A block hybrid integrator for numerically solving fourth-order Initial Value Problems

Modebei, Mark I.; Adeniyi, R. B.; Jator, S. N.; Ramos, Higinio

doi:10.1016/j.amc.2018.10.080

Cited by 10 publications

(9 citation statements)

References 12 publications

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“…where y∈C 2 [a, b] is an arbitrary function. Expanding (17) in Taylor's series about the point x n , the expression is obtained as:…”

Section: Order and Local Truncation Error (Lte)mentioning

confidence: 99%

“…To establish hat the methods are zero stable, each of the method in block form are solved simultaneously to obtain all the y i and y i 's for appropriate index i, (see Modebei et al [17]). For the method (9) and its additional methods in (10), they are taken in block form and solved simultaneously to obtain y i 3 for i = 1(1)6 to obtain the following block method.…”

Section: Zero Stabilitymentioning

confidence: 99%

“…For the method (9) and its additional methods in (10), they are taken in block form and solved simultaneously to obtain y i 3 for i = 1(1)6 to obtain the following block method. For block method (9)- (10) y n+ 1 A numerical method is zero-stable if the solutions remain bounded as h→0, which means that the method does not provide solutions that grow unbounded as the number of steps increases, Modebei et al [17]. To show the zero-stability of the block method (36), we take h→0 the method may be rewritten in matrix form as…”

Section: Zero Stabilitymentioning

confidence: 99%

See 2 more Smart Citations

Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Olaiya¹,

Azeez²,

Modebei³

2021

J. Nig. Soc. Phys. Sci.

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This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

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“…where y∈C 2 [a, b] is an arbitrary function. Expanding (17) in Taylor's series about the point x n , the expression is obtained as:…”

Section: Order and Local Truncation Error (Lte)mentioning

confidence: 99%

Section: Zero Stabilitymentioning

confidence: 99%

Section: Zero Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Olaiya¹,

Azeez²,

Modebei³

2021

J. Nig. Soc. Phys. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…, where p andč p+5 are respectively known as the order and local principal error constant of the corresponding formula (see [4,6,9,10]). The local truncation errors and the principal error constants of the main formulas are obtained to be:…”

Section: Additional Formulasmentioning

confidence: 99%

Development and Implementation of a Tenth-Order Hybrid Block Method for Solving Fifth-Order Boundary Value Problems

Ramos

Momoh

2021

Mathematical Modelling and Analysis

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A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

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“…Modebei, et al [11] constructed a block hybrid integrator for numerically solving fourth-order Initial Value Problems, which are developed with the presence of higher derivatives with same order. The resulting block methods are used to solve fourth order ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

Ukpebor

Omole

Adoghe

2020

International Journal of Mathematical Research

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From the time immemorial, researchers have been beaming their search lights round the numerical solution of ordinary differential equation of initial value problems. This was as a result of its large applications in the area of Sciences, Engineering, Medicine, Control System, Electrical Electronics Engineering, Modeled Equations of Higher order, Thin flow, Fluid Mechanics just to mention few. There are a lot of differential equations which do not have theoretical solution; hence the use of numerical solution is very imperative. This paper presents the derivation, analysis and implementation of a class of new numerical schemes using Lucas polynomial as the approximate solution for direct solution of fourth order ODEs. The new schemes will bridge the gaps of the conventional methods such as reduction of order, Runge-kutta's and Euler's methods which has been reported to have a lot of setbacks. The schemes are chosen at the integration interval of seven-step being a perfection interval. The even grid-points are interpolated while the odd grid-points are collocated. The discrete scheme, additional schemes and derivatives are combined together in block mode for the solution of fourth order problems including special, linear as well as application problems from Ship Dynamics. The analysis of the schemes shows that the schemes are Reliable, P-stable and Efficient. The basic properties of the schemes were examined. Numerical results were presented to demonstrate the accuracy, the convergence rate and the speed advantage of the schemes. The schemes perform better in terms of accuracy when compared with other methods in the literature. Contribution/Originality: The study uses Lucas polynomial for the derivation of a new class of numerical schemes. The schemes were implemented in block mode for approximating fourth order ODEs directly without reduction. It solves variety of problems including problem in Electrical Engineering. The schemes performs excellently better than other schemes in the literatures.

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A block hybrid integrator for numerically solving fourth-order Initial Value Problems

Cited by 10 publications

References 12 publications

Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Development and Implementation of a Tenth-Order Hybrid Block Method for Solving Fifth-Order Boundary Value Problems

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

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