Efficient Numerical Algorithm for the Solution of Eight Order Boundary Value Problems by Haar Wavelet Method

Amin, Rohul; Shah, Kamal; Al‐Mdallal, Qasem M.; Khan, Imran; Asif, Muhammad

doi:10.1007/s40819-021-00975-x

Cited by 10 publications

(7 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, Amin et al successfully obtained effective approximate solutions for 8th and 6th order differential equations using only first-order weak derivative Haar wavelets as trial functions. [155,156] It is worth mentioning that these two points have been a part of the reasons that have long plagued the development of traditional methods for solving high-order differential equations.…”

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

View full text Add to dashboard Cite

The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

show abstract

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

View full text Add to dashboard Cite

show abstract

“…El-Gamel and Abdrabou, (2019) have used sinc-Galerkin method for the numerical solution of eight order boundary value problems. In Amin et al, (2021) the Authors study Haar technique for the solution of both nonlinear and linear eight-order boundary value problems. Haq.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

Mutawakilu,

Bello

2024

USci

View full text Add to dashboard Cite

A Taylor series method which is taught at undergraduate levels and which has been hitherto used for the solution of initial value problem is successfully used in this work for the solutions of eighth-order two point boundary-value problems. The method is based on successive differentiation of the governing equation to obtain high order derivatives and then evaluated at the boundary point x=a. The solution is expressed inform a Taylor series with the unknown coefficients at a point x=a. By applying boundary conditions at x=b in the Taylor series solution, the system of unknown coefficient is obtained. After solving the system, then unknown coefficient are determined. The procedure is applied on both linear and nonlinear boundary-value problems. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is simple, effective and also is in good agreement with the previous result and exact solution as showing in the tables and figures.

show abstract

“…, 2012), Runge–Kutta (R–K) methods (Hussain et al. , 2016, 2017; You and Chen, 2013), finite difference methods (Chawla and Katti, 1979), classical orthogonal polynomials (Legendre polynomials, Chebyshev polynomials and Jacobi polynomials), wavelet methods (Haar wavelets (Amin et al. , 2021; Islam et al.…”

Section: Introductionmentioning

confidence: 99%

“…The numerical solutions on the IVPs and BVPs of high-order ODEs and SODEs have attracted the attention of many scholars. Most of these investigations are focused on two main aspects: one is to simplify the higher order ODE into an equivalent system of first-order ODEs, which involves a lot of computer time and more human effort and the other is to solve directly high-order ODEs, such as block methods (Majid et al, 2012), Runge-Kutta (R-K) methods (Hussain et al, 2016(Hussain et al, , 2017You and Chen, 2013), finite difference methods (Chawla and Katti, 1979), classical orthogonal polynomials (Legendre polynomials, Chebyshev polynomials and Jacobi polynomials), wavelet methods (Haar wavelets (Amin et al, 2021;Islam et al, 2010), Shannon wavelets (Shi and Li, 2014)), Adomian decomposition method with Green's function (Al-Hayani, 2011), spectral algorithms (Doha and Abd-Elhameed, 2009;Doha and Bhrawy, 2002;Doha et al, 2012) etc.…”

mentioning

confidence: 99%

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Weng

Sun

2022

View full text Add to dashboard Cite

PurposeThis paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).Design/methodology/approachThe proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.FindingsTo demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.Originality/valueThe H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

show abstract

Efficient Numerical Algorithm for the Solution of Eight Order Boundary Value Problems by Haar Wavelet Method

Cited by 10 publications

References 30 publications

Application of Wavelet Methods in Computational Physics

Application of Wavelet Methods in Computational Physics

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

Contact Info

Product

Resources

About