Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems
In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence <inline-formula> <math display=&qu...
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MDPI AG
2020-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/3/452 |
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| author | Giro Candelario Alicia Cordero Juan R. Torregrosa |
| author_facet | Giro Candelario Alicia Cordero Juan R. Torregrosa |
| author_sort | Giro Candelario |
| collection | DOAJ |
| description | In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare it with the existing fractional Newton method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </semantics> </math> </inline-formula>. Moreover, we also introduce a multipoint fractional Traub-type method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the first step of the class) and classical Traub’s scheme (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages. |
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| last_indexed | 2024-12-13T06:32:58Z |
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| spelling | doaj.art-63cb0eb4ccc24841888d035ec7cd00cf2022-12-21T23:56:33ZengMDPI AGMathematics2227-73902020-03-018345210.3390/math8030452math8030452Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear ProblemsGiro Candelario0Alicia Cordero1Juan R. Torregrosa2Área de Ciencias Básicas y Ambientales, Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican RepublicInstitute for Multidisciplinary Mathematics, Universitat Politècnica de Valenència, Camino de Vera s/n, 46022 València, SpainInstitute for Multidisciplinary Mathematics, Universitat Politècnica de Valenència, Camino de Vera s/n, 46022 València, SpainIn the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare it with the existing fractional Newton method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </semantics> </math> </inline-formula>. Moreover, we also introduce a multipoint fractional Traub-type method with order <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of the first step of the class) and classical Traub’s scheme (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub’s methods do not converge and the proposed methods do, among other advantages.https://www.mdpi.com/2227-7390/8/3/452nonlinear equationsfractional derivativesmultistep methodsconvergencestability |
| spellingShingle | Giro Candelario Alicia Cordero Juan R. Torregrosa Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems Mathematics nonlinear equations fractional derivatives multistep methods convergence stability |
| title | Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems |
| title_full | Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems |
| title_fullStr | Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems |
| title_full_unstemmed | Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems |
| title_short | Multipoint Fractional Iterative Methods with (2<i>α</i> + 1)th-Order of Convergence for Solving Nonlinear Problems |
| title_sort | multipoint fractional iterative methods with 2 i α i 1 th order of convergence for solving nonlinear problems |
| topic | nonlinear equations fractional derivatives multistep methods convergence stability |
| url | https://www.mdpi.com/2227-7390/8/3/452 |
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