New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
Fractional calculus and fractional differential equations (FDE) have many applications in different branches of sciences. But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on general...
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Online Access: | https://doi.org/10.1515/nleng-2018-0095 |
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doaj-9afb61c01f5741bfbedac117c9219d852021-09-06T19:21:07ZengDe GruyterNonlinear Engineering2192-80102192-80292019-01-018143844810.1515/nleng-2018-0095nleng-2018-0095New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kindParand K.0Nikarya M.1Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran, Member of research group of Scientific Computing.Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, IranFractional calculus and fractional differential equations (FDE) have many applications in different branches of sciences. But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on generalized Bessel function of the first kind (GBF), spectral methods and Newton–Krylov subspace method to solve nonlinear FDEs. In this paper, we use the GBFs as the basis functions. Then, we introduce explicit formulas to calculate Riemann–Liouville fractional integral and derivative of GBFs that are very helpful in computation and saving time. In the presented method, a nonlinear FDE will be converted to a nonlinear system of algebraic equations using collocation method based on GBF, then the solution of this nonlinear algebraic system will be achieved by using Newton-generalized minimum residual (Newton–Krylov) method. To illustrate the reliability and efficiency of the proposed method, we apply it to solve some examples of nonlinear Abel FDE.https://doi.org/10.1515/nleng-2018-0095first kind of nonlinear abel fdegeneralized bessel functionsspectral collocation methodnonlinear system of equationsnewton–krylov methods |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Parand K. Nikarya M. |
spellingShingle |
Parand K. Nikarya M. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind Nonlinear Engineering first kind of nonlinear abel fde generalized bessel functions spectral collocation method nonlinear system of equations newton–krylov methods |
author_facet |
Parand K. Nikarya M. |
author_sort |
Parand K. |
title |
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind |
title_short |
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind |
title_full |
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind |
title_fullStr |
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind |
title_full_unstemmed |
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind |
title_sort |
new numerical method based on generalized bessel function to solve nonlinear abel fractional differential equation of the first kind |
publisher |
De Gruyter |
series |
Nonlinear Engineering |
issn |
2192-8010 2192-8029 |
publishDate |
2019-01-01 |
description |
Fractional calculus and fractional differential equations (FDE) have many applications in different branches of sciences. But, often a real nonlinear FDE has not the exact or analytical solution and must be solved numerically. Therefore, we aim to introduce a new numerical algorithm based on generalized Bessel function of the first kind (GBF), spectral methods and Newton–Krylov subspace method to solve nonlinear FDEs. In this paper, we use the GBFs as the basis functions. Then, we introduce explicit formulas to calculate Riemann–Liouville fractional integral and derivative of GBFs that are very helpful in computation and saving time. In the presented method, a nonlinear FDE will be converted to a nonlinear system of algebraic equations using collocation method based on GBF, then the solution of this nonlinear algebraic system will be achieved by using Newton-generalized minimum residual (Newton–Krylov) method. To illustrate the reliability and efficiency of the proposed method, we apply it to solve some examples of nonlinear Abel FDE. |
topic |
first kind of nonlinear abel fde generalized bessel functions spectral collocation method nonlinear system of equations newton–krylov methods |
url |
https://doi.org/10.1515/nleng-2018-0095 |
work_keys_str_mv |
AT parandk newnumericalmethodbasedongeneralizedbesselfunctiontosolvenonlinearabelfractionaldifferentialequationofthefirstkind AT nikaryam newnumericalmethodbasedongeneralizedbesselfunctiontosolvenonlinearabelfractionaldifferentialequationofthefirstkind |
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