Exact three-point difference scheme for singular nonlinear boundary value problems

Król, Marta; Kunynets, Andrii; Кутнів, М. В.

doi:10.1016/j.cam.2015.12.003

Cited by 7 publications

(2 citation statements)

References 4 publications

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“…The shooting strategies are important for using IVP algorithms to solve BVPs, but the computational mechanisms involve additional costs to find out high order initial conditions [2][3][4]. A wellknown direct method for BVPs is the finite difference method widely used in the literature [5][6][7][8][9][10]. Even finite difference methods produce acceptable results for many BVPs, their local order refinement (p-refinement) is not easy task due to the direct disconnection of the higher order formulations.…”

Section: Introductionmentioning

confidence: 99%

A new implicit-explicit local differential method for boundary value problems

Tunc¹,

Sarı²

2021

Turk J Math

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In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

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Section: Introductionmentioning

confidence: 99%

A new implicit-explicit local differential method for boundary value problems

Tunc¹,

Sarı²

2021

Turk J Math

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“…The special interest concerning the problems () and () is also connected with the fact that three‐point difference schemes of high order for singular boundary‐value problems in ODEs 21,22 require solving associated IVPs. From other side, any stationary diffusion equation in the cylindrical or spherical coordinate systems in the case of the axial or central symmetry accordingly are reduced to singular boundary value problems (BVPs) and IVPs for Equation ().…”

Section: Introductionmentioning

confidence: 99%

New explicit high‐order one‐step methods for singular initial value problems

Datsko

Кутнів

Kunynets

et al. 2020

Comp and Math Methods

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Summary New explicit one‐step high‐order numerical methods (Taylor series and Runge‐Kutta methods) for singular initial value problems for second‐order nonlinear ordinary differential equations are constructed. These methods allow to calculate an approximate solution in the neighborhood of a singular point x=0 with a high‐order of accuracy. By the substitution of the independent variable, we transform the original singular initial value problem into a nonsingular one at an infinite interval. To solve this nonsingular problem, standard numerical methods with initial conditions obtained near a singular point can be used. The effectiveness of the presented approach is confirmed by numerical experiments.

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