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Handbook of Mathematics for Engineers and Scientists > Contents > 14. Linear Partial Differential Equations
14. Linear Partial Differential Equations
 14.1. Classification of SecondOrder 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)w_{t} = [p(x)w_{x}]_{x} − q(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)w_{tt} = [p(x)w_{x}]_{x} − q(x)w + Φ(x, t)
 14.8.3. Problems for Equation w_{tt} + a(t)w_{t} = b(t){[p(x)w_{x}]_{x} − q(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)w_{xx} + w_{yy} + b(x)w_{x} + 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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