1Introduction to Partial Differential Equations
109Integral Transforms
21.1 Classification of PDEs
11011.1 Laplace Transform
31.2 Boundary and Initial Conditions
111Definition:
41.3 Well-Posed Problems
112Properties:
51.4 Physical Applications
113Solving ODEs and PDEs:
61.5 Separation of Variables
114Inverse Laplace Transform:
71.6 Dimensional Analysis
115Applications:
8First–Order PDEs
11611.2 Fourier-Bessel Transform
92.1 Linear and Nonlinear PDEs
117Definition:
102.2 Method of Characteristics
118Properties:
112.3 Lagrange’s Method
119Applications:
122.4 Charpit’s Method
12011.3 Hankel Transform
132.5 Integral Surfaces
121Definition:
142.6 Applications in Physics
122Properties:
15Wave Equation
123Applications:
163.1 Derivation and Properties
12411.4 Mellin Transform
173.2 D’Alembert’s Solution
125Definition:
18One-Dimensional Wave Equation:
126Properties:
19D’Alembert’s Solution:
127Solving Differential Equations:
20Interpretation:
128Applications:
21Initial Conditions:
12911.5 Applications in Solving PDEs
223.3 Boundary Value Problems
130Asymptotic Methods
23General Formulation:
13112.1 Regular and Singular Perturbation Theory
24Separation of Variables:: Numerical Methods:
132Regular Perturbation Theory:: Singular Perturbation Theory:
253.4 Fourier Series Method
13312.2 Boundary Layer Theory: Blasius Boundary Layer:
263.5 Separation of Variables
13412.3 WKB Approximation
27One-Dimensional Wave Equation:
13512.4 Multiple Scale Analysis: Multiple scale analysis has numerous applications in various fields, including:
28Separation of Variables Ansatz:
13612.5 Applications in Fluid Mechanics and Quantum Mechanics
29Solving the ODEs:
137Applications in Fluid Mechanics:
30General Solution:
138Applications in Quantum Mechanics:
31Advantages and Limitations:
139Calculus of Variations
323.6 Applications in Acoustics and Electromagnetics
14013.1 Euler-Lagrange Equation
33Applications in Acoustics:: Applications in Electromagnetics:
14113.2 Variational Principles
34Heat Equation
142Hamilton’s Principle:: Fermat’s Principle of Least Time:
354.1 Derivation and Properties
14313.3 Rayleigh-Ritz Method
364.2 Separation of Variables
14413.4 Finite Element Method: Implementation of the Finite Element Method
374.3 Fourier Series Method
14513.5 Applications in Mechanics and Optimization
384.4 Boundary Value Problems: Solving Boundary Value Problems:
146Applications in Mechanics:
394.5 Steady-State Solutions
147Applications in Optimization:
404.6 Applications in Heat Transfer
148Limitations and challenges of the FEM:
411. Transient Heat Conduction:
149Elliptic PDEs
422. Steady-State Heat Conduction:
15014.1 Properties and Classification
433. Thermal Stress Analysis:
151Definition and General Form:
444. Heat Exchanger Design:
152Properties of Elliptic PDEs:
455. Furnace and Kiln Design:
153Classification of Elliptic PDEs:
466. Electronic Cooling:
154Applications of Elliptic PDEs:
477. Geothermal Energy:
15514.2 Maximum Principles
488. Biomedical Applications:
156Basic Maximum Principle:
49Laplace’s Equation
157Consequences and Applications:
505.1 Properties and Applications
158Strong Maximum Principle:
515.2 Separation of Variables
15914.3 Harmonic Functions
525.3 Boundary Value Problems
160The Laplace Equation:
535.4 Green’s Functions
161Harmonic Functions:
545.5 Potential Theory
162Applications of Harmonic Functions:
55Harmonic Functions:: Potential Theory Applications:
163Solving the Laplace Equation:
565.6 Applications in Electrostatics and Fluid Mechanics
16414.4 Dirichlet and Neumann Problems
57Electrostatics:: Fluid Mechanics:
165Dirichlet Problem:
58Fourier Transform Methods
166Existence and Uniqueness:
596.1 Fourier Transform Pairs: Fourier Transform Pairs:
167Numerical Solutions:
606.2 Convolution Theorem
168Neumann Problem:
616.3 Solving PDEs with Fourier Transforms
169Existence and Uniqueness:
626.4 Discrete Fourier Transforms
170Numerical Solutions:
63Fast Fourier Transform (FFT):: Applications of DFT and FFT:
171Mixed Boundary Value Problems:
64Sturm–Liouville Problems
17214.5 Applications in Potential Theory and Elasticity
657.1 Sturm-Liouville Theory
173Applications in Potential Theory:
667.2 Regular and Singular Problems
174Applications in Elasticity:
671. Regular Sturm-Liouville Problems :: 2. Singular Sturm-Liouville Problems :
175Numerical Solutions:
687.3 Eigenvalue Problems
176Hyperbolic PDEs
697.4 Orthogonal Functions
17715.1 Characteristics and Domains of Dependence
707.5 Applications in Quantum Mechanics
178Characteristics:
71Green’s Functions
179Domains of Dependence:
728.1 Definition and Properties: Properties of Green’s Functions:
18015.2 Shock Waves and Discontinuities
738.2 Green’s Function for the Wave Equation
181Shock Waves:
748.3 Green’s Function for the Heat Equation
182Discontinuities:
758.4 Green’s Function for the Laplace Equation
183Handling Discontinuities:
768.5 Applications in Boundary Value Problems
18415.3 Conservation Laws
77Numerical Methods for PDEs
185Conservation Law Form:
789.1 Finite Difference Methods
186Shock Waves and Entropy Conditions:: Numerical Methods for Conservation Laws:
799.2 Finite Element Methods
187Applications of Conservation Laws:
809.3 Spectral Methods
18815.4 Riemann Problems
819.4 Stability and Convergence
189Definition:
82Stability:: Convergence:
190Solution Structure:
839.5 Error Analysis
191Importance of Riemann Problems:
849.6 Applications in Computational Fluid Dynamics
192Analytical and Numerical Solutions:
85Finite Difference Methods in CFD:
19315.5 Applications in Fluid Dynamics and Relativity
86Finite Element Methods in CFD:
194Applications in Fluid Dynamics:
87Spectral Methods in CFD:
195Applications in Relativity:
88Applications of CFD:
196Parabolic PDEs
89Nonlinear PDEs
19716.1 Properties and Classification
9010.1 Classification of Nonlinear PDEs
198Classification of Second-Order PDEs:
9110.2 Travelling Wave Solutions
199Properties of Parabolic PDEs:: Examples of Parabolic PDEs:
921. Solitary Waves :
20016.2 Maximum Principles
932. Kink Solutions :
201Maximum Principle for the Heat Equation:: Maximum Principle for General Parabolic PDEs:
943. Shock Waves :
202Implications and Applications:
954. Periodic Travelling Waves :
20316.3 Diffusion Equations
9610.3 Shock Waves and Conservation Laws
204General Form:
97Conservation Laws :
205Physical Interpretation:
98Rankine-Hugoniot Jump Conditions :
206Initial and Boundary Conditions:
99Numerical Treatment of Shock Waves :
207Examples and Applications:
10010.4 Burgers’ Equation
208Analytical and Numerical Solutions:
101Analytical Solutions:
20916.4 Boundary Value Problems
102Numerical Solutions:
210General Formulation:: Well-Posedness and Existence of Solutions:
10310.5 Korteweg-de Vries Equation
211Analytical and Numerical Solutions:
104Soliton Solutions:
21216.5 Applications in Heat Transfer and Population Dynamics
105Multiple Soliton Solutions:: Numerical Solutions:
213Applications in Heat Transfer:
10610.6 Applications in Fluid Dynamics and Plasma Physics
214Applications in Population Dynamics:
107Fluid Dynamics Applications:
215Glossary
108Plasma Physics Applications:
216Index
