A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions
This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/3872 |
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| author | Mohammed M. Al-Shamiri V. Rexma Sherine G. Britto Antony Xavier D. Saraswathi T. G. Gerly P. Chellamani Manal Z. M. Abdalla N. Avinash M. Abisha |
| author_facet | Mohammed M. Al-Shamiri V. Rexma Sherine G. Britto Antony Xavier D. Saraswathi T. G. Gerly P. Chellamani Manal Z. M. Abdalla N. Avinash M. Abisha |
| author_sort | Mohammed M. Al-Shamiri |
| collection | DOAJ |
| description | This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>th-fractional sum of a function <i>f</i> has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to <i>n</i>th-sum for a class of delta integrable functions, that is, functions with both discrete integration and <i>n</i>th-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing <i>∞</i>-order delta integrable functions, the discrete integration related to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>th-fractional sum of <i>f</i> can be obtained by applying Newton’s formula. In addition, this concept is extended to <i>h</i>-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions. |
| first_indexed | 2024-03-10T22:29:34Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:29:34Z |
| publishDate | 2023-09-01 |
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| series | Mathematics |
| spelling | doaj.art-0101f6fe2248494c982dd8118d24b8e92023-11-19T11:48:46ZengMDPI AGMathematics2227-73902023-09-011118387210.3390/math11183872A New Approach to Discrete Integration and Its Implications for Delta Integrable FunctionsMohammed M. Al-Shamiri0V. Rexma Sherine1G. Britto Antony Xavier2D. Saraswathi3T. G. Gerly4P. Chellamani5Manal Z. M. Abdalla6N. Avinash7M. Abisha8Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi ArabiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaDepartment of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai 600119, IndiaDepartment of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi ArabiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaDepartment of Mathematics, Sacred Heart College, Tirupattur 635601, IndiaThis research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>th-fractional sum of a function <i>f</i> has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to <i>n</i>th-sum for a class of delta integrable functions, that is, functions with both discrete integration and <i>n</i>th-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing <i>∞</i>-order delta integrable functions, the discrete integration related to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>th-fractional sum of <i>f</i> can be obtained by applying Newton’s formula. In addition, this concept is extended to <i>h</i>-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.https://www.mdpi.com/2227-7390/11/18/3872closed formsummation formNewton’s formuladiscrete integrationdelta integrable functionfractional sum |
| spellingShingle | Mohammed M. Al-Shamiri V. Rexma Sherine G. Britto Antony Xavier D. Saraswathi T. G. Gerly P. Chellamani Manal Z. M. Abdalla N. Avinash M. Abisha A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions Mathematics closed form summation form Newton’s formula discrete integration delta integrable function fractional sum |
| title | A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions |
| title_full | A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions |
| title_fullStr | A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions |
| title_full_unstemmed | A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions |
| title_short | A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions |
| title_sort | new approach to discrete integration and its implications for delta integrable functions |
| topic | closed form summation form Newton’s formula discrete integration delta integrable function fractional sum |
| url | https://www.mdpi.com/2227-7390/11/18/3872 |
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