Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials
This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fow...
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MDPI AG
2023-01-01
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| author | Waleed Mohamed Abd-Elhameed Mohamed Salem Al-Harbi Amr Kamel Amin Hany M. Ahmed |
| author_facet | Waleed Mohamed Abd-Elhameed Mohamed Salem Al-Harbi Amr Kamel Amin Hany M. Ahmed |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>5</mn></msub><msub><mi>F</mi><mn>4</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms. |
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| language | English |
| last_indexed | 2024-03-11T09:09:51Z |
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| spelling | doaj.art-4ea6b7b9abce40d6a39e5f964ca27acd2023-11-16T19:05:13ZengMDPI AGAxioms2075-16802023-01-011229910.3390/axioms12020099Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev PolynomialsWaleed Mohamed Abd-Elhameed0Mohamed Salem Al-Harbi1Amr Kamel Amin2Hany M. Ahmed3Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, College of Science, University of Jeddah, Jeddah 23218, Saudi ArabiaDepartment of Basic Sciences, Adham University College, Umm AL-Qura University, Makkah 28653, Saudi ArabiaDepartment of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 12612, EgyptThis paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>5</mn></msub><msub><mi>F</mi><mn>4</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms.https://www.mdpi.com/2075-1680/12/2/99Chebyshev polynomialsmodified Chebyshev polynomialsinitial value problemssingular equationsconvergence analysis |
| spellingShingle | Waleed Mohamed Abd-Elhameed Mohamed Salem Al-Harbi Amr Kamel Amin Hany M. Ahmed Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials Axioms Chebyshev polynomials modified Chebyshev polynomials initial value problems singular equations convergence analysis |
| title | Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials |
| title_full | Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials |
| title_fullStr | Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials |
| title_full_unstemmed | Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials |
| title_short | Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials |
| title_sort | spectral treatment of high order emden fowler equations based on modified chebyshev polynomials |
| topic | Chebyshev polynomials modified Chebyshev polynomials initial value problems singular equations convergence analysis |
| url | https://www.mdpi.com/2075-1680/12/2/99 |
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