Enhancing the Accuracy of Solving Riccati Fractional Differential Equations
In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing <i>x</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inlin...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/5/275 |
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| author | Antonela Toma Flavius Dragoi Octavian Postavaru |
| author_facet | Antonela Toma Flavius Dragoi Octavian Postavaru |
| author_sort | Antonela Toma |
| collection | DOAJ |
| description | In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing <i>x</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>x</mi><mi>α</mi></msup></semantics></math></inline-formula>, with positive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T03:52:39Z |
| format | Article |
| id | doaj.art-dba013c3d49445d1bc5d78044bb048d7 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T03:52:39Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-dba013c3d49445d1bc5d78044bb048d72023-11-23T11:03:52ZengMDPI AGFractal and Fractional2504-31102022-05-016527510.3390/fractalfract6050275Enhancing the Accuracy of Solving Riccati Fractional Differential EquationsAntonela Toma0Flavius Dragoi1Octavian Postavaru2Center for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnica of Bucharest, 060042 Bucharest, RomaniaCenter for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnica of Bucharest, 060042 Bucharest, RomaniaCenter for Research and Training in Innovative Techniques of Applied Mathematics in Engineering, University Politehnica of Bucharest, 060042 Bucharest, RomaniaIn this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing <i>x</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>x</mi><mi>α</mi></msup></semantics></math></inline-formula>, with positive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2504-3110/6/5/275hybrid functionsCaputo derivativeRiemann–Liouville integral |
| spellingShingle | Antonela Toma Flavius Dragoi Octavian Postavaru Enhancing the Accuracy of Solving Riccati Fractional Differential Equations Fractal and Fractional hybrid functions Caputo derivative Riemann–Liouville integral |
| title | Enhancing the Accuracy of Solving Riccati Fractional Differential Equations |
| title_full | Enhancing the Accuracy of Solving Riccati Fractional Differential Equations |
| title_fullStr | Enhancing the Accuracy of Solving Riccati Fractional Differential Equations |
| title_full_unstemmed | Enhancing the Accuracy of Solving Riccati Fractional Differential Equations |
| title_short | Enhancing the Accuracy of Solving Riccati Fractional Differential Equations |
| title_sort | enhancing the accuracy of solving riccati fractional differential equations |
| topic | hybrid functions Caputo derivative Riemann–Liouville integral |
| url | https://www.mdpi.com/2504-3110/6/5/275 |
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