Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
List of Tables
Notes: Generalised Equation of Mathematical Physics and Engineering (GEMPE)
List of Figures and Diagrams
Preface
About the Authors
Acknowledgements
Abbreviations for Mathematical Equations
List of Physical Quantities
Chapter 3: Linear Partial Differential and Difference Equations and Simultaneous Systems with Constant or Homogeneous Coefficients
3.1. Method of Separation of Variables
3.1.1. Two-Dimensional Cartesian Harmonic Equation (Laplace 1820)
3.1.2. One-Dimensional Cartesian Wave Equation (Dβalembert 1747, Lagrange 1760)
3.1.3. Permanent Propagating Waves and Standing Modes
3.1.4. One-Dimensional Cartesian Diffusion Equation (Fourier 1818)
3.1.5. Space-Time Decay of Non-Permanent Solutions
3.1.6. Comparison of the Harmonic, Wave, and Diffusion Equations
3.1.7. Elliptic, Hyperbolic, and Parabolic P.D.E
3.1.8. Separable and Non-Separable P.D.E
3.2. Method of Similarity Functions
3.2.1. Similarity Solutions for the Harmonic Equation
3.2.2. Similarity Solutions for the Wave Equation
3.2.3. Non-Similarity Solution for the Diffusion Equation
3.2.4. Similarity Method for Similarity and Non-Similarity Solutions
3.2.5. Telegraph or Wave-Diffusion Equation (Maxwell 1873, Heaviside 1892)
3.2.6. Transverse Vibrations of an Elastic Bar (Bernoulli 1744, Euler 1744)
3.2.7. Stiff Bar and Beam Equations
3.3. Method of Symbolic Differentiation Operators
3.3.1. Symbolic Differential Operators for the Harmonic Equation
3.3.2. Symbolic Differential Operators for the Wave Equation
3.3.3. Two Initial Conditions for Waves
3.3.4. Two Boundary Conditions for Waves
3.3.5. One Initial Condition for the Diffusion Equation
3.3.6. Two Boundary Conditions for the Diffusion Equation
3.3.7. Equivalence of One Initial and Two Boundary Conditions
3.3.8. Analytic and Singular Solutions of the Diffusion Equation
3.3.9. First-Order Linear Partial Differential Equations with Constant Coefficients
3.3.10. Unidirectional and Bidirectional Damped Waves
3.3.11. Extension to Higher-Order Partial Differential Equations via Change of Dependent Variable
3.3.12. Second-Order Partial Differential Equation
3.3.13. Second-Order Decomposition into First-Order Factors
3.3.14. Second-Order as the Square of the First-Order
3.3.15. Five Methods of Solution of Partial Differential Equations
3.4. Method of Characteristic Polynomial of Exponentials of Several Variables
3.4.1. Unforced Linear Partial Differential Equations with Constant Coefficients
3.4.2. Characteristic Polynomial of Several Variables
3.4.3. Existence of a Characteristic Polynomial
3.4.4. Single Roots of the Characteristic Polynomial
3.4.5. Multiple Roots of the Characteristic Polynomial
3.4.6. Characteristic Polynomial for the Harmonic Equation
3.4.7. Characteristic Polynomial for the Wave Equation
3.4.8. Characteristic Polynomial for the Diffusion Equation
3.4.9. Influence Function due to Forcing by a Unit Impulse
3.4.10. Convolution Integral and Forcing by an Integrable Function
3.4.11. Forcing by a Unit Jump and Error Function
3.4.12. Decay of a Signal by Dissipation
3.4.13. Characteristic Polynomial for the Wave-Diffusion Equation
3.4.14. Characteristic Polynomial for the Elastic Bar Equation
3.4.15. Characteristic Polynomial for the Elastic Beam Equation
3.4.16. Resonant and Non-Resonant Forcing by an Exponential
3.4.17. Non-Resonant Exponential Forcing of Six Partial Differential Equations
3.4.18. Single Resonance for Exponential Forcing
3.4.19. Single Resonance with Regard to Time or Position
3.4.20. Doubly Resonant Exponential Forcing
3.4.21. Comparison of Non-Resonant with Single/Double Resonant Solutions
3.4.22. Forcing by Circular/Hyperbolic Cosines/Sines
3.4.23. Forcing of the Harmonic Equation by the Circular Cosine
3.4.24. Forcing of the Wave Equation by the Hyperbolic Sine
3.4.25. Diffusion Equation Forced by the Product of Circular and Hyperbolic Functions
3.4.26. Doubly Resonant Solutions of Forced Harmonic and Wave Equations
3.4.27. Complete Integral of the Forced Wave Equation
3.4.28. Phase Variables for the Wave Equation
3.4.29. General, Particular, and Complete Integrals
3.4.30. Method of Complex Variables for the Harmonic Equation
3.4.31. Inverse Characteristic Polynomial of Partial Derivatives
3.4.32. Factorisation and Interpretation of the Inverse Polynomial of Partial Derivatives
3.4.33. Polynomial Forcing of Harmonic/Wave/Diffusion Equations
3.4.34. Polynomial Solutions of the Harmonic/Wave/Diffusion Equations
3.4.35. Forcing by the Product of a Smooth Function by an Exponential
3.4.36. Forcing by Product of Smooth, Exponential, Circular, and Hyperbolic Functions
3.4.37. Harmonic Equation Forced by the Product of Power and Exponential
3.4.38. Wave Equation Forced by Product of Power, Exponential, and Circular Sine
3.4.39. Diffusion Equation Forced by the Product of Power, Exponential, and Circular and Hyperbolic Functions
3.4.40. Combination of the Methods of Factorisation and Inversion of the Polynomial of Partial Derivatives
3.4.41. Forcing of a Linear Equation with Second-Order Partial Derivatives
3.4.42. Forcing of the Harmonic and Wave Equations by Inverse Variables
3.4.43. Multiple Roots of the Factorised Characteristic Polynomial of Partial Derivatives
3.4.44. Partial Differential Equation with Second-Order Derivatives and Double Root
3.4.45. Complete Integrals by the Method of Characteristic Polynomial
3.4.46. Other Equations with Characteristic Polynomials
3.5. Simultaneous Systems of Partial Differential Equations with Constant Coefficients
3.5.1. Simultaneous Linear Systems with Constant Coefficients
3.5.2. Conditions for a Complex Holomorphic Function (Cauchy 1821, Riemann 1851)
3.5.3. Two Simultaneous Partial Differential Equations Reducible to the Harmonic Equation
3.5.4. Unicity of Solutions and Boundary Conditions
3.5.5. Comparison of Cauchy-Riemann and Alternate Harmonic System
3.5.6. Coupling of the Wave and Diffusion Equations through Gradients
3.5.7. Non-Resonant and Resonant Forcing by Exponentials
3.5.8. Forcing of the Alternate Harmonic System
3.5.9. Non-Resonant and Singly and Doubly Resonant Solutions
3.6. Linear Equation with Homogeneous Partial Derivatives
3.6.1. Characteristic Polynomial and Power and Logarithmic Solutions
3.6.2. Analogue Wave-Diffusion Equation with Homogeneous Powers
3.6.3. Ordinary/Homogeneous Derivatives of Exponentials/Powers/Logarithms
3.6.4. Analogue Wave-Diffusion Equation with Homogeneous Derivatives
3.6.5. Similarity Solutions for the Original (Homogeneous) Wave Equation
3.6.6. Initial Conditions and Unicity of Solution of the Homogeneous Wave Equation
3.6.7. Non-Resonant and Resonant Forcing by Powers
3.6.8. Non-Resonant, Singly, and Doubly Resonant Solutions
3.6.9. Inverse Characteristic Polynomial of Homogeneous Partial Derivatives
3.6.10. Forcing by the Product of Powers and Logarithms
3.7. Simultaneous System of Partial Differential Equations with Power Coefficients
3.7.1. Matrix of Polynomials of Homogeneous Derivatives
3.7.2. Characteristic Polynomial of Homogeneous Derivatives
3.7.3. System of Homogeneous Derivatives Analogous to the Wave Equation
3.7.4. Corresponding Wave and Coupled System of Homogeneous Derivatives
3.7.5. Boundary Conditions on One Function and its Derivative
3.7.6. Boundary Conditions Applied to Two Functions
3.7.7. Similarity Solutions and Homogeneous Analogues
3.7.8. Forcing of System of Homogeneous Derivatives by Powers
3.7.9. Non-Resonant and Resonant Forcing of the Homogeneous Wave System
3.7.10. Inverse Matrix of Polynomials of Homogeneous Partial Derivatives
3.7.11. One (Two) Pair(s) of Solutions of the Unforced (Forced) System
3.8. Partial Finite Difference Equations.
3.8.1. Characteristic Polynomial of Partial Finite Differences
3.8.2. General Solution of the Unforced Equation
3.8.3. Finite Difference Analogue of the Wave Diffusion Equation
3.8.4. Starting Conditions and Unicity of Solution
3.8.5. Analogue Finite Difference Wave Equation
3.8.6. Resonant/Non-Resonant Forcing by Integral Powers
3.8.7. Non-Resonant and Singly/Doubly Resonant Solutions
3.9. Simultaneous System of Partial Finite Difference Equations
3.9.1. Characteristic Polynomial for Simultaneous Partial Finite Differences
3.9.2. Natural Sequences of Powers with Integer Exponents
3.9.3. Finite Difference System Analogous to the Wave Equation
3.9.4. Three Starting Conditions on One Function
3.9.5. One (Two) Starting Condition(s) for One (Another) Function
3.9.6. Simultaneous Partial Finite Difference Equations Forced by Powers
3.9.7. Non-Resonant and Singly Resonant Solutions
3.9.8. Six Classes of Partial Differential Equations/Partial Finite Difference Equations and Simultaneous Systems with Characteristic Polynomials
Notes: Generalised Equation of Mathematical Physics and Engineering (GEMPE)
References
Bibliography
Index