Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies
A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of th...
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MDPI AG
2023-09-01
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| author | Ezekiel Olaoluwa Omole Emmanuel Oluseye Adeyefa Victoria Iyadunni Ayodele Ali Shokri Yuanheng Wang |
| author_facet | Ezekiel Olaoluwa Omole Emmanuel Oluseye Adeyefa Victoria Iyadunni Ayodele Ali Shokri Yuanheng Wang |
| author_sort | Ezekiel Olaoluwa Omole |
| collection | DOAJ |
| description | A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples. |
| first_indexed | 2024-03-10T23:03:01Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:03:01Z |
| publishDate | 2023-09-01 |
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| spelling | doaj.art-e7c734b02ae34570a9189f885831e0112023-11-19T09:33:08ZengMDPI AGAxioms2075-16802023-09-0112989110.3390/axioms12090891Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation StrategiesEzekiel Olaoluwa Omole0Emmanuel Oluseye Adeyefa1Victoria Iyadunni Ayodele2Ali Shokri3Yuanheng Wang4Department of Mathematics, Federal University Oye-Ekiti, P.M.B. 373, Oye-Ekiti 370112, Ekiti State, NigeriaDepartment of Mathematics, Federal University Oye-Ekiti, P.M.B. 373, Oye-Ekiti 370112, Ekiti State, NigeriaDepartment of Computer Science and Mathematics, Nigeria Police Academy, Wudil-Kano 713101, Kano State, NigeriaDepartment of Mathematics, Faculty of Sciences, University of Maragheh, Maragheh 83111-55181, IranCollege of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, ChinaA computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.https://www.mdpi.com/2075-1680/12/9/891multistep collocation formulasconvergence analysis of the formulasorthogonal approximating functionconvection diffusion reaction equationsfluid dynamics problemspartial differential equations |
| spellingShingle | Ezekiel Olaoluwa Omole Emmanuel Oluseye Adeyefa Victoria Iyadunni Ayodele Ali Shokri Yuanheng Wang Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies Axioms multistep collocation formulas convergence analysis of the formulas orthogonal approximating function convection diffusion reaction equations fluid dynamics problems partial differential equations |
| title | Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies |
| title_full | Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies |
| title_fullStr | Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies |
| title_full_unstemmed | Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies |
| title_short | Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies |
| title_sort | ninth order multistep collocation formulas for solving models of pdes arising in fluid dynamics design and implementation strategies |
| topic | multistep collocation formulas convergence analysis of the formulas orthogonal approximating function convection diffusion reaction equations fluid dynamics problems partial differential equations |
| url | https://www.mdpi.com/2075-1680/12/9/891 |
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