Appl. Math. Inf. Sci. 14, No. 4, 563-575 (2020) 563 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/140405 Adaptation Of Residual Power Series Approach For Solving Time-Fractional Nonlinear Kline-Gordon Equations With Conformable Derivative Rasha Amryeen 1 , Fatimah Noor Harun 1 , Mohammed Al-Smadi 2,∗ and Azwani Alias 1 1 Pusat Pengajian Informatik & Matematik Gunaan, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia 2 Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan Received: 12 Feb. 2019, Revised: 2 Apr. 2020, Accepted: 18 Apr. 2020 Published online: 1 Jul. 2020 Abstract: In this paper, the time-fractional nonlinear Kline-Gordon equations are considered and solved using the adaptive of residual power series method. The fractional derivative is considered in a conformable sense. Analytical solutions are obtained based on conformable Taylor series expansion by substituting the truncated conformable series solutions to residual error functions. This adaptation can be implemented as a novel alternative technique to handle many nonlinear issues occurring in physics and engineering. Effectiveness, validity, and feasibility of the proposed method are demonstrated by testing some numerical applications. Tabular and graphic results indicate that the method is superior, accurate and appropriate for solving these fractional partial differential models with compatible derivatives. Keywords: Fractional partial differential equations, Klein-Gordon equations, Conformable derivative, Residual errors, Approximate solutions 1 Introduction The theory of fractional calculus has gained much attention in various fields of science and engineering due to vast array of applications and the critical role it plays to describe the complex dynamic behavior of real-world problems such as fluid flow, traffic, biological populations, and diffusion system [[1]-[6]]. The fractional state has many advantages over the classical order, which helps simplify control over the modeling of nature without any lack of genetic characteristics and memory effort. The fractional operator is a powerful mathematical tool that plays an important role in simulating many nonlinear problems, including electrical circuits, electromagnetic waves, damping laws, signal processing, and rheology [[7]-[11]]. The Klein-Gordon equation (KGE) is considered one of the most popular nonlinear partial differential equations that gained much attention in describing relativistic electrons, solitons, quantum, fluid dynamics, and mechanics [[12]-[15]]. It also plays an important role in many other applications, including optics, plasma ions, and solid-state problems [[16]-[18]]. On the other hand, several effective numeric-analytic methods have recently been used to obtain approximate solutions to nonlinear fractional Klein-Gordon equations. For instance, the homotopy perturbation method has been applied for solving a class of nonlinear FKGEs [15]. In [16], the homotopy analysis method has been implemented to approximate solutions of the nonlinear FKGEs. The Riccati expansion method has been employed for solving nonlinear space-time FKGEs [17]. In [18], the modified reduced differential transform method has been introduced for providing numeric solutions for nonlinear space-time FKGEs. However, other categories of advanced numerical methods for different topics can be found in [[19]-[28]]. In this work, we extend the scope of application of the residual power series method in the sense of conformable derivative to construct multiple time-fractional power series solutions to time-fractional nonlinear Klein–Gordon equations with conformable derivative in ∗ Corresponding author e-mail: mhm.smadi@bau.edu.jo c  2020 NSP Natural Sciences Publishing Cor.