AUTHOR=Kumar Sachin , Baleanu Dumitru 

TITLE=A New Numerical Method for Time Fractional Non-linear Sharma-Tasso-Oliver Equation and Klein-Gordon Equation With Exponential Kernel Law

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00136

DOI=10.3389/fphy.2020.00136

ISSN=2296-424X

ABSTRACT=<p>In this work, we derived a novel numerical scheme to find out the numerical solution of fractional PDEs having Caputo-Fabrizio (C-F) fractional derivatives. We first find out the formula of approximation for the C-F derivative of the function <italic>f</italic>(<italic>t</italic>) = <italic>t</italic><sup><italic>k</italic></sup>. We approximate the C-F derivative in time direction with the help of Legendre spectral method and approximation formula of <italic>t</italic><sup><italic>k</italic></sup>. The unknown function and their derivatives in spatial direction are approximated with the help of the method which is based on a quasi wavelet. We implement this newly derived method to solve the non-linear Sharma-Tasso-Oliver equation and non-linear Klein-Gordon equation in which time-fractional derivative is of C-F type. The accuracy and validity of this new method are depicted by giving the numerical solution of some numerical examples. The numerical results for the particular cases of Klein-Gordon equation are compared with the existing exact solutions and from the obtained error we can conclude that our proposed numerical method achieves accurate results. The effect of time-fractional exponent α on the solution profile is characterized by figures. The comparison of solution profile <italic>u</italic>(<italic>x, t</italic>) for different type time-fractional derivative (C-F vs. Caputo) is depicted by figures.</p>