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.
Inhaltsverzeichnis (Table of Contents)
- Chapter 1: Introduction
- 1.1 Origin of the fractional derivative
- 1.2 Historical Background
- 1.3 Applications
- Chapter 2: Definitions and Preliminary Notion
- 2.1 The Gamma Function
- Lemma 2.1.1
- 2.2 The Laplace Transform:
- Property 2.2.1
- Property 2.2.2
- 2.3 The Riemann‐Liouville
- 2.4 Caputo fractional derivative
- Linearity Property
- Lemma 2.4.1
- 2.5 Ideal of Fractional Laplace - new novel analytic method
- 2.6 Generating an analytical approach:
- 2.7 Convergence analysis:
- Theorem 2.7.1
- Definition 2.7.2
- Corollary 2.7.3
- 2.8 Some test problems:
- Example 1.
- Example 2.
- Example 3.
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to present a new analytical method for solving systems of nonlinear fractional Korteweg-de Vries (KdV) partial differential equations (FPDEs). The method utilizes the Laplace transform and a novel analytical approach, incorporating the Caputo fractional derivative.
- Fractional Calculus and its applications to nonlinear PDEs.
- The Laplace transform as a tool for solving FPDEs.
- Development and application of a new novel analytical method.
- Convergence analysis of the proposed method.
- Testing the method's efficacy on various nonlinear KdV systems.
Zusammenfassung der Kapitel (Chapter Summaries)
Chapter 1: Introduction: This chapter introduces the concept of fractional calculus, contrasting it with integer-order calculus. It traces the historical development of fractional derivatives, starting with Leibniz and L'Hopital's correspondence and highlighting key contributions from Laplace, Liouville, and others. The chapter also discusses the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation describing solitary waves, and its applications in various scientific fields, emphasizing the importance of efficient methods for solving fractional-order PDEs relevant to these applications.
Chapter 2: Definitions and Preliminary Notion: This chapter lays the groundwork for the novel analytical method by defining essential concepts and properties. It begins by defining the Gamma function, crucial for fractional calculus, and proves a key lemma related to its properties. The Laplace transform and its properties, including convolution, are then established. The chapter proceeds to define the Riemann-Liouville fractional integral and the Caputo fractional derivative, along with their linearity property and other essential lemmas. The core of the chapter introduces the "Laplace - new novel analytic method" which forms the basis for the subsequent analysis and application to several test problems. This method is explained conceptually and its convergence is rigorously analyzed.
Schlüsselwörter (Keywords)
Fractional Calculus, Fractional Partial Differential Equations (FPDEs), Korteweg-de Vries (KdV) equation, Laplace Transform, Caputo derivative, Novel Analytical Method, Convergence Analysis, Nonlinear systems.
Inhaltsverzeichnis (Table of Contents)
- Chapter 1: Introduction
- 1.1 Origin of the fractional derivative
- 1.2 Historical Background
- 1.3 Applications
- Chapter 2: Definitions and Preliminary Notion
- 2.1 The Gamma Function
- Lemma 2.1.1
- 2.2 The Laplace Transform:
- Property 2.2.1
- Property 2.2.2
- 2.3 The Riemann‐Liouville
- 2.4 Caputo fractional derivative
- Linearity Property
- Lemma 2.4.1
- 2.5 Ideal of Fractional Laplace - new novel analytic method
- 2.6 Generating an analytical approach:
- 2.7 Convergence analysis:
- Theorem 2.7.1
- Definition 2.7.2
- Corollary 2.7.3
- 2.8 Some test problems:
- Example 1.
- Example 2.
- Example 3.
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to present a new analytical method for solving systems of nonlinear fractional Korteweg-de Vries (KdV) partial differential equations (FPDEs). The method utilizes the Laplace transform and a novel analytical approach, incorporating the Caputo fractional derivative.
- Fractional Calculus and its applications to nonlinear PDEs.
- The Laplace transform as a tool for solving FPDEs.
- Development and application of a new novel analytical method.
- Convergence analysis of the proposed method.
- Testing the method's efficacy on various nonlinear KdV systems.
Zusammenfassung der Kapitel (Chapter Summaries)
Chapter 1: Introduction: This chapter introduces the concept of fractional calculus, contrasting it with integer-order calculus. It traces the historical development of fractional derivatives, starting with Leibniz and L'Hopital's correspondence and highlighting key contributions from Laplace, Liouville, and others. The chapter also discusses the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation describing solitary waves, and its applications in various scientific fields, emphasizing the importance of efficient methods for solving fractional-order PDEs relevant to these applications.
Chapter 2: Definitions and Preliminary Notion: This chapter lays the groundwork for the novel analytical method by defining essential concepts and properties. It begins by defining the Gamma function, crucial for fractional calculus, and proves a key lemma related to its properties. The Laplace transform and its properties, including convolution, are then established. The chapter proceeds to define the Riemann-Liouville fractional integral and the Caputo fractional derivative, along with their linearity property and other essential lemmas. The core of the chapter introduces the "Laplace - new novel analytic method" which forms the basis for the subsequent analysis and application to several test problems. This method is explained conceptually and its convergence is rigorously analyzed.
Schlüsselwörter (Keywords)
Fractional Calculus, Fractional Partial Differential Equations (FPDEs), Korteweg-de Vries (KdV) equation, Laplace Transform, Caputo derivative, Novel Analytical Method, Convergence Analysis, Nonlinear systems.
Häufig gestellte Fragen (Frequently asked questions)
Worum geht es in diesem Text?
Dieser Text ist eine umfassende Sprachvorschau, die den Titel, das Inhaltsverzeichnis, die Ziele und Themenschwerpunkte, Kapitelzusammenfassungen und Schlüsselwörter enthält. Er konzentriert sich auf die Anwendung eines neuen analytischen Verfahrens zur Lösung von Systemen nichtlinearer fraktionaler Korteweg-de Vries (KdV) partieller Differentialgleichungen (FPDEs).
Was sind die Hauptziele dieses Werkes?
Das Hauptziel ist die Vorstellung einer neuen analytischen Methode zur Lösung von Systemen nichtlinearer fraktionaler Korteweg-de Vries (KdV) partieller Differentialgleichungen (FPDEs). Die Methode verwendet die Laplace-Transformation und einen neuartigen analytischen Ansatz unter Einbeziehung der Caputo-fraktionalen Ableitung.
Welche Themen werden in diesem Text behandelt?
Zu den Hauptthemen gehören:
- Fraktionale Infinitesimalrechnung und ihre Anwendungen auf nichtlineare PDEs.
- Die Laplace-Transformation als Werkzeug zur Lösung von FPDEs.
- Entwicklung und Anwendung einer neuen neuartigen analytischen Methode.
- Konvergenzanalyse der vorgeschlagenen Methode.
- Testen der Wirksamkeit der Methode an verschiedenen nichtlinearen KdV-Systemen.
Was behandelt Kapitel 1?
Kapitel 1 führt in das Konzept der fraktionalen Infinitesimalrechnung ein und stellt es der Infinitesimalrechnung ganzzahliger Ordnung gegenüber. Es zeichnet die historische Entwicklung fraktionaler Ableitungen nach und diskutiert die Korteweg-de Vries (KdV)-Gleichung und ihre Anwendungen.
Was behandelt Kapitel 2?
Kapitel 2 legt den Grundstein für die neuartige analytische Methode, indem es wesentliche Konzepte und Eigenschaften definiert. Es beginnt mit der Definition der Gammafunktion und der Laplace-Transformation. Es definiert das Riemann-Liouville fraktionale Integral und die Caputo fraktionale Ableitung. Es stellt auch die neuartige analytische Methode mit Hilfe der Laplace-Transformation vor und analysiert deren Konvergenz.
Welche Schlüsselwörter sind mit diesem Text verbunden?
Die Schlüsselwörter sind: Fraktionale Infinitesimalrechnung, Fraktionale partielle Differentialgleichungen (FPDEs), Korteweg-de Vries (KdV)-Gleichung, Laplace-Transformation, Caputo-Ableitung, Neuartige analytische Methode, Konvergenzanalyse, Nichtlineare Systeme.
- Quote paper
- Danish Ali Raza (Author), Ijlal Hussain (Author), 2021, An Analytical Technique to Solve the System of Non-Linear Korteweg–De Vries Equation (KdV), FPDE, Munich, GRIN Verlag, https://www.grin.com/document/1127621
