An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials
The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satis...
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MDPI AG
2025-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/2/78 |
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| author | Youssri Hassan Youssri Waleed Mohamed Abd-Elhameed Amr Ahmed Elmasry Ahmed Gamal Atta |
| author_facet | Youssri Hassan Youssri Waleed Mohamed Abd-Elhameed Amr Ahmed Elmasry Ahmed Gamal Atta |
| author_sort | Youssri Hassan Youssri |
| collection | DOAJ |
| description | The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. |
| format | Article |
| id | doaj-art-ddbda5e0d1944b9d9da4e14010c048f1 |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-ddbda5e0d1944b9d9da4e14010c048f12025-08-20T02:44:56ZengMDPI AGFractal and Fractional2504-31102025-01-01927810.3390/fractalfract9020078An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev PolynomialsYoussri Hassan Youssri0Waleed Mohamed Abd-Elhameed1Amr Ahmed Elmasry2Ahmed Gamal Atta3Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptFaculty of Engineering, Egypt University of Informatics, Knowledge City, New Administrative Capital 19519, EgyptDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, EgyptThe current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques.https://www.mdpi.com/2504-3110/9/2/78Euler–Bernoulli beam equationPetrov–Galerkin methodspectral solutionpinned–pinned boundary conditionsclamped–clamped boundary conditionsconvergence analysis |
| spellingShingle | Youssri Hassan Youssri Waleed Mohamed Abd-Elhameed Amr Ahmed Elmasry Ahmed Gamal Atta An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials Fractal and Fractional Euler–Bernoulli beam equation Petrov–Galerkin method spectral solution pinned–pinned boundary conditions clamped–clamped boundary conditions convergence analysis |
| title | An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials |
| title_full | An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials |
| title_fullStr | An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials |
| title_full_unstemmed | An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials |
| title_short | An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials |
| title_sort | efficient petrov galerkin scheme for the euler bernoulli beam equation via second kind chebyshev polynomials |
| topic | Euler–Bernoulli beam equation Petrov–Galerkin method spectral solution pinned–pinned boundary conditions clamped–clamped boundary conditions convergence analysis |
| url | https://www.mdpi.com/2504-3110/9/2/78 |
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