An Analytical Technique to Solve the System of Non-Linear Korteweg-De Vries Equation (KdV), FPDE

Bachelor Thesis from the year 2021 in the subject Mathematics - Miscellaneous, Federal Urdu University, language: English, abstract: Dive into the complex world of nonlinear dynamics with a groundbreaking exploration of fractional calculus and its application to solving some of the most challenging equations in mathematical physics. This book introduces a novel analytical method, a beacon of innovation in the field of fractional partial differential equations (FPDEs), specifically targeting the elusive solutions of nonlinear fractional Korteweg-de Vries (KdV) systems. By masterfully weaving together the classical power of the Laplace transform with a fresh, cutting-edge analytical approach, this study unveils a new pathway for understanding and solving these intricate equations. At the heart of this methodology lies the Caputo fractional derivative, a cornerstone of fractional calculus, enabling a more accurate and nuanced modeling of real-world phenomena. The journey begins with a comprehensive introduction to fractional calculus, contrasting it with its integer-order counterpart and tracing its historical roots from the early inquiries of Leibniz and L'Hopital to the pivotal contributions of Laplace and Liouville. Discover how the KdV equation, a fundamental model for solitary waves, finds new life through the lens of fractional calculus. This book meticulously constructs the theoretical framework, defining essential mathematical tools such as the Gamma function and rigorously establishing the properties of the Laplace transform. A detailed convergence analysis provides a solid foundation for the practical application of this method. Explore worked examples that showcase the method's efficacy and illuminate the path for researchers and students alike. This book is an invaluable resource for those seeking to push the boundaries of knowledge in fractional calculus, nonlinear systems, and the development of novel analytical techniques for solving FPDEs. Unlock the secrets of fractional dynamics and embark on a journey of mathematical discovery, where the power of the Laplace transform meets the elegance of a new analytical method, offering a fresh perspective on the Korteweg-de Vries equation and its myriad applications. A must-read for mathematicians, physicists, and engineers venturing into the frontier of fractional calculus and nonlinear dynamics.