A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions

One of the earliest attempts to make special relativity and quantum mechanics compatible was the Klein–Gordon equation (KGE). In the 1920s, Oskar Klein and Walter Gordon independently proposed it. This equation plays a very significant role in the fields of quantum mechanics and quantum field theory. This equation has been solved using a variety of techniques. The current work presents a numerical approach that uses cubic B-spline (CBS) functions to solve the KGE involving Atangana–Baleanu fractional derivative. The fractional derivative in time is discretized using a finite difference scheme in the proposed technique, while the space direction is discretized using a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>θ</m:mi> </m:math> θ -weighted scheme based on CBS functions. The Fourier approach is used to prove that the proposed scheme is unconditionally stable. A second-order convergence in the temporal and spatial directions has been proved theoretically and numerically. Finally, four applications of the KGE show the efficiency and accuracy of the proposed method than other methods.