Abstract
Here, the Deoxyribo-Nucleic Acid (DNA) dynamic equation that arises from the oscillator chain named the Peyrard-Bishop model for plenty of solitary wave solutions is presented. The efficacy of newly designed algorithms are investigated, namely, the extended Auxiliary equation method and Kudryashov expansion method for constructing the new solitary wave solutions of the DNAdynamic Peyrard-Bishop model with beta-derivative. Here, the proposed methods contribute to a range of accurate solutions for soliton, including light, dark, and other solutions are obtained. In addition, some results are also clarified by computer simulations demonstrating the uniqueness of our work relative to the existing literature on the classic Peyrard-Bishop model. These solutions lead to the issue of the possibility to expand the method to deal with other non-linear equations of fractional space-time derivatives in non-linear science. It is noted that the newly proposed approach is accurate and is used to create new general closed-form solutions for all other fractional NPDEs.