TY - JOUR
T1 - Local Gaussian-Collocation Scheme to Approximate the Solution of Nonlinear Fractional Differential Equations Using Volterra Integral Equations
AU - Assari , Pouria
AU - Asadi-Mehregan , Fatemeh
AU - Dehghan , Mehdi
JO - Journal of Computational Mathematics
VL - 2
SP - 261
EP - 282
PY - 2020
DA - 2020/11
SN - 39
DO - http://doi.org/10.4208/jcm.1912-m2019-0072
UR - https://global-sci.org/intro/article_detail/jcm/18374.html
KW - Nonlinear fractional differential equation, Volterra integral equation, Gaussian-collocation method, Meshless method, Error analysis.
AB - <p style="text-align: justify;">This work describes an accurate and effective method for numerically solving a class of
nonlinear fractional differential equations. To start the method, we equivalently convert
these types of differential equations to nonlinear fractional Volterra integral equations of
the second kind by integrating from both sides of them. Afterward, the solution of the
mentioned Volterra integral equations can be estimated using the collocation method based
on locally supported Gaussian functions. The local Gaussian-collocation scheme estimates
the unknown function utilizing a small set of data instead of all points in the solution
domain, so the proposed method uses much less computer memory and volume computing
in comparison with global cases. We apply the composite non-uniform Gauss-Legendre
quadrature formula to estimate singular-fractional integrals in the method. Because of the
fact that the proposed scheme requires no cell structures on the domain, it is a meshless
method. Furthermore, we obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high. Illustrative examples
clearly show the reliability and efficiency of the new technique and confirm the theoretical
error estimates.</p>