A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation

Abstract

In this paper, a novel approach comprising the modified decomposition method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation utt-RDxαu+sinu=0 where RDxα is the Riesz space fractional derivative, 1≤α≤2. For α = 2, it becomes classical sine-Gordon equation utt − uxx + sin u = 0 and corresponding to α = 1, it becomes nonlocal sine-Gordon equation utt − Hu + sin u = 0 which arises in Josephson junction theory, where H is the Hilbert transform. The fractional sine-Gordon equation is considered as an interpolation between the classical sine-Gordon equation (corresponding to α = 2) and nonlocal sine-Gordon equation (corresponding to α = 1). Here the analytic solution of fractional sine-Gordon equation is derived by using the modified decomposition method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

Keywords

• Modified decomposition method with Fourier transform (MDM-FT)
• Riesz space fractional derivative
• Nonlocal sine-Gordon equation
• Hurwitz zeta function
• Riemann zeta function
• Generalized zeta function