential equations, differential–algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

Cited by 9 publications

References 5 publications

New numerical methods for blow-up problems

New numerical methods for blow-up problems

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

On a problem for a delay differential equation

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