Harmonic Wave Systems: Partial Differential Equations of the Helmholtz Decomposition

Harmonic Wave Systems is the first textbook about the computational method of Decomposition in Invariant Structures (DIS) that generalizes the analytical methods of separation of variables, undetermined coefficients, asymptotic expansions, and series expansions. In recent years, there has been a boom in publications on propagation of nonlinear waves described by a fascinating list of partial differential equations (PDEs). The vast majority of wave problems are reducible to one-dimensional ones in propagation variables. However, a list of publications with two- and three-dimensional applications of the DIS method is brief.

The book offers a comprehensive and rigorous treatment of the DIS method in two and three dimensions using the PDE approach to the Helmholtz decomposition that provides the most general background for mathematical modelling of harmonic waves in fluid dynamics, electrodynamics, heat transfer, and other numerous areas of science and engineering, which are dealing with propagation and interaction of N internal waves.

Components 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)
• BACK MATTER
• Bibliography
• Index
Readership: 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
 1 FRONT MATTER Dr. Victor A. Miroshnikov PDF (3748 KB) 18 I. Harmonic Wave Systems in Two Dimensions PDF (121 KB) 19 Chapter 1. The Scalar Laplace PDE Dr. Victor A. Miroshnikov 49 Chapter 2. The Vector Laplace PDE Dr. Victor A. Miroshnikov 67 Chapter 3. The Homogeneous Helmholtz PDE in Vector Harmonic Variables Dr. Victor A. Miroshnikov 89 Chapter 4. The Inhomogeneous Helmholtz PDE in Scalar Harmonic Variables Dr. Victor A. Miroshnikov 101 Chapter 5. The Inhomogeneous Helmholtz PDE in Vector Harmonic Variables Dr. Victor A. Miroshnikov 131 Chapter 6. The Lamb-Helmholtz PDE in Scalar-Vector Harmonic Variables Dr. Victor A. Miroshnikov 164 II. Harmonic Wave Systems in Three Dimensions Dr. Victor A. Miroshnikov PDF (121 KB) 165 Chapter 7. The Scalar Laplace PDE Dr. Victor A. Miroshnikov 214 Chapter 8. The Vector Laplace PDE Dr. Victor A. Miroshnikov 242 Chapter 9. The Homogeneous Helmholtz PDE in Vector Harmonic Variables Dr. Victor A. Miroshnikov 278 Chapter 10. The Inhomogeneous Helmholtz PDE in Scalar Harmonic Variables Dr. Victor A. Miroshnikov 300 Chapter 11. The Inhomogeneous Helmholtz PDE in Vector Harmonic Variables Dr. Victor A. Miroshnikov 349 Chapter 12. The Lamb-Helmholtz PDE in Scalar-Vector Harmonic Variables Dr. Victor A. Miroshnikov 398 Appendix. Independent General Solutions 438 Answers to Exercises Dr. Victor A. Miroshnikov 454 BACK MATTER PDF (6903 KB)
Dr. Victor A. Miroshnikov (Biography), Department of Mathematics, College of Mount Saint Vincent

This Book

464pp. Published December 2017

Scientific Research Publishing, Inc.,USA.

Category:Physics & Mathematics

ISBN: 978-1-61896-413-7

(Hardcover) USD 149.00

ISBN: 978-1-61896-405-2

(Paperback) USD 129.00

ISBN: 978-1-61896-406-9

(E-Book) USD 59.00

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