Block Hybrid Method for the Numerical solution of Fourth order Boundary Value Problems

Modebei, Mark I.; Jator, S. N.; Ramos, Higinio

doi:10.1016/j.cam.2020.112876

Cited by 11 publications

(5 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently, the ISBS has error constants � C pþ2 ¼ C 9 and is of order p = [7, 7, 7, 7, 7, 7, 7, 7, 7, 7] T respectively. � C 9 ¼ ð6:65 � 10 À 3 ; 1:64 � 10 À 2 ; 2:57 � 10 À 2 ; 3:50 � 10 À 2 ; 4:48 � 10 À 2 ; 5:14 � 10 À 2 ; Þ T ð1:14 � 10 À 2 ; 8:47 � 10 À 3 ; 1:00 � 10 À 2 ; 8:47 � 10 À 3 ; 1:14 � 10 À 2 ; 0Þ 29)-( 34) together with the derivative ( 35)- (40). should the method's order be larger than or equal to one, it is considered consistent.…”

Section: Plos Onementioning

confidence: 99%

“…Ngwane and Jator [ 39 ] employed the sophisticated trigonometrically fitted block technique to tackle a complex oscillating system of equations, exhibiting both Hamiltonian dynamics and second-order initial conditions. Modebei et al [ 40 ] proposed a novel numerical approach for simulating fourth-order differential equations involving partial derivatives, leveraging uniform-order block formulae outlined in their reference. Jator [ 41 ] delved into research on block algorithms to provide precise and efficient solutions for sine-Gordon partial differential equations (PDEs) with varying parameters.…”

Section: Background Informationmentioning

confidence: 99%

See 1 more Smart Citation

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Olaoluwa Omole,

Olusheye Adeyefa,

Iyabo Apanpa

et al. 2024

PLoS ONE

View full text Add to dashboard Cite

In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

show abstract

Section: Plos Onementioning

confidence: 99%

Section: Background Informationmentioning

confidence: 99%

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Olaoluwa Omole,

Olusheye Adeyefa,

Iyabo Apanpa

et al. 2024

PLoS ONE

View full text Add to dashboard Cite

show abstract

“…Theorem 1 [ see [7]]. If f (x, V) verifies a Lipschitz condition on the variable V = y, y , y , y , y (4) with Lipschitz constants L i > 0 for each component, it holds that the system (3.2) has a unique solution whenever…”

Section: Existence and Uniqueness Of The Discrete Solutionmentioning

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Al-Said [14] presented a method to approximate the solution of linear second order BVP and also to determine first, second and third order derivatives at every point of the range of integration. Ramos and others [15][16][17] used different types of block methods to the direct approximation of the solution of fourth order Boundary Value Problems. Recently, differential or integral equation is discretized by using a novel fractional step in [18].…”

Section: Introductionmentioning

confidence: 99%

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Chand

Tyada

Navascués

2022

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

Block Hybrid Method for the Numerical solution of Fourth order Boundary Value Problems

Cited by 11 publications

References 25 publications

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

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

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Contact Info

Product

Resources

About