Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

Abstract

The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

Existence, uniqueness, and galerkin shifted Legendre's approximation of time delays integrodifferential models by adapting the Hilfer fractional attitude

“…Practically most of the pre-existing approaches have several strengths and limitations, and the choice of the scheme used depends on the specific shape concerning the problem. Likewise, one of the significant advantages of this algorithm is its capability of handling both linear/nonlinear fractional models, resulting in an approximation solution that closely matches the exact one exploiting only a few OSLPs, and the GSLM follows a straightforward numerical procedure, making it easy to obtain the required approximation solution [ 25 , 26 , [38] , [39] , [40] ].…”

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

“…Nevertheless, new reliable mathematical algorithms including numerical ones, solitons, and approximations via polynomials and wavelets are found more effective. Several attempts have been made to handle nonlinear problems numerically and a wide range of these efforts have been reported earlier in literature [6][7][8][9][10][11][12]. The approximation through polynomials and wavelet-based algorithms is an emerging area and has attracted the attention from research community.…”

“…The listed techniques cover some areas from chemical physics, quantum chemistry, and many engineering disciplines as well but still need further investigations to make progress in other fields of science. Studies in the recent past [6–8, 10, 18–23] reveal that wavelets methods are more precise in numerous situations to tackle problems containing higher nonlinearity factor. Alongside, various attempts [11, 14, 24–27] in order to improve the efficiency and accuracy levels of these methods have been reported, but such methods also carry certain disadvantages that need to be overcome.…”

“…Alongside, various attempts [11, 14, 24–27] in order to improve the efficiency and accuracy levels of these methods have been reported, but such methods also carry certain disadvantages that need to be overcome. Different wavelet methods; named as Chebyshev, CAS (cosine and sine), Gegenbauer, Haar, Laguerre, Legendre and so on, have been proposed or extended to attain approximate solutions for dynamical or other relevant physical problems [6–12, 20, 28].…”