61.2.3 Number of Independent Variables
1327.5 Inhomogeneous PDEs: Mathematical Question and Solution:
71.3 Applications of PDEs
1337.6 Applications
81.4 Boundary and Initial Conditions
1341. Electrostatics and Magnetostatics:
91.4.1 Boundary Conditions: 1.4.2 Initial Conditions
1352. Heat Transfer:
101.5 Well-Posed Problems
1363. Fluid Dynamics:
111.6 Analytical and Numerical Solutions
1374. Quantum Mechanics:
121.6.1 Analytical Solutions
1385. Elasticity and Structural Analysis:
131.6.2 Numerical Solutions
1396. Electromagnetic Wave Propagation:
141.6.3 Advantages and Limitations of Analytical and Numerical Solutions
1407. Integral Equation Formulations:
15Advantages of Analytical Solutions:
1418. Inverse Problems:
16Limitations of Analytical Solutions:
142Conclusion
17Advantages of Numerical Solutions:
143Numerical Methods for PDEs
18Limitations of Numerical Solutions:
1448.1 Finite Difference Methods
19Conclusion
145Finite Difference Approximations:
20First-Order PDEs
146Finite Difference Schemes:: Stability, Consistency, and Convergence:
212.1 Linear First-Order PDEs
1478.2 Finite Element Methods
222.1.1 Classification of Linear First-Order PDEs: 2.1.2 Solution Methods for Linear First-Order PDEs
1488.3 Spectral Methods: Mathematical Question and Solution:
232.2 Nonlinear First-Order PDEs
1498.4 Stability and Convergence
242.2.1 Classification of Nonlinear First-Order PDEs: 2.2.2 Solution Methods for Nonlinear First-Order PDEs
150Stability:
252.3 Method of Characteristics
151Convergence:: Mathematical Question and Solution:
262.3.1 Concept of Characteristic Curves
1528.5 Error Analysis
272.3.2 Steps in the Method of Characteristics
153Types of Errors:
282.3.3 Advantages and Limitations
154Error Estimation and Control:: Mathematical Question and Solution:
292.3.4 Numerical Methods for First-Order PDEs
1558.6 Software Packages
302.4 Cauchy Problem
156Conclusion
312.4.1 Formulation of the Cauchy Problem
1571. Finite Difference Methods:
322.4.2 Cauchy Problem for Linear and Nonlinear PDEs
1582. Finite Element Methods:
33Linear First-Order PDEs:: Nonlinear First-Order PDEs:
1593. Spectral Methods:
342.4.3 Examples and Mathematical Questions
1604. Stability and Convergence:
352.4.4 Ill-Posed Cauchy Problems and Regularization
1615. Error Analysis:: 6. Software Packages:
362.5 Applications of First-Order PDEs
162Nonlinear PDEs
372.5.1 Traffic Flow Models
1639.1 Burgers’ Equation: Numerical Solution of Burgers’ Equation using Finite Differences:
382.5.2 Wave Propagation
1649.2 Korteweg-de Vries Equation
392.5.3 Geometric Optics
1651. Soliton Solution:
402.5.4 Other Applications
1662. Periodic Solution:: Numerical Solution of the KdV Equation:
412.6 Numerical Methods for First-Order PDEs
1679.3 Sine-Gordon Equation
422.6.1 Finite Difference Methods (FDM)
168Analytical Solutions to the Sine-Gordon Equation:
432.6.2 Finite Volume Methods (FVM)
1691. Soliton Solution:
442.6.3 Finite Element Methods (FEM)
1702. Periodic Solution:: Numerical Solution of the Sine-Gordon Equation:
452.6.4 Other Numerical Methods
1719.4 Shock Waves: Numerical Simulation of Shock Waves:
46Second-Order Linear PDEs
1729.5 Solitons: Numerical Simulation of Solitons:
473.1 Classification of Second-Order Linear PDEs
1739.6 Applications of Nonlinear PDEs
48Elliptic PDEs:
1741. Fluid dynamics:
49Parabolic PDEs:
1752. Nonlinear optics:
50Hyperbolic PDEs:: Practice Question:
1764. Condensed matter physics:
513.2 Separation of Variables: Practice Question:
1775. Quantum field theory:: 6. Astrophysics and cosmology:
523.3 Heat Equation
178Conclusion
53Analytical Solutions:
179Calculus of Variations
54Numerical Solutions:
18010.1 Functionals and Euler-Lagrange Equations
55Practice Question:: Mathematical Questions and Solutions:
18110.2 Variational Problems
563.4 Wave Equation
18210.3 Geodesics and Minimal Surfaces
57Analytical Solutions:
183Geodesics: Minimal Surfaces
58Numerical Solutions:: Practice Question:
18410.4 Isoperimetric Problems
593.5 Laplace’s Equation
18510.5 Ritz Method
60Analytical Solutions:
18610.6 Applications to PDEs
61Numerical Solutions:: Practice Question:
1871. Boundary Value Problems:
623.6 Boundary Value Problems
1882. Variational Principles:
63Analytical Solutions:
1893. Optimal Control Problems:: 4. Isoperimetric Problems:
64Numerical Solutions:: Practice Question:
190Conclusion
65Conclusion
191Elliptic PDEs
66Fourier Series
19211.1 Laplace’s Equation
674.1 Periodic Functions: Practice Question:
19311.2 Poisson’s Equation
684.2 Fourier Series Representation: Practice Question:
19411.3 Maximum Principles
694.3 Convergence of Fourier Series
19511.4 Dirichlet and Neumann Problems
70Dirichlet’s Conditions:
196The Dirichlet Problem:: The Neumann Problem:
71Convergence of Fourier Series for Discontinuous Functions:
19711.5 Green’s Functions
72Practice Question:: Mathematical Questions and Solutions:
19811.6 Applications
734.4 Applications of Fourier Series
1991. Electrostatics:
741. Signal Processing:
2002. Heat Conduction:
752. Communication Systems:
2013. Quantum Mechanics:: 4. Fluid Mechanics:
763. Control Systems:
202Conclusion
774. Electrical Engineering:
203Parabolic PDEs
785. Mechanical Engineering:
20412.1 Heat Equation
796. Quantum Mechanics:
20512.2 Diffusion Equations
807. Image and Signal Processing:
20612.3 Maximum Principles: Mathematical Questions and Solutions:
818. Numerical Analysis:: Practice Question:
20712.4 Initial and Boundary Value Problems
824.5 Fourier Integrals and Transforms
208Initial Conditions:: Boundary Conditions:
83Fourier Integrals:
20912.5 Finite Difference Methods
84Fourier Transforms:
21012.6 Applications
851. Signal Processing:
2111. Heat Transfer:
862. Image Processing:
2122. Diffusion Processes:
873. Partial Differential Equations:
2133. Finance and Economics:
884. Quantum Mechanics:
2144. Image Processing:
895. Spectroscopy:: Practice Question:
2155. Fluid Dynamics:
904.6 Generalized Fourier Series
2166. Population Dynamics:
911. Legendre Polynomials:
2177. Quantum Mechanics:
922. Bessel Functions:
2188. Materials Science:: Mathematical Questions and Solutions:
933. Spherical Harmonics:
219Conclusion
944. Chebyshev Polynomials:
220Hyperbolic PDEs
95Practice Question:: Mathematical Questions and Solutions:
22113.1 Wave Equation
96Separation of Variables
222Physical Interpretation:
975.1 Cartesian Coordinates
223Initial and Boundary Conditions:
98Mathematical Question 1:: Mathematical Question 2:
224Initial Conditions:
995.2 Cylindrical Coordinates
225Boundary Conditions (for a string of length L):: Mathematical Questions and Solutions:
100Applying the boundary conditions:
22613.2 Characteristics
101Mathematical Question 1:: Mathematical Question 2:
227Definition of Characteristics:
1025.3 Spherical Coordinates: Mathematical Question 1:
228Properties of Characteristics:: Mathematical Questions and Solutions:
1035.4 Sturm-Liouville Problems
22913.3 Domain of Dependence and Influence
104Example: Consider the Sturm-Liouville problem:
230Domain of Dependence:
105Properties of Sturm-Liouville Problems:
231Domain of Influence:
106Mathematical Question 1:
232Significance and Applications:: Mathematical Questions and Solutions:
107Mathematical Question 2:: Consider the Sturm-Liouville problem:
23313.4 Initial and Boundary Value Problems
1085.5 Eigenvalue Problems: Properties of Eigenvalue Problems:
234Initial Conditions:
109Conclusion
235Boundary Conditions:: Mathematical Questions and Solutions:
110Integral Transforms
23613.5 Finite Difference Methods
1116.1 Fourier Transforms
237Finite Difference Approximations:
1126.1.1 Fourier Series
238Explicit and Implicit Schemes:
1136.1.2 Fourier Transform
2391. Explicit Schemes:
1146.1.3 Fourier Transform Methods for PDEs: Mathematical Question 1:
2402. Implicit Schemes:
1156.2 Laplace Transforms
241Stability and Convergence:: Mathematical Questions and Solutions:
1166.2.1 Definition and Properties
24213.6 Applications
1176.2.2 Laplace Transform Methods for PDEs: Mathematical Question 1:
2431. Acoustics and Sound Waves:
1186.3 Inverse Transforms
2442. Electromagnetic Waves:
1196.3.1 Inverse Fourier Transform
2453. Seismology and Earthquake Waves:
1206.3.2 Inverse Laplace Transform
2464. Fluid Dynamics:
121Mathematical Question 1:: Mathematical Question 2:
2475. Quantum Mechanics:
1226.4 Convolution Theorem
2486. Numerical Simulations:
1236.5 Applications to PDEs
2497. Signal Processing:: Mathematical Questions and Solutions:
1246.6 Numerical Inversion Techniques
250Conclusion
125Conclusion
251Glossary
126Green’s Functions
252Index
