Chapter 9 focuses on solving boundary value problems in mathematical physics using the methods introduced in the preceding chapters. The primary solution method is Fourier's method of separation of variables, complemented by Sturm–Liouville theory. The chapter is divided into seven sections: 1. **9.1 Heat Diffusion in One Dimension**: Discusses transient heat diffusion problems in one spatial dimension, solving eight specific problems. 2. **9.2 Boundary Value Problems for Vibrating Strings**: Treats boundary value problems for the wave equation in one spatial dimension. 3. **9.3 Fourier's Method for Steady-State Temperatures in Plates**: Develops Fourier's method for solving the two-dimensional Laplace equation in steady-state temperature problems. 4. **9.4 Transient Temperatures in Plates**: Analyzes transient temperature problems in two dimensions. 5. **9.5 Vibrations of Circular Drums**: Examines vibrations of circular drums using the wave equation. 6. **9.6 Steady-State Heat Diffusion in Solids**: Solves steady-state heat diffusion problems in three dimensions. 7. **9.7 Alternative to Fourier's Method**: Introduces an alternative method based on the Laplace transform. Each section provides detailed solutions to specific problems, illustrating the principal types of boundary value problems discussed in Chapter 7.Chapter 9 focuses on solving boundary value problems in mathematical physics using the methods introduced in the preceding chapters. The primary solution method is Fourier's method of separation of variables, complemented by Sturm–Liouville theory. The chapter is divided into seven sections: 1. **9.1 Heat Diffusion in One Dimension**: Discusses transient heat diffusion problems in one spatial dimension, solving eight specific problems. 2. **9.2 Boundary Value Problems for Vibrating Strings**: Treats boundary value problems for the wave equation in one spatial dimension. 3. **9.3 Fourier's Method for Steady-State Temperatures in Plates**: Develops Fourier's method for solving the two-dimensional Laplace equation in steady-state temperature problems. 4. **9.4 Transient Temperatures in Plates**: Analyzes transient temperature problems in two dimensions. 5. **9.5 Vibrations of Circular Drums**: Examines vibrations of circular drums using the wave equation. 6. **9.6 Steady-State Heat Diffusion in Solids**: Solves steady-state heat diffusion problems in three dimensions. 7. **9.7 Alternative to Fourier's Method**: Introduces an alternative method based on the Laplace transform. Each section provides detailed solutions to specific problems, illustrating the principal types of boundary value problems discussed in Chapter 7.