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A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions
Author(s) -
Yan Rian,
MA Qiang,
Ding Xiaohua,
Qin Wendi
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6821
Subject(s) - mathematics , uniqueness , lemma (botany) , nonlinear system , convergence (economics) , mathematical analysis , fractional calculus , singular solution , initial value problem , collocation method , smoothing , gronwall's inequality , spectral method , collocation (remote sensing) , differential equation , ordinary differential equation , inequality , physics , remote sensing , quantum mechanics , geology , ecology , statistics , poaceae , economics , biology , economic growth
A general class of nonlinear fractional differential equations is considered. Some sufficient conditions for the existence and uniqueness of exact solution are established by using Weissinger's fixed point theorem and the Gronwall‐Bellman lemma. A spectral collocation method based on the smoothing technique is presented to solve the problem numerically. Then the rigorous error estimates under the L 2 and L ∞ norms are derived. The most remarkable feature of the method is its capability to achieve spectral convergence for weakly singular solutions. Finally, numerical results are given to support the theoretical conclusions with smooth and weakly singular solutions.