The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation
Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is ca...
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| Format: | Article |
| Language: | English |
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Universitas Muhammadiyah Mataram
2023-07-01
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| Series: | JTAM (Jurnal Teori dan Aplikasi Matematika) |
| Subjects: | |
| Online Access: | http://journal.ummat.ac.id/index.php/jtam/article/view/14193 |
| _version_ | 1797753826051096576 |
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| author | Muhammad Taufik Abdillah Berlian Setiawaty Sugi Guritman |
| author_facet | Muhammad Taufik Abdillah Berlian Setiawaty Sugi Guritman |
| author_sort | Muhammad Taufik Abdillah |
| collection | DOAJ |
| description | Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used. |
| first_indexed | 2024-03-12T17:23:29Z |
| format | Article |
| id | doaj.art-7e86626cc9124a5e90a457af5ed39db7 |
| institution | Directory Open Access Journal |
| issn | 2597-7512 2614-1175 |
| language | English |
| last_indexed | 2024-03-12T17:23:29Z |
| publishDate | 2023-07-01 |
| publisher | Universitas Muhammadiyah Mataram |
| record_format | Article |
| series | JTAM (Jurnal Teori dan Aplikasi Matematika) |
| spelling | doaj.art-7e86626cc9124a5e90a457af5ed39db72023-08-05T16:03:09ZengUniversitas Muhammadiyah MataramJTAM (Jurnal Teori dan Aplikasi Matematika)2597-75122614-11752023-07-017363164210.31764/jtam.v7i3.141937018The Solution of Generalization of the First and Second Kind of Abel’s Integral EquationMuhammad Taufik Abdillah0Berlian Setiawaty1Sugi Guritman2Departement of Mathematics, Bogor Agricultural UniversityDepartement of Mathematics, Bogor Agricultural UniversityDepartement of Mathematics, Bogor Agricultural UniversityIntegral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used.http://journal.ummat.ac.id/index.php/jtam/article/view/14193fractional calculusgeneralization of abel’s integral equationlaplace transformsuccessive approximations. |
| spellingShingle | Muhammad Taufik Abdillah Berlian Setiawaty Sugi Guritman The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation JTAM (Jurnal Teori dan Aplikasi Matematika) fractional calculus generalization of abel’s integral equation laplace transform successive approximations. |
| title | The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation |
| title_full | The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation |
| title_fullStr | The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation |
| title_full_unstemmed | The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation |
| title_short | The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation |
| title_sort | solution of generalization of the first and second kind of abel s integral equation |
| topic | fractional calculus generalization of abel’s integral equation laplace transform successive approximations. |
| url | http://journal.ummat.ac.id/index.php/jtam/article/view/14193 |
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