CUBIC-BSPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTIONS OF THE NONLINEAR FRACTIONAL ORDER KLEIN–GORDON EQUATION

CUBIC-BSPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTIONS OF THE NONLINEAR FRACTIONAL ORDER KLEIN–GORDON EQUATION

J. Damirchi, S. Shagholi, S. Foadian

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Abstract

This research paper focuses on utilizing the cubic-Bspline collocation method to obtain numerical solutions for the time-fractional nonlinear Klein–Gordon (TFNKG) equation. The Klein–Gordon (KG) equation, which characterizes nonlinear wave propagation, is extended by replacing the time derivative in Caputo sense of order derivative of order α, (1 < α ≤ 2). The L2 discretization formula is employed to approximate the time-fractional derivative. The spatial variable is discretized using cubic B-spline basis functions, and the nonlinear terms are linearized using the quasilinearization technique. Through the proposed method, the main problem is transformed into a more computationally manageable problem. Numerical examples involving different types of nonlinearities are tested to demonstrate the accuracy of the developed scheme. The simulations confirm the high accuracy of the proposed method when compared to analytical solutions, as well as other methods such as the Sinc-Chebyshev collocation method (SCCM) and the variational iteration method (VIM). The accuracy of the developed scheme is also evaluated using error norms L∞ and L2. The research findings of this study substantiate the efficacy and credibility of the proposed methodologies in the analysis of fractional differential equations.

Keywords

Fractional Klein–Gordon equation, Caputo derivative, Cubic-Bspline method, Quasilinearization.