New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations
It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differe...
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Wiley
2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/3821870 |
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| author | Jian Rong Loh Chang Phang Abdulnasir Isah |
| author_facet | Jian Rong Loh Chang Phang Abdulnasir Isah |
| author_sort | Jian Rong Loh |
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| description | It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods. |
| format | Article |
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| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-c5d3fbfa3411435586df157ebe6fc58f2025-02-03T06:00:12ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/38218703821870New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential EquationsJian Rong Loh0Chang Phang1Abdulnasir Isah2Department of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Johor, MalaysiaDepartment of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Johor, MalaysiaDepartment of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Johor, MalaysiaIt is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.http://dx.doi.org/10.1155/2017/3821870 |
| spellingShingle | Jian Rong Loh Chang Phang Abdulnasir Isah New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations Advances in Mathematical Physics |
| title | New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations |
| title_full | New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations |
| title_fullStr | New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations |
| title_full_unstemmed | New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations |
| title_short | New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations |
| title_sort | new operational matrix via genocchi polynomials for solving fredholm volterra fractional integro differential equations |
| url | http://dx.doi.org/10.1155/2017/3821870 |
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