Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation

In the present article, we study a new version of the physical model, namely the Klein -Gordon equation involved with tempered fractional derivative. This model is numerically simulated for the tempered and the non-tempered fractional derivatives. Two-dimensional Gegenbauer Wavelets are implemented for the solution of the tempered fractional Klein-Gordon equation. The proposed method is a combination of different operational matrices of the two-dimensional Gegenbauer wavelets and the collocation nodes. The applicability and the accuracy of the suggested method has been examined by four illustrative examples and the obtained simulation results are in good agreement with the exact solution. The maximum absolute error, L-2 norm error, and the root mean square error have been evaluated to analyze the precision of the proposed method. This method gives better and comparable results to the existing methods given in the literature. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

In this article, a new version of the physical model, involving tempered fractional derivatives in the Klein-Gordon equation, was studied. The solution of the tempered fractional Klein-Gordon equation was obtained using two-dimensional Gegenbauer Wavelets. The proposed method, combining operational matrices of the Gegenbauer wavelets and collocation nodes, exhibited good applicability and accuracy as demonstrated by numerical simulations and comparison with the exact solution.