Received: 30 January 2018 DOI: 10.1002/mma.5375 RESEARCH ARTICLE Stability analysis and a numerical scheme for fractional Klein-Gordon equations Hasib Khan 1,2 Aziz Khan 3 Wen Chen 1 Kamal Shah 4 1 College of Engineering, Mechanics and Materials, Hohai University, Nanjing, China 2 Department of Mathematics, Shaheed Benazir Bhutto University Sheringal, Upper Dir District, Pakistan 3 Department of Mathematics, University of Peshawar, Peshawar, Pakistan 4 Department of Mathematics, University of Malakand, Lower Dir, Pakistan Correspondence Wen Chen, College of Engineering, Mechanics and Materials, Hohai University, Nanjing, China. Email: chenwen@hhu.edu.cn Communicated by: S. Messaoudi Funding information China Government Yong Talent Program MSC Classification: 34A25; 34A34; 34A45 Fractional order nonlinear Klein-Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order non- linear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications. KEYWORDS Adomian decomposition, Caputo's fractional derivative, Klein-Gordon equation, Sumudu transform 1 INTRODUCTION In last few decades, analysis, computation, error estimation, and stabilities of fractional differential equations (FDEs) have gained a considerable attention of researchers in different fields. Several physical and biological problems were mathemat- ically modeled via FDEs, which gave high quality accuracy than the models by integer order differential equations. One can see in the references of the paper some useful applications of FDEs in viscoelasticity, bioengineering, damping laws, rheology, thermodynamics, synchronization, dynamical system, electrical circuits, signal processing, and fluid mechan- ics, see. 1-13 The fractional order Klein-Gordon equation (KGE) is derived from the KGE of the integer order by switching time derivative by noninteger order ( ∈[0, 1]) derivative. The fractional order KGE can be illustrated as below:   t  (x, t)=  2 x  (x, t)+  (x, t)+ ( (x, t)) = (x, t), t ≥ 0, (1) with conditions  t  (x, 0)= g 1 (x),  (x, 0)= g 0 (x), where (x, t), g 0 (x), g 1 (x) are analytical functions, ,  are constants, ( ) is a nonlinear function and  is a function of x and t to be determined. The nonlinear KGEs are arising from quantum mechanics and classical relativistic, such type Math Meth Appl Sci. 2018;1–10. wileyonlinelibrary.com/journal/mma © 2018 John Wiley & Sons, Ltd. 1