Trigonometrically fitted block Numerov type method for y′′ = f(x, y, y′)

Jator, S. N.; Swindell, S.; French, Richard H.

doi:10.1007/s11075-012-9562-1

Cited by 34 publications

(21 citation statements)

References 24 publications

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“…The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-byblock manner. In this way, the method is self-starting and implemented without predictors (see Jator and Oladejo [22], Jator et al [9], Jator [7], and Ngwane and Jator [6,23,24]). It is shown that the method can also be used to directly solve non-Hamiltonian systems with embedded first derivatives as well as partial differential equations via the method of lines.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

“…, . The methods in (8) and (9) specified by the coefficients from Tables 1 and 2 are combined to give the BHM, which is expressed as Mathematical Problems in Engineering where 0 , 1 , 0 , and 1 are square matrices of dimension twenty whose elements characterize the method and are given by the coefficients of (8) and (9). The linear stability of the BHM is discussed by applying the method to the test equation…”

Section: Remarkmentioning

confidence: 99%

“…where : R × R × R → R , > 0 is an integer, { , } R , is the dimension of the system, ℵ is a real nonsingular diagonalizable × constant matrix having large eigenvalues, and ‖ℵ‖ ≫ 1 has been studied (see [1][2][3][4][5][6][7][8][9]). Special cases of (1) have also been extensively discussed in the literature ( [10][11][12][13][14], Hairer [15]).…”

Section: Introductionmentioning

confidence: 99%

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Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Jator

King

2018

Mathematical Problems in Engineering

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In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

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Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Jator

King

2018

Mathematical Problems in Engineering

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“…In order to solve differential equations whose solutions are known to oscillate, methods based on trigonometric polynomials have been proposed (see [8]- [13]). However, little attentions have been paid to the Block Differentiation Formula using the trigonometric polynomial as the basis function for solving IVP whose solution oscillate.…”

Section: Introductionmentioning

confidence: 99%

L0 Stable Trigonometrically Fitted Block Backward Differentiation Formula of Adams Type for Autonomous Oscillatory Problems

Okunuga¹,

Abdulganiy²

2017

IJAPM

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Abstract:In this paper, a 0 Stable Second Derivative Trigonometrically Fitted Block Backward Differentiation Formula of Adams Type (SDTFF) of algebraic order 4 is presented for the solution of autonomous oscillatory problems. A Continuous Second Derivative Trigonometrically Fitted (CSDTF) whose coefficients depend on the frequency and step size is constructed using trigonometric basis function. The CSDTF is used to generate the main method and one additional method which are combined and applied in block form as simultaneous numerical integrators. The stability properties of the method are investigated using boundary locus plot. It is found that the method is zero stable, consistent and hence converges. The method is applied on some numerical examples and the result show that the method is accurate and efficient.

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“…In 2008, the trigonometrically fitted explicit Numerov-type method was proposed [24]. The explicit adapted Numerov method [25] and the trigonometrically fitted block Numerov-type method (TBNM) [26] were proposed in 2013.…”

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confidence: 99%

Ordinary differential equation algorithms for a frequency-domain water wave Green’s function

Shen

Duan

et al. 2015

J Eng Math

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Our study investigated an algorithm for a second-order ordinary differential equation for the frequencydomain Green's function of free-surface waves in water of infinite depth. A power-series method is introduced if the wave frequency ω < 1; otherwise, if ω ≥ 1, then the trigonometrically fitted block Numerov-type method (TBNM) is employed. The calculation precision of the power-series method and the TBNM reached 10 −7 and 10 −6 , respectively. The two methods have a high calculation efficiency compared with calculating the Green's function using the series expansion representation approach. The calculation speed for these two methods is 15 times faster using the same computing codes.

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Trigonometrically fitted block Numerov type method for y′′ = f(x, y, y′)

Cited by 34 publications

References 24 publications

Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

L0 Stable Trigonometrically Fitted Block Backward Differentiation Formula of Adams Type for Autonomous Oscillatory Problems

Ordinary differential equation algorithms for a frequency-domain water wave Green’s function

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