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- Title
New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind.
- Authors
Parand, K.; Nikarya, M.
- Abstract
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.
- Subjects
BESSEL functions; FRACTIONAL differential equations; FRACTIONAL calculus; COLLOCATION methods; NONLINEAR systems
- Publication
Nonlinear Engineering, 2019, Vol 8, Issue 1, p438
- ISSN
2192-8010
- Publication type
Article
- DOI
10.1515/nleng-2018-0095