Residual power series scheme treatments for fractional Klein-Gordon problem arising in soliton theory

The Klein-Gordon problem (KGP) is one of the interesting models that appear in many scientific phenomena. These models are characterized by memory effects, which provide insight into complex phenomena in the fields of physics. In this regard, we propose a new robust algorithm called the confluent Bernoulli approach with residual power series scheme (CBCA-RPSS) to give an approximate solution for the fractional nonlinear KGP. The convergence, uniqueness and error analysis of the proposed method are discussed in detail. A comparison of the numerical results obtained by CBCA-RPSS with the results obtained by some well-known algorithms is presented. Numerical simulations using base errors indicate that CBCA-RPSS is an accurate and efficient technique and thus can be used to solve linear and nonlinear fractional models in physics and engineering. All the numerical results for the studied problems were obtained through implementation codes in Matlab R2017b.