Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives

Abstract

We present and analyze two numerical schemes for solving a nonlinear Klein‐Gordon equation with time‐space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier‐spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of scriptOfalse(normalΔt+Nβ−rfalse), and the nonlinearized method converge with the order scriptOfalse(normalΔt3−α+Nβ−rfalse), where normalΔt, N, β, and r are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of u. Some numerical experiments are performed to demonstrate the theoretical results.

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], advection-dispersion process in anomalous diffusion [30][31][32][33][34], and bioprocesses with genetic attribute [35,36]. As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives [73][74][75][76], Caputo-Fabrizio derivatives [77,78], Hilfer derivatives …”