Solutions Oftwodimensionalnewell-Whiteheedsegalequationof Time Fractionalorderwithnaturaltransformmethod

Abstract

This thesis delves into the application of the natural transform method for solving the intricate two-dimensional Newell-Whitehead Segal equation of time fractional order. The exploration begins with a thorough examination of the natural transform method, showcasing its efficacy in address ing fundamental mathematical challenges such as one-dimensional time diffusion equations and the nonlinear Burger equation. These foundational problems serve as benchmarks, demonstrating the method’s prowess and preparing the groundwork for addressing the more complex two-dimensional Newell-Whitehead Segal equation. The primary focus of this research is to unravel the complexi ties involved in solving the two-dimensional Newell-Whitehead Segal equation, particularly when incorporating time fractional orders. Through meticulous mathematical derivations and formula tions, the natural transform method emerges as a robust and innovative approach capable of navi gating the intricacies embedded in such equations. The inclusion of test problems further validates the proposed methodology, highlighting its versatility and reliability across various mathematical scenarios. This not only strengthens the theoretical foundation of the natural transform method but also emphasizes its practical utility in addressing real-world challenges. The natural transform method presented in this thesis proves to be a simple yet highly accurate tool for obtaining approx imate and analytical solutions to both fractional order ordinary and partial differential equations. Its efficiency is evident in producing highly accurate solutions with minimal iterations, making it suitable for solving uni- or multi-dimensional problems