Solution Techniques for Elementary Partial Differential Equations

1. Ordinary Differential Equations: Brief Review. 1.1. First-Order Equations. 1.2. Homogeneous Linear Equations with Constant Coefficients. 1.3. Nonhomogeneous Linear Equations with Constant Coefficients. 1.4. Cauchy-Euler Equations. 1.5. Functions and Operators. 2. Fourier Series. 2.1. The Full Fourier Series. 2.2. Fourier Sine and Cosine Series. 2.3. Convergence and Differentiation. 2.4. Series Expansion of More General Functions. 3. Sturm-Liouville Problems. 3.1. Regular Sturm-Liouville Problems. 3.2. Other Problems. 3.3. Bessel Functions. 3.4. Legendre Polynomials. 3.5. Spherical Harmonics. 4. Some Fundamental Equations of Mathematical Physics. 4.1. The Heat Equation. 4.2 The Laplace Equation. 4.3. The Wave Equation. 4.4. Other Equations. 5. The Method of Separation of Variables. 5.1. The Heat Equation. 5.2. The Wave Equation. 5.3. The Laplace Equation. 5.4. Other Equations. 5.5. Equations with More than Two Variables. 6. Linear Nonhomogeneous Problems. 6.1. Equilibrium Solutions. 6.2. Nonhomogeneous Problems. 7. The Method of Eigenfunction Expansion. 7.1. The Nonhomogeneous Heat Equation. 7.2. The Nonhomogeneous Wave Equation. 7.3. The Nonhomogeneous Laplace Equation. 7.4. Other Nonhomogeneous Equations. 8. The Fourier Transformations. 8.1. The Full Fourier Transformation. 8.2. The Fourier Sine and Cosine Transformations. 8.3. Other Applications. 9. The Laplace Transformation. 9.1. Definition and Properties. 9.2. Applications. 10. The Method of Green's Functions. 10.1. The Heat Equation. 10.2. The Laplace Equation. 10.3. The Wave Equation. 11. General Second-Order Linear Equations. 11.1. The Canonical Form. 11.2. Hyperbolic Equations. 11.3. Parabolic Equations. 11.4. Elliptic Equations. 11.5. Other Problems. 12. The Method of Characteristics. 12.1. First-Order Linear Equations. 12.2. First-Order Quasilinear Equations. 12.3. The One-Dimensional Wave Equation. 12.4. Other Hyperbolic Equations. 13. Perturbation and Asymptotic Methods. 13.1. Asymptotic Series. 13.2. Regular Perturbation Problems. 13.3. Singular Perturbation Problems. 14. Complex Variable Methods. 14.1. Elliptic Equations. 14.2. Systems of Equations. Appendices.