Analytical Approaches to Nonlinear Systems: The Itoh-Kupershmidt Equation via HPM and VIM

Modern approaches like the HPM and the VIM are investigated by researchers because non-linear ODEs and PDEs frequently elude simple analytical solutions. We investigate a well-known non-linear PDE, the Itoh-Kupershmidt equation, which has numerous applications in mathematical physics. Although HPM is not necessarily compact, it is well known for its ability to provide approximations of solutions by series expansion. It is still a helpful tool, though, particularly for non-linear ODE treatments and semi-analytical PDE. The Itoh-Kupershmidt equation has both exact and approximate solutions, which may or may not be in non-compact forms. Our objective is to solve the equation using HPM. Simultaneously, we examine the VIM, a widely used method for reliably and accurately solving a wide range of nonlinear PDEs. Applying VIM to the Itoh-Kupershmidt problem is expected to yield more convenient solutions than HPM. There will be a comparison study to assess how well the HPM and VIM techniques solve the Itoh-Kupershmidt equation. Predictably, both methods should yield promising results demonstrating their effectiveness on a wide range of non-linear PDEs. Additionally, numerical simulations will be performed to show the benefits and drawbacks of each method. Lastly, an accuracy assessment will be performed by contrasting the approaches of the two methods to exact solutions, with particular focus on how well the solutions fulfill the specified boundary criteria. We'll look closely at the underlying causes of any variation or errors. In summary, this work aims to show how effective VIM and HPM are at solving the Itoh-Kupershmidt equation, stressing their applicability to a larger class of non-linear PDEs and providing a critical evaluation of their performance relative to exact solutions and numerical benchmarks.