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.
Highlights
Introduction with regard to jurisdictional claims inThis research deals with an effective approximative technique based on (generalized) novel Bessel bases together with the quasilinearization technique to obtain the solution of a class of nonlinear fractional-order differential equations of the form: published maps and institutional affiliations
A novel matrix method in terms of generalized Bessel functions, which is based on some suitable collocation points, was developed for the approximate solutions of integer- and fractional-order nonlinear Bratu and Lane–Emden-type differential equations
By applying the direct Bessel collocation method, the governing equations are transformed into a nonlinear fundamental matrix equation, which may be solved ineffectively for a large number of Bessel functions
Summary
This research deals with an effective approximative technique based on (generalized) novel Bessel bases together with the quasilinearization technique to obtain the solution of a class of nonlinear fractional-order differential equations of the form: published maps and institutional affiliations. Numerical and approximation algorithms are an indispensable tool for the investigation of the solutions of such fractional-order equations. The following approximative and numerical schemes have recently been proposed for Model Problem (a) and closely related problems: the ADM and complex transform methods [19], the factional differential transform approach [20], the Legendre reproducing kernel method [21], the homotopy perturbation transform method [22], the Legendre spectral method [23], and the Laplace decomposition method [24].
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