Citation: Sunthrayuth, P.; Naeem, M.; Shah, N.A.; Shah, R.; Chung, J.D. On the Solution of Fractional Biswas–Milovic Model via Analytical Method. Symmetry 2023, 15, 210. https://doi.org/10.3390/ sym15010210 Academic Editor: Dumitru Baleanu Received: 7 December 2022 Revised: 5 January 2023 Accepted: 9 January 2023 Published: 11 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article On the Solution of Fractional Biswas–Milovic Model via Analytical Method Pongsakorn Sunthrayuth 1,† , Muhammad Naeem 2 , Nehad Ali Shah 3,† , Rasool Shah 4 and Jae Dong Chung 3, * 1 Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathumthani 12110, Thailand 2 Department of Mathematics, Deanship of Applied Sciences, Umm Al-Qura University, Makkah 517, Saudi Arabia 3 Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea 4 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan * Correspondence: jdchung@sejong.ac.kr † These authors contributed equally to this work and are co-first authors. Abstract: Through the use of a unique approach, we study the fractional Biswas–Milovic model with Kerr and parabolic law nonlinearities in this paper. The Caputo approach is used to take the fractional derivative. The method employed here is the homotopy perturbation transform method (HPTM), which combines the homotopy perturbation method (HPM) and Yang transform (YT). The HPTM combines the homotopy perturbation method, He’s polynomials, and the Yang transform. He’s polynomial is a wonderful tool for dealing with nonlinear terms. To confirm the validity of each result, the technique was substituted into the equation. The described techniques can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give a precise solution. Graphs are used to show the derived numerical results. The maple software package is used to carry out the numerical simulation work. The results of this research are highly positive and demonstrate how effective the suggested method is for mathematical modeling of natural occurrences. Keywords: Yang transform; homotopy perturbation method; Caputo operator; time-fractional Biswas–Milovic model 1. Introduction Due to its numerous applications in numerous nonlinear phenomena, fractional cal- culus (FC) has gained the attention of academics. To describe the memory and heredity characteristics of many phenomena, FC is a reliable source. The expansion of integer to non- integer order of differentiation is known as fractional differentiation. Few phenomenons including quantum mechanics, viscoelasticity, diffusion processes, fluid mechanics, etc., are effectively described by fractional differential equations (FDEs). FC is connected to practical endeavours and is frequently used in human diseases, nanotechnology, chaos theory, optics, and other disciplines, as noted in Refs. [1–4]. A helpful tool for repre- senting nonlinear events in scientific and engineering models is the fractional differential equation. In applied mathematics and engineering, partial differential equations, particu- larly nonlinear ones, have been utilised to simulate a wide range of scientific phenomena. Fractional-order partial differential equations (FPDEs) allowed researchers to recognise and model a wide range of significant and real-world physical issues in parallel with their work in the physical sciences. It has always been claimed how important it is to obtain approximations for scientists by using either numerical or analytical methods. Because of this, symmetry analysis is a fantastic tool for comprehending partial differential equations, especially when looking at equations generated from mathematical concepts connected Symmetry 2023, 15, 210. https://doi.org/10.3390/sym15010210 https://www.mdpi.com/journal/symmetry