Partial Differential Equations of the Helmholtz Decomposition

Victor A. Miroshnikov

Harmonic Wave Systems is the first textbook to consider the method of decomposition in invariant structures (DIS) for solving systems of partial differential equations (PDEs) in two and three dimensions. This novel method of Computational Mathematics generalizes analytical methods of separation of variables, undetermined coefficients, asymptotic expansions, and series expansions. In recent years, there has been a boom in the number of publications on propagation of nonlinear waves described by various PDEs reducible to one-dimensional problems. However, a list of publications with two-dimensional (2-D) and three-dimensional (3-D) applications of the DIS method is so far brief.

The DIS method is systematically used in the textbook to compute families of exact solutions to the Helmholtz system of PDEs with interrelated components, to find families of exact solutions to the Laplace system of PDEs with separated components, and to show a principal difference between these families. The scalar-vector duality of the Helmholtz decomposition for external and internal forcings, the two-component decomposition, and the structural invariance of the stationary kinematic Euler-Fourier structures are meticulously studied. Fruitfulness of the DIS method is exhibited by 16 scalar, 112 vector, and 36 scalar-vector 3-D general solutions and 12 scalar, 24 vector, and 8 scalar-vector 2-D general solutions considered in the textbook.

Partial cases of the Helmholtz decomposition theorem for curl-free and divergence-free fields are examined and summarized, topology of general solutions is analyzed, and wave lattices with slanted, rectangular, and stepped patterns are visualized. A unique synthesis of mathematical and computational aspects of its field makes Harmonic Wave Systems an ideal text and reference source for undergraduate students, graduate students, and researchers in the various areas of Science and Engineering, where solving PDEs is required, including Applied Mathematics, Mechanical, Aeronautical, and Chemical Engineering, Computational Mathematics, Computational Physics, and Computational Chemistry. The textbook is composed of two parts: I. Harmonic Wave Systems in Two Dimensions (Chapters 1-6), II. Harmonic Wave Systems in Three Dimensions (Chapters 7-12). Each chapter is followed by several examples and numerous exercises, which are designed for analytical problem solving. Totally, 25 examples of calculating general solutions and boundary-value-problem solutions to PDEs and PDE systems are provided, which are complemented by 246 exercises.

Enjoy amazing robustness and simplicity of solving systems of PDEs by the DIS method!

Target audiences

1. Undergraduate students majoring in Mathematics, Engineering, Physics, and other Sciences who are interested in PDE models and wave systems

2. Researchers whose interests are in development of new computational and mathematical methods for investigation of propagation and interaction of internal waves

3. Researchers who are interested in various applications of the novel wave solutions

About the Author

Victor A. Miroshnikov received his B.S. and M.S. degrees in engineering physics from Moscow Institute of Physics and Technology, his Ph.D. in applied mathematics with specialization in fluid mechanics from Moscow Institute of Physics and Technology, and his Dr. in physics with specialization in magnetohydro-dynamics from Institute of Physics of the Latvian Academy of Sciences.

During four decades of his carrier in leading scientific centers of Russia, Latvia, France, and USA, he published numerous papers in fluid dynamics, magnetohydrodynamics, and mass transfer. He has made seminal contributions to the theory of surface waves, internal waves, and coherent structures. He also pioneered in development of the experimental and theoretical computation in invariant structures.

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Contents of the Book:

FRONT MATTER

Introduction

I. Harmonic Wave Systems in Two Dimensions

Chapter 1. The Scalar Laplace PDE

1.1. SKEF Functions
1.2. Fundamental Scalar Solutions
1.3. General Scalar Solutions
1.4. The Scalar Dirichlet Boundary Problem with Surface Forcing
1.5. The Scalar Neumann Boundary Problem with Surface Forcing
1.6. Examples and Exercises

Chapter 2. The Vector Laplace PDE

2.1. General Vector Solutions
2.2. The Vector Dirichlet Boundary Problem with Surface Forcing
2.3. The Vector Neumann Boundary problem with Surface Forcing
2.4. Examples and Exercises

Chapter 3. The Homogeneous Helmholtz PDE in Vector Harmonic Variables

3.1. General Vector Solutions Compelled by Surface Forcing
3.2. The Quasi-Scalar Dirichlet Boundary Problem with Surface Forcing
3.3. The Quasi-Scalar Neumann Boundary Problem with Surface Forcing
3.4. Examples and Exercises

Chapter 4. The Inhomogeneous Helmholtz PDE in Scalar Harmonic Variables

4.1. General Scalar Solutions Induced by the 𝑥-Component of External Forcing
4.2. General Scalar Solutions Compelled by the 𝑧-Component of External Forcing
4.3. Examples and Exercises

Chapter 5. The Inhomogeneous Helmholtz PDE in Vector Harmonic Variables

5.1. General Vector Solutions Generated by External Forcing
5.2. The Quasi-Scalar Dirichlet Boundary Problem with External Forcing
5.3. The Quasi-Scalar Neumann Boundary Problem with External Forcing
5.4. Examples and Exercises

Chapter 6. The Lamb-Helmholtz PDE in Scalar-Vector Harmonic Variables

6.1. General Scalar-Vector Solutions Produced by Internal Forcing
6.2. The Quasi-Scalar Dirichlet Boundary Problem with Internal Forcing
6.3. The Quasi-Scalar Neumann Boundary Problem with Internal Forcing
6.4. The Helmholtz Decomposition of a Divergence-Free and Curl-Free Field
6.5. Examples and Exercises

II. Harmonic Wave Systems in Three Dimensions

Chapter 7. The Scalar Laplace PDE

7.1. SKEF Functions
7.2. Fundamental Scalar Solutions
7.3. General Scalar Solutions
7.4. The Scalar Dirichlet Boundary Problem with Surface Forcing
7.5. The Scalar Neumann Boundary Problem with Surface Forcing
7.6. Examples and Exercises

Chapter 8. The Vector Laplace PDE

8.1. General Vector Solutions
8.2. The Vector Dirichlet Boundary Problem with Surface Forcing
8.3. The Vector Neumann Boundary Problem with Surface Forcing
8.4. Examples and Exercises

Chapter 9. The Homogeneous Helmholtz PDE in Vector Harmonic Variables

9.1. General Vector Solutions Compelled by Surface Forcing
9.2. The Quasi-Scalar Dirichlet Boundary Problem with Surface Forcing
9.3. The Quasi-Scalar Neumann Boundary Problem with Surface Forcing
9.4. Examples and Exercises

Chapter 10. The Inhomogeneous Helmholtz PDE in Scalar Harmonic Variables

10.1. General Scalar Solutions Induced by the 𝑥-Component of External Forcing
10.2. General Scalar Solutions Produced by the 𝑦-Component of External Forcing
10.3. General Scalar Solutions Compelled by the 𝑧-Component of External Forcing
10.4. Examples and Exercises

Chapter 11. The Inhomogeneous Helmholtz PDE in Vector Harmonic Variables

11.1. General Vector Solutions Generated by External Forcing
11.2. The Quasi-Scalar Dirichlet Boundary Problem with External Forcing
11.3. The Quasi-Scalar Neumann Boundary Problem with External Forcing
11.4. Examples and Exercises

Chapter 12. The Lamb-Helmholtz PDE in Scalar-Vector Harmonic Variables

12.1. General Scalar-Vector Solutions Produced by Internal Forcing
12.2. The Quasi-Scalar Dirichlet Boundary Problem with Internal Forcing
12.3. The Quasi-Scalar Neumann Boundary Problem with Internal Forcing
12.4. The Helmholtz Decomposition Theorem
12.5. Examples and Exercises

Appendix. Independent General Solutions

A.1. General Solutions in Variables (𝑥, 𝑦, 𝑧, 𝑡)
A.2. General Solutions in Variables (𝑥, 𝑧, 𝑡)
A.3. General Solutions in Variables (𝑦, 𝑧, 𝑡)

Answers to Exercises

BACK MATTER

Bibliography
Index

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