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14. Linear Partial Differential Equations .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 585
14.1. Classification of Second-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 585
14.1.1. Equations with TwoIndependent Variables . .. .. .. .. ... .. .. .. .. .. .. .. .. 585
14.1.2. Equations with Many Independent Variables . .. .. .. ... .. .. .. .. .. .. .. .. 589
14.2. Basic Problems of Mathematical Physics . .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 590
14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590
14.2.2. First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . . . . . . . 593
14.3. Properties and Exact Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
14.3.1. Homogeneous Linear Equations and Their Particular Solutions . . . . . . . . . . . . 594
14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions . . . . . . . . . 598
14.3.3. General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . 600
14.4. Method of Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
14.4.1. Description of the Method of Separation of Variables. General Stage of
Solution . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 602
14.4.2. Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . . . . . . 605
14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . . . . . . . 607
14.4.4. Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . . . . . . . 609
14.5. Integral Transforms Method .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 611
14.5.1. Laplace Transform and Its Application in Mathematical Physics . . . . . . . . . . . 611
14.5.2. Fourier Transform and Its Application in Mathematical Physics . . . . . . . . . . . 614
14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution . . 615
14.6.1. Cauchy Problem for Parabolic Equations .. .. .. .. .. ... .. .. .. .. .. .. .. .. 615
14.6.2. Cauchy Problem for Hyperbolic Equations . . . . . . . . . ... .. .. .. .. .. .. .. .. 617
14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s
Function . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 618
14.7.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 618
14.7.2. Problems for Equation s(x) ∂w = ∂ p(x) ∂w –q(x)w + Φ(x, t) . .. .. .. .. .. 620
∂t ∂x ∂x
14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s
Function. Goursat Problem .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 623
14.8.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 623
2
∂
14.8.2. Problems for Equation s(x) ∂ w 2 = ∂x p(x) ∂w –q(x)w + Φ(x, t) . .. .. .. .. . 624
∂x
∂t
2 ∂w ∂ ∂w
∂ w
14.8.3. Problems for Equation ∂t 2 + a(t) ∂t = b(t) ∂x p(x) ∂x – q(x)w + Φ(x, t) 626
14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve . . . . . 627
14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) . . . . . . . . . . 629
14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables . . . . . . . . . 631
14.9.1. Problems and the Green’s Functions for Equation
2
2
∂y 2 + b(x)
∂x 2 +
a(x) ∂ w ∂ w ∂w + c(x)w =–Φ(x, y) . .. .. .. ... .. .. .. .. .. .. .. .. 631
∂x
14.9.2. Representation of Solutions to Boundary Value Problems via the Green’s
Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 633
14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions
via the Green’s Function . .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 634
14.10.1. Problems for Parabolic Equations .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 634
14.10.2. Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
14.10.3. Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
14.10.4. Comparison of the Solution Structures for Boundary Value Problems for
Equations of Various Types . .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 638