Review of Adams-Bashforth method for numerical solution of first order ordinary differential equations

This project work presents a comprehensive review of the Adams-Bashforth method for the numerical solution of explicit first-order ordinary differential equations (ODEs). The study begins with a historical overview of the development of ordinary differential equations (ODEs), tracing back to the seminal works of Newton, Leibniz, and their contemporaries. The evolution of differential equations as a distinct mathematical discipline and their wide-ranging applications across various fields are discussed. Furthermore, the paper provides an in-depth analysis of the Adams-Bashforth method, a prominent numerical technique for solving ODEs. The method is derived and analyzed using Mathematica Software, demonstrating its efficacy and accuracy in approximating solutions to ODEs. A comparison with existing methods such as the Euler method and the Runge-Kutta method highlights the advantages of the Adams-Bashforth method, particularly in terms of computational efficiency and accuracy. Moreover, fundamental concepts related to ODEs, including the initial value problem, existence, and uniqueness of solutions, are explored in the context of numerical solution methods. The properties of linear multistep methods and their relevance to solving ODEs efficiently are also discussed. In conclusion, this project work provides valuable insights into the Adams-Bashforth method as a powerful tool for the numerical solution of first-order ODEs. By leveraging modern computational tools such as Mathematica Software, the method offers a practical and efficient approach to solving complex differential equations encountered in various scientific and engineering applications.