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

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.