A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions
This article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two...
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| Format: | Article |
| Language: | English |
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Elsevier
2024-01-01
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| Series: | Alexandria Engineering Journal |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110016823010566 |
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| author | Muhammad Ahsan Weidong Lei Amir Ali Khan Masood Ahmed Maher Alwuthaynani Ayesha Amjad |
| author_facet | Muhammad Ahsan Weidong Lei Amir Ali Khan Masood Ahmed Maher Alwuthaynani Ayesha Amjad |
| author_sort | Muhammad Ahsan |
| collection | DOAJ |
| description | This article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two-point boundary conditions and two-point integral boundary conditions. Managing these types of boundary conditions can be challenging in numerical methods. However, in this study, these types of equations are handled in a simple manner using the Haar wavelet expressions, as provided in the given information. In the case of nonlinear problems, the quasi-linearization technique is introduced to linearize the equation. Nonlinear fourth-order differential equations are transformed into a simple linear system of algebraic equations using the quasi-linearization technique and Haar wavelets. These equations are then solved very easily to find the solution of the differential equations. The convergence rate and stability of both the methods are studied in details. The convergence rate of the proposed HCMHW is faster than the CMHW (2+2s>2,s=1,2…). Some of the examples are given to indicate the better performance and accuracy of the proposed HCMHW. |
| first_indexed | 2024-03-08T11:54:16Z |
| format | Article |
| id | doaj.art-3ad935ef4a27463f8ef110becc07a2bb |
| institution | Directory Open Access Journal |
| issn | 1110-0168 |
| language | English |
| last_indexed | 2024-03-08T11:54:16Z |
| publishDate | 2024-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Alexandria Engineering Journal |
| spelling | doaj.art-3ad935ef4a27463f8ef110becc07a2bb2024-01-24T05:17:15ZengElsevierAlexandria Engineering Journal1110-01682024-01-0186230242A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditionsMuhammad Ahsan0Weidong Lei1Amir Ali Khan2Masood Ahmed3Maher Alwuthaynani4Ayesha Amjad5School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, China; Department of Mathematics, University of Swabi, Swabi, Pakistan; Corresponding author.School of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen, 518055, ChinaDepartment of Mathematics, University of Swabi, Swabi, PakistanElementary & Secondary Education Department, Khyber Pakhtunkhwa, 19110, PakistanDepartment of Mathematics, College of Khurma University College, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaFaculty of Organization and Management, Silesian University of Technology, Gliwice 44-100, Poland; Centre for Mechanical Engineering, Materials and Processes (CEMMPRE) University of Coimbra, Polo II, Coimbra, 3030-788, PortugalThis article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two-point boundary conditions and two-point integral boundary conditions. Managing these types of boundary conditions can be challenging in numerical methods. However, in this study, these types of equations are handled in a simple manner using the Haar wavelet expressions, as provided in the given information. In the case of nonlinear problems, the quasi-linearization technique is introduced to linearize the equation. Nonlinear fourth-order differential equations are transformed into a simple linear system of algebraic equations using the quasi-linearization technique and Haar wavelets. These equations are then solved very easily to find the solution of the differential equations. The convergence rate and stability of both the methods are studied in details. The convergence rate of the proposed HCMHW is faster than the CMHW (2+2s>2,s=1,2…). Some of the examples are given to indicate the better performance and accuracy of the proposed HCMHW.http://www.sciencedirect.com/science/article/pii/S1110016823010566Nonlinear differential equationHaar waveletCollocation methodQuasi-linearization approach |
| spellingShingle | Muhammad Ahsan Weidong Lei Amir Ali Khan Masood Ahmed Maher Alwuthaynani Ayesha Amjad A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions Alexandria Engineering Journal Nonlinear differential equation Haar wavelet Collocation method Quasi-linearization approach |
| title | A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions |
| title_full | A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions |
| title_fullStr | A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions |
| title_full_unstemmed | A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions |
| title_short | A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions |
| title_sort | higher order collocation technique based on haar wavelets for fourth order nonlinear differential equations having nonlocal integral boundary conditions |
| topic | Nonlinear differential equation Haar wavelet Collocation method Quasi-linearization approach |
| url | http://www.sciencedirect.com/science/article/pii/S1110016823010566 |
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