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14. Linear Partial Differential Equations

• 14.1. Classification of Second-Order Partial Differential Equations
• 14.1.1. Equations with Two Independent Variables
• 14.1.2. Equations with Many Independent Variables
• 14.2. Basic Problems of Mathematical Physics
• 14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems
• 14.2.2. First, Second, Third, and Mixed Boundary Value Problems
• 14.3. Properties and Exact Solutions of Linear Equations
• 14.3.1. Homogeneous Linear Equations and Their Particular Solutions
• 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions
• 14.3.3. General Solutions of Some Hyperbolic Equations
• 14.4. Method of Separation of Variables (Fourier Method)
• 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution
• 14.4.2. Problems for Parabolic Equations: Final Stage of Solution
• 14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution
• 14.4.4. Solution of Boundary Value Problems for Elliptic Equations
• 14.5. Integral Transforms Method
• 14.5.1. Laplace Transform and Its Application in Mathematical Physics
• 14.5.2. Fourier Transform and Its Application in Mathematical Physics
• 14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution
• 14.6.1. Cauchy Problem for Parabolic Equations
• 14.6.2. Cauchy Problem for Hyperbolic Equations
• 14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green's Function
• 14.7.1. Representation of Solutions via the Green's Function
• 14.7.2. Problems for Equation s(x)wt = [p(x)wx]xq(x)w + Φ(x, t)
• 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green's Function. Goursat Problem
• 14.8.1. Representation of Solutions via the Green's Function
• 14.8.2. Problems for Equation s(x)wtt = [p(x)wx]xq(x)w + Φ(x, t)
• 14.8.3. Problems for Equation wtt + a(t)wt = b(t){[p(x)wx]xq(x)w} + Φ(x, t)
• 14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve
• 14.8.5. Goursat Problem (a Problem with Initial Data at Characteristics)
• 14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables
• 14.9.1. Problems for Equation a(x)wxx + wyy + b(x)wx + c(x)w = −Φ(x, t)
• 14.9.2. Representation of Solutions to Boundary Value Problems via the Green's Functions
• 14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green's Function
• 14.10.1. Problems for Parabolic Equations
• 14.10.2. Problems for Hyperbolic Equations
• 14.10.3. Problems for Elliptic Equations
• 14.10.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types
• 14.11. Construction of the Green's Functions. General Formulas and Relations
• 14.11.1. Green's Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains
• 14.11.2. Green's Functions Admitting Incomplete Separation of Variables
• 14.11.3. Construction of Green's Functions via Fundamental Solutions
• 14.12. Duhamel's Principles in Nonstationary Problems
• 14.12.1. Problems for Homogeneous Linear Equations
• 14.12.2. Problems for Nonhomogeneous Linear Equations
• 14.13. Transformations Simplifying Initial and Boundary Conditions
• 14.13.1. Transformations That Lead to Homogeneous Boundary Conditions
• 14.13.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions
• References for Chapter 14

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