An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

Khalsaraei, Mohammad Mehdizadeh; Shokri, Ali

doi:10.1007/s11075-019-00784-w

Cited by 6 publications

(6 citation statements)

References 33 publications

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“…The development of a two-stage optimisation procedure is based on parameter estimation using nonparametric estimators [11], taking into account time-varying parameters [12] and their Bayesian estimation [13], three-and four-stage modifications of a two-stage optimisation procedure [14][15][16], consideration of errors in the autoregressive model [17] and applications to chemical kinetics models [18]. This variety of approaches is largely determined by the task specifics of estimating the parameters of differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

Tsitsiashvili,

Gudimenko,

Osipova

2023

Mathematics

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In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.

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Section: Introductionmentioning

confidence: 99%

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

Tsitsiashvili,

Gudimenko,

Osipova

2023

Mathematics

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“…Ordinary differential conditions are extensively used in demonstrating numerous natural and physical applications. Mathematical strategies dependent on finite differences [1,2], Taylor series [3], interpolation, such as Runge-Kutta, Euler, and multistep techniques [4,5], and some different strategies [6,7] are broadly utilized. A large number of problems in mathematical epidemiology are modeled by autonomous systems of nonlinear ordinary differential equations, which implies that the boundaries of the model are autonomous, regarding time.…”

Section: Introductionmentioning

confidence: 99%

Efficient Numerical Solutions to a SIR Epidemic Model

et al. 2022

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Two non-standard predictor-corrector type finite difference methods for a SIR epidemic model are proposed. The methods have useful and significant features, such as positivity, basic stability, boundedness and preservation of the conservation laws. The proposed schemes are compared with classical fourth order Runge–Kutta and non-standard difference methods (NSFD). The stability analysis is studied and numerical simulations are provided.

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“…Many existing numerical methods for solving the class of problem in () have been developed; see, for example, previous studies 2–12 . Those strategies include Runge‐Kutta type, linear multistep, Numerov‐type, P‐stable Obrechkoff, or collocation methods.…”

Section: Introductionmentioning

confidence: 99%

“…Many existing numerical methods for solving the class of problem in (1) have been developed; see, for example, previous studies. [2][3][4][5][6][7][8][9][10][11][12] Those strategies include Runge-Kutta type, linear multistep, Numerov-type, P-stable Obrechkoff, or collocation methods. One standard approach is to transform problem (1) into an equivalent system of first-order ODEs.…”

Section: Introductionmentioning

confidence: 99%

An adaptive optimized Nyström method for second‐order IVPs

Rufai

Mazzia

Ramos

2022

Math Methods in App Sciences

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This research work deals with the development, analysis, and implementation of an adaptive optimized one-step Nyström method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points. An embedding-like procedure is used to estimate the error of the proposed optimized method. The current approach has been used to compute efficiently approximate solutions to general second-order IVPs. The numerical experiments demonstrate that the introduced error estimation and step-size control strategy presented in this manuscript have produced a good performance compared to some of the other existing numerical methods. KEYWORDS collocation method, error estimation and control, ordinary and time-dependent partial differential equations, optimized Nyström method, variable stepsize formulation MSC CLASSIFICATION 65L05, 65L20

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An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

Cited by 6 publications

References 33 publications

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

Efficient Numerical Solutions to a SIR Epidemic Model

An adaptive optimized Nyström method for second‐order IVPs

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