This book is an invaluable resource for applied researchers to find the analytical solution of differential equations describing the dynamical system with less computational effort and time. It describes the basic concepts of the differential transform method and solution of various real-world problems described by simple to complicated differential equations. It provides a computational technique that is not only conceptually simple and easy to use but also readily adaptable for computer coding. Different chapters of the book deal with the basic differential equations involved in the physical phenomena as well as a complicated system of differential equations described by the mathematical model.
The book offers comprehensive coverage of the most essential topics, including
- Basic concepts and fundamental properties of the proposed technique with proof
- The solution of linear, nonlinear, homogeneous, and nonhomogeneous ordinary differential equations (ODEs) and partial differential equations (PDEs)
- The initial and boundary value problems
- Real-world ODE and PDE problems are also discussed
Applications of Differential Transform to Real World Problems is primarily aimed at undergraduates, graduates, and researchers studying differential equations. Scientists dealing with complicated differential equations or systems of differential equations will also find this book useful.
Author(s): Yogeshwari F. Patel, Jayesh M. Dhodiya
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 306
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Authors
Chapter 1: One-Dimensional Differential Transform Method
1.1 Introduction
1.2 ODDTM
1.2.1 Basic Definition of the ODDTM
1.2.2 Fundamental Properties of the ODDTM
1.3 Solution of a Linear Differential Equation by the ODDTM
1.4 Solution of a Nonlinear Differential Equation by the ODDTM
1.4.1 Riccati Differential Equation
1.4.2 Lane–Emden Equation
1.5 Higher-Order Nonlinear Singular Boundary Value Problems
1.6 Solution of an Integrodifferential Equation
Exercises
References
Chapter 2: Application of the One-Dimensional Differential Transform Method to Solve Real-World Problems
2.1 Introduction
2.2 Application of the ODDTM to an ODE Based on Real-World Problems
2.2.1 Problem of Population Growth and Decay
2.2.2 Mixing Problem
2.2.3 Problem of Management Growth
2.2.4 Problem of Heat Transfer
2.2.5 Problem of Atmospheric Sciences
2.2.6 Problem of Physical Sciences
2.2.7 Problem of Electronic Engineering
2.2.8 Problem of Civil Engineering
2.2.9 Problem of Electrical Engineering
Exercises
References
Chapter 3: Solution of a System of Differential Equations by the One Dimensional Differential Transform Method
3.1 Introduction
3.2 Definition of a System of Differential Equations
3.3 Solution of a System of Differential Equations by the ODDTM
3.4 System of Integrodifferential Equations
Exercises
References
Chapter 4: A Coupled System of Ordinary Differential Equations–Based Real-World Problems and Its Solution by the One Dimensional Differential Transform Method
4.1 Introduction
4.2 Compartment-Based Mixing Problem
4.3 Compartment-Based Chemical Risk Assessment Problem
4.3.1 Mathematical Formula of Chemical Risk Assessment Problem
4.3.2 Solution of the CRAP by the ODDTM
4.3.3 Numerical Simulation of the CRAP
4.4 Surface Water Quality Problem in a Stratified Lake
4.4.1 Mathematical Formulation of the Surface Water Quality Problem in a Stratified Lake
4.4.2 Numerical Simulation of the Water Quality Problem in Stratified Lake
4.5 Transmission of a Computer Virus Problem (SAEIQRS Model)
4.5.1 Numerical Simulation of the Transmission of a Computer Virus Problem (SAEIQRS Model)
4.6 Nonlinear Two-Compartment Model of the Drug Delivery Problem
4.6.1 Using the ODDTM to Solve a Nonlinear Two-Compartment Model of Drug Delivery
4.7 Susceptible–Infected–Recovered Epidemic Spread Model
4.7.1 Mathematical Formulation of the SIR Epidemic Spread Model
4.7.2 Numerical Simulation of SIR Epidemic Spread
Exercises
References
Chapter 5: Two-Dimensional Differential Transform Method
5.1 Introduction
5.2 TDDTM
5.2.1 Basic Definition of the TDDTM
5.2.2 Fundamental Properties of the TDDTM
5.3 Solution of PDE by the TDDTM
5.3.1 Solution of the Heat Equation
5.3.2 Solution of the Wave Equation
5.3.3 Solution of Laplace Equation
5.3.4 Solution of Diffusion–Convection Equation
5.3.5 Solution of Reaction–Convection–Diffusion Equation
5.3.6 Solution of Reaction–Diffusion Equation
5.3.7 Solution of Nonlinear Gas Dynamics Equation
5.3.8 Solution of Klein–Gordon Equation
5.3.9 Solution of Time-Dependent Emden–Fowler-Type Equation
Exercises
References
Chapter 6: Application of the Differential Transform Method to a System of Partial Differential Equations
6.1 Introduction
6.2 Solution of a System of Linear PDEs by the TDDTM
6.3 Solution of a System of Nonlinear PDEs by the TDDTM
Exercises
References
Chapter 7: Solution of a Partial Differential Equation Describing Transport Phenomena by the Differential Transform Method
7.1 Introduction
7.2 Coupled System of Reaction Diffusion
7.3 Coupled Burgers’ Equation
7.4 Fokker–Planck Equation
7.5 Coupled KdV Equation
7.6 Jaulent–Miodek Equation
7.7 Whitham–Broer–Kaup Equation
Exercises
References
Chapter 8: A Compartment Model of the Reaction–Diffusion Mechanism in Cell 42 and Its Solution Using the Differential Transform Method
8.1 Introduction
8.2 Compartment-Based Reaction and Diffusion Mechanism in a Cell with Protein Binding
8.2.1 Numerical Simulation
Exercises
References
Chapter 9: An Application of the Differential Transform Method to Solve Partial Differential Equation–Based Real-World Problems
9.1 Introduction
9.2 OTIS Model with Decay
9.2.1 Two-Compartment OTIS Model Neglecting Decay Rate in River and Storage Compartments
9.2.2 OTIS Model with Decay
9.3 Compartment-Based Double-Pipe Heat Exchanger Model
9.3.1 Compartment-Based Double-Pipe Heat Exchanger Model
Exercises
References
Chapter 10: Analytical Solutions Using the Differential Transform Method and Their Convergence
10.1 Introduction
10.2 Convergence Analysis for ODE-Based Series by the DTM
10.3 Convergence Analysis for PDE-Based Series by the DTM
Exercises
References
Index
