Handbook of Fractional Calculus for Engineering and Science

Harendra Singh, H. M. Srivastava, Juan J. Nieto, Harendra Singh, H. M. Srivastava, Juan J. Nieto

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Print ISBN:
  9781032204307
  9781032047799
• Edition: 1
• eBook ISBN:
  9781000540109
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

Fractional calculus is used to model many real-life situations from science and engineering. The book includes different topics associated with such equations and their relevance and significance in various scientific areas of study and research. In this book readers will find several important and useful methods and techniques for solving various types of fractional-order models in science and engineering. The book should be useful for graduate students, PhD students, researchers and educators interested in mathematical modelling, physical sciences, engineering sciences, applied mathematical sciences, applied sciences, and so on.

This Handbook:

• Provides reliable methods for solving fractional-order models in science and engineering.
• Contains efficient numerical methods and algorithms for engineering-related equations.
• Contains comparison of various methods for accuracy and validity.
• Demonstrates the applicability of fractional calculus in science and engineering.
• Examines qualitative as well as quantitative properties of solutions of various types of science- and engineering-related equations.

Readers will find this book to be useful and valuable in increasing and updating their knowledge in this field and will be it will be helpful for engineers, mathematicians, scientist and researchers working on various real-life problems.

1 Analytical and Numerical Methods to Solve the Fractional Model of the Vibration Equation

Temirkhan S. Aleroev

Moscow State University of Civil Engineering, Moscow

Asmaa M. Elsayed

Moscow State University of Civil Engineering, Moscow

Zagazig University, Zagazig, Arab Republic of Egypt/Egypt

DOI: 10.1201/9781003263517-1

Contents

1. 1.1 Introduction
2. 1.2 The Problem Formulation for FVEs with Viscoelastic Damping and Its Application in Science and Engineering
3. 1.3 Mathematical Model of FVEs
4. 1.4 Notations and Preliminaries
5. 1.5 Solving a FVEs by Fourier Method
6. 1.6 Solving FVEs Numerically by Finite Difference Scheme
7. 1.7 Stability and Convergence Analysis
8. 1.8 Numerical Examples
9. 1.9 Conclusion
10. Author Contributions
11. Conflicts of Interest
12. Acknowledgment
13. References

1.1 Introduction

Fractional partial differential equations (FPDEs) are found in many different branches of science and engineering, such as hydrodynamics, electroanalytical chemistry, quantum science, viscoelastic mechanics, signal image processing, chain-breaking of polymer materials, molecular spectrum, and anomalous diffusion process of ions in nerve cells [1–4]. Also, PDEs with a fractional order have been used to simulate the flow and filtration of a fluid in a porous fractal medium. The use of fractional derivatives (FDs) for modeling real physical processes or environments leads to the appearance of equations containing derivatives and integrals of fractional order in addition to the classical ones. Researchers have focused their efforts on fractional-order physical models [5, 6] becau...

Table of contents

• Cover Page
• Half Title Page
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• Editors
• Contributors
• 1 Analytical and Numerical Methods to Solve the Fractional Model of the Vibration Equation
• 2 Analysis of a Nonlinear System Arising in a Helium-Burning Network with Mittag–Leffler Law
• 3 Computational Study of Constant and Variable Coefficients Time-Fractional PDEs via Reproducing Kernel Hilbert Space Method
• 4 Spectral Collocation Method Based Upon Special Functions for Fractional Partial Differential Equations*
• 5 On the Wave Properties of the Conformable Generalized Bogoyavlensky–Konopelchenko Equation
• 6 Analytical Solution of a Time-Fractional Damped Gardner Equation Arising from a Collisional Effect on Dust-ion-acoustic Waves in a Dusty Plasma with Bi-Maxwellian Electrons
• 7 An Efficient Numerical Algorithm for Fractional Differential Equations
• 8 Generalization of Fractional Kinetic Equations Containing Incomplete I-Functions*
• 9 Behavior of Slip Effects on Oscillating Flows of Fractional Second-Grade Fluid
• 10 A Novel Fractional-Order System Described by the Caputo Derivative, Its Numerical Discretization, and Qualitative Properties
• 11 Extraction of Deeper Properties of the Conformable Gross–Pitaevskii Equation via Two Powerful Approaches
• 12 New Fractional Integrals and Derivatives Results for the Generalized Mathieu-Type and Alternating Mathieu-Type Series*
• Index