Linear partial Differential Equations

1. Basic Concepts and Definitions, Mathematical Problems, Linear Operators, Superposition Principle.

2. Geometrical Interpretation of a First-Order Equation , Method of Characteristics and General Solutions, Canonical Forms of First-Order Linear Equations, Method of Separation of Variables.

3. Classification of Second-Order Linear Equations,  Second-Order Equations in Two Independent Variables,  Canonical Forms,General Solutions.

4.  The Cauchy Problem,  The Cauchy–Kowalewskaya Theorem, HomogeneousWave Equations, Initial Boundary-Value Problems,  Equations with Nonhomogeneous Boundary Conditions,  NonhomogeneousWave Equations. 

5. Fourier Series and Integrals with Applications, Systems of Orthogonal Functions, Fourier Series, Convergence of Fourier Series, Examples and Applications of Fourier Series, Examples and Applications of Cosine and Sine Fourier Series, The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem, Uniform Convergence, Differentiation, and Integration,  Fourier Integrals.

6. Method of Separation of Variables, The Vibrating String Problem, Existence and Uniqueness of Solution of the Vibrating String Problem, Existence and Uniqueness of Solution of the Heat Conduction Problem, The Laplace and BeamEquations, Nonhomogeneous Problems.

7. Eigenvalue Problems and Special Functions,  Sturm–Liouville Systems, Eigenvalues and Eigenfunctions, Eigenfunction Expansions, Convergence in theMean, Completeness and Parseval’s Equality, Bessel’s Equation and Bessel’s Function, Adjoint Forms and Lagrange Identity,  Singular Sturm–Liouville Systems, Legendre’s Equation and Legendre’s Function, Green’s Functions for Ordinary Differential Equations.

8. Boundary-Value Problems and Applications, Maximum and MinimumPrinciples, Uniqueness and Continuity Theorems, Dirichlet Problem,  Neumann Problem,

9. Higher-Dimensional Boundary-Value Problems, Dirichlet Problem for a Cube, a Cylinder and a Sphere.