Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations
The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is str...
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| Language: | English |
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Elsevier
2024-12-01
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| Series: | Alexandria Engineering Journal |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110016824008251 |
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| author | Soheyla Ansari Mohammad Hossein Akrami |
| author_facet | Soheyla Ansari Mohammad Hossein Akrami |
| author_sort | Soheyla Ansari |
| collection | DOAJ |
| description | The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions. |
| format | Article |
| id | doaj-art-d988f30281854774b2be36348f2181e9 |
| institution | Kabale University |
| issn | 1110-0168 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Alexandria Engineering Journal |
| spelling | doaj-art-d988f30281854774b2be36348f2181e92024-11-22T07:36:15ZengElsevierAlexandria Engineering Journal1110-01682024-12-01108261272Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equationsSoheyla Ansari0Mohammad Hossein Akrami1Department of Mathematical Sciences, Yazd University, Yazd, IranCorresponding author.; Department of Mathematical Sciences, Yazd University, Yazd, IranThe main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.http://www.sciencedirect.com/science/article/pii/S1110016824008251Shifted Jacobi polynomialsNumerical simulationsPicard iteration methodGauss–Seidel techniqueCaputo fractional derivative |
| spellingShingle | Soheyla Ansari Mohammad Hossein Akrami Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations Alexandria Engineering Journal Shifted Jacobi polynomials Numerical simulations Picard iteration method Gauss–Seidel technique Caputo fractional derivative |
| title | Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations |
| title_full | Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations |
| title_fullStr | Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations |
| title_full_unstemmed | Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations |
| title_short | Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations |
| title_sort | fractional jacobi picard iteration method using gauss seidel technique for solving a system of nonlinear fractional differential equations |
| topic | Shifted Jacobi polynomials Numerical simulations Picard iteration method Gauss–Seidel technique Caputo fractional derivative |
| url | http://www.sciencedirect.com/science/article/pii/S1110016824008251 |
| work_keys_str_mv | AT soheylaansari fractionaljacobipicarditerationmethodusinggaussseideltechniqueforsolvingasystemofnonlinearfractionaldifferentialequations AT mohammadhosseinakrami fractionaljacobipicarditerationmethodusinggaussseideltechniqueforsolvingasystemofnonlinearfractionaldifferentialequations |