Generalized Bessel quasilinearization technique applied to Bratu and Lane-Emden-type equations of arbitrary order
Date
2021
Authors
Izadi, Mohammad
Srivastava, H.M.
Journal Title
Journal ISSN
Volume Title
Publisher
Fractal and Fractional
Abstract
The ultimate goal of this study is to develop a numerically effective approximation technique
to acquire numerical solutions of the integer and fractional-order Bratu and the singular
Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary
conditions were considered and the fractional derivative being considered in the Liouville–Caputo
sense. In the direct approach, the generalized Bessel matrix method based on collocation points was
utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the
technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered
model problems. Consequently, the quasilinearization method was utilized to transform the original
nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation
scheme was employed to solve the resulting linear equations iteratively. In particular, to convert
the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the
derivative of the basis functions is presented. The error analysis of the quasilinear approach is
also discussed. The effectiveness of the present linearized approach is illustrated through several
simulations with some test examples. Comparisons with existing well-known schemes revealed that
the presented technique is an easy-to-implement method while being very effective and convenient
for the nonlinear Bratu and Lane–Emden equations.
Description
Keywords
Bessel functions, Bratu's problem, collocation method, error analysis, Lane-Emden equation, Liouville-Caputo fractional derivative, quasilinearization technique
Citation
Izadi, M. & Srivastava, H. M. (2021). “Generalized Bessel quasilinearization technique applied to Bratu and Lane-Emden-type equations of arbitrary order.” Fractal and Fractional, 5(4), 179. https://doi.org/10.3390/fractalfract5040179