Mathematical Handbook for Scientists and Engineers

10.1-1. Sections 10.2-1 to 10.2-7 deal with partial differential equations of the first order and their geometrical interpretation and include an outline of the Hamilton-Jacobi theory of canonical equations. Sections 10.3-1 to 10.3-3 introduce the characteristics and boundary-value problems of hyperbolic, parabolic, and elliptic second-order equations. Sections 10.4-1 to 10.5-4 present the solutions of the most important linear partial differential equations of physics (heat conduction, wave equation, etc.) from the heuristic point of view of an elementary course and outline the use of integral-transform methods. A more sophisticated theory of linear boundary-value problems and eigenvalue problems is described in Chap.15.

10.1-2. Partial Differential Equations (see also Sec. 9.1-2). (a) A partial differential equation of order r is a functional equation of the form

which involves at least one r^th-order partial derivative of the unknown function Φ = Φ(x₁, x₂,. . . , x_n) of two or more independent variables x_l, x₂,. . .,x_n. A function Φ(x₁, x₂ ,. . . , x_n) which satisfies the given partial differential equation on a specified region of “points” (x₁, x₂, . . . , x_n) is called a solution or integral of the partial differential equation.

The general solution (general integral) of a given r^th-order equation (1) will, in general, involve arbitrary functions. Substitution of specific functions yields particular integrals corresponding to given accessory conditions, e.g., given conditions on Φ(x₁, x₂, . . . , x_n) and/or its derivatives on a curve, surface, etc., in the space of “points” (x₁, x₂, . . . , x_n) (boundary conditions, initial conditions). Many partial differential equations admit additional solutions (singular integrals) which are not obtainable through substitution of specific functions for the arbitrary functions in the general integral (Sec. 10.2-lc).

(b)A partial differential equation is homogeneous if and only if every constant multiple Φa of any solution Φ is a solution. A partial differential equation (1) is linear if and only if F is a linear function of Φ and its derivatives (see also Secs. 10.4-1 and 10.4-2).

(c)Systems of Partial Differential Equations. Compatibility Conditions. A system of partial differential equations

involves a set of unknown functions Φ₁(x₁, x₂, . . . , x_n), Φ₂(x₁, x₂, . . . , x_n), . . . and their partial derivatives. One can reduce every partial differential equation or system of partial differential equations to a system of first-order equations by introducing suitable derivatives as new variables (see also Sec. 9.1-3).

A system of partial differential equations (2) may admit a solution Φ₁, Φ₂, . . . only if the given functions F_i and their derivatives satisfy a set of compatibility conditions (integrability conditions) which ensure that differentiations of two or more equations (2) yield compatible higher-order derivatives. To derive a compatibility condition, eliminate the Φ_k and their derivatives from a set of equations obtained by differentiation of the given partial differential equations (2).

EXAMPLE: Given, differentiation yields , so that the given partial differential equations are compatible only if

(d) Existence of Solutions. As with ordinary differential equations (Sec. 9.1-4), the actual existence and uniqueness of solutions for a given partial differential equation or system of partial differential equations require a proof in each case, even if all compatibility conditions are satisfied. See Refs. 10.5 and 10.18 for a number of existence theorems.

10.1-3. Solution of Partial Differential Equations: Separation of Variables (see also Secs. 10.4-2 to 10.4-9). In many important applications, an attempt to write solutions of the form

permits one to rewrite a given partial differential equation (1) in the “separated” form

Then the unknown functions φ₁(x₁) and φ₀(x₂, x₃, . . . ,x_n) ^{must satisfy the differential equations}

where C is a constant of integration (separation constant) to be determined in accordance with suitably given boundary conditions or other accessory conditions. Note that Eq. (4a) is an ordinary differential equation for the unknown function φ₁(x₁); it may be possible to repeat the separation process with Eq. (4b).

Separation of variables applies particularly well to many linear homogeneous partial differential equations of physics; sometimes separation becomes possible after an appropriate change of variables (see Secs. 10.4-3 to 10.4-9 for examples).

10.2. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

10.2-1. First-order Partial Differential Equations with Two Independent Variables. Geometrical Interpretation (see also Secs. 9.2-2 and 17.3.11). (a) Given a first-order partial differential equation

for the unknown function z = z(x, y), let the given function F be single-valued and twice continuously differentiable, and consider x, y, z as rectangular cartesian coordinates. Then every solution z = z(x, y) of

the partial differential equation (1) represents a surface whose normal has the direction numbers p, q, —1 at every surface point (x, y, z); the solution surface must touch a Monge cone of characteristic directions defined by

at every point (x, y, z).

Fig. 10.2-1. The initial strip and one characteristic strip of the solution surface. Note the Monge cone at P0. {From Burington, R. S.f and C. C. Torrance, Higher Mathematics, Mc-Graw-Hill, New York, 1939.)

A set of values (x, y, z, p, q) is said to describe a planar element associating the direction numbers p, q, –1 (and thus the tangent plane of a hypothetical surface) with a point (x, y, z). A given partial differential equation (1) selects a “field” of planar elements (x, y, z, p, q) tangent to the Monge cones (Fig. 10.2-1).

If F_p and F_q do not depend explicitly on p and q (quasilinear partial differential equation of the first order), then each Monge cone degenerates into a straight line (Monge axis).

(b) Strips and Characteristic Equations. A set of suitably differentiable functions

represents the planar elements (points and tangent planes) along a strip of a regular surface if the functions (2) satisfy the strip condition

Given a first-order partial differential equation (1), every set of functions (2) which satisfies the ordinary differential equations

together with Eq. (1) is said to describe a characteristic strip. A characteristic strip touches a Monge cone at each point (x, y, z); the associated curve x = x(t), y = y(t), z = z(t) (characteristic curve, characteristic) lies on a solution surface and has a characteristic direction at each point. Solution surfaces can touch one another only along characteristics.

(c) Singular Integrals (see also Secs. 9.2-2b, 10.1-2a, and 10.2-3c). Solutions z = z(x, y) of Eq. (1) derived by elimination of p and q from

are singular integrals; they do not satisfy the condition F_p² + F_q² ≠ 0 and are not contained in a general integral of the partial differential equation (1).

10.2-2. The Initial-value Problem It is desired to find a solution z = z{x, y) of Eq. (1) subject to the initial conditions (Cauchy-type boundary conditions)

with

which specify an “initial strip” of points and tangent planes of the solution surface along a regular curve C₀; the projection of C₀ onto the xy plane is to be a simple curve (Sec. 3.1-13). To solve this initial-value problem, find the solution

of the system of characteristic equations (3) subject to the initial conditions (5) for t = 0. The resulting functions (6) satisfy Eq. (1); find the solution z = z(x, y) or an implicit solution φ(x, y, z) =0 by eliminating the parameters t and τ.

The initial-value problem has a unique solution if the given initial conditions (5) imply

Otherwise the problem has a solution only if the given initial conditions (5) describe a characteristic strip; in this case there are infinitely many solutions.

NOTE: If the functions (5a) are left arbitrary subject to the conditions (5b) and (7), then the solution obtained above constitutes a general integral of the given partial differential equation (1).

10.2-3. Complete Integrals. Derivation of General Integrals, Particular Integrals, Singular Integrals, and Solutions of the Characteristic Equations, (a) A complete integral of the first-order partial differential equation (1) is a two-parameter family of solutions

having single-valued derivatives

and continuous second-order derivatives with respect to x, y, λ, μ, except possibly for ∂²Φ/∂λ² and ∂²Φ/∂μ²; a given set of values (x, y, z, p, q) satisfying Eq. (1) must define a unique set of parameters λ, μ. A complete integral (7) yields a general integral if one introduces an arbitrary function μ = μ(λ) and eliminates λ from

(envelope of a one-parameter family of solutions, Sec. 17.3-11).

(b) Derivation of Particular Integrals. To obtain the particular integral corresponding to a given set of initial conditions (5), one must find the correct function μ(λ) to be substituted in the general integral derived in Sec. 10.2-3a. Obtain μ = μ(λ) by eliminating τ from

may yield a singular integral (envelope of a two-parameter family of solutions).

(d) Solution of the Characteristic Equations. Every complete integral (8) of a first-order partial differential equation (1) yields the complete solution of the system of ordinary differential equations (3). x = x(t), y = y(t) are obtained from

where λ, μ, and β are arbitrary constants of integration; z = z(t), p = p(t), and q = q(t) are then obtained by substitution of x = x(t, λ, μ, β) and y = y(t, λ, μ, β) into

(e) Special Cases. Table 10.2-1 lists complete integrals for a number of frequently encountered types of first-order partial differential equations and permits one to apply the methods of Sec. 10.2-3 to many problems. See also Ref. 10.2 for a general method of deriving complete integrals.

Table 10.2-1. Complete Integrals for Special Types of First-order Partial Differential Equations

10.2-4. Partial Differential Equations of the First Order with n Independent Variables, (a) The Initial-value Problem (see also Secs. 10.2-1 and 10.2-2). It is desired to find the solution

of the first-order partial differential equation

* Where g(z, λ) is a solution of p = ƒ(z, λ p).

subject to the initial conditions

where x_i = x_i0(τ₁, τ₂, . . ., , τ_n-1) (i = 1, 2, . . . , n) is to represent a hypersurface free of multiple points, and

To obtain the correct relation between x₁, x₂, . . . , x_n, and z, solve the system of ordinary differential equations

subject to the initial conditions (15) for t = 0, and eliminate the n parameters τ₁, τ₂, . . . , τ_n-1, and t.

The initial-value problem has a unique solution if the given initial conditions (15) imply

(b)Complete Integrals and Solution of the Characteristic Equations (see also Sec. 10.2-3). A complete integral of the first-order partial differential equation (14) is an n-parameter family of solutions

having single-valued derivatives

and continuous second-order derivatives ∂²Φ/∂x_i∂x_k and ∂²Φ/∂x_i∂α_k; a given set of values (x₁ x₂, . . . , x_n; z; p₁ p₂, . . . , p_n) satisfying Eq. (14) must define a unique set of parameters α₁, α₂, . . . , α_n. A complete integral (18) yields a general integral if one introduces n arbitrary functions α_k = α_k(λ₁, λ₂, . . . , λ_n-1) (k = 1, 2, . . . , n) and eliminates the n – 1 parameters λ_j from the n equations

Every complete integral (18) yields the complete solution of the system of ordinary differential equations (16). Obtain x_i = x_i(t) (i = 1, 2, . . . , n) from

where α₁, α₂, . . . , α_n, β₁, β₂, . . . , βn-1 are 2n – 1 arbitrary constants of integration; then find z = z(t) and p_i = P_i(t) (i = 1, 2, . . . , n) by-substituting x_i = x_i(t; α₁, α₂, . . . , α_n; β₁, β₂, . . . , β_n-1) into

(c) Singular Integrals (see also Secs. 10.2-lc and 10.2-3c). Singular integrals of the partial differential equation (14) are solutions z = Φ(x₁, x₂, . . . , x_n) obtained by elimination of the p_i from the n + 1 equations

subject to ∂F/∂x_i + p_i∂F/∂z =0. A given complete integral (18) may yield a singular integral by elimination of α₁, α₂, . . . , α_n from the n + 1 equations

10.2-5. Contact Transformations (see also Sec. 9.2-3b). Some problems involving first-order partial differential equations can be simplified by a twice continuously differentiable transformation

chosen so that every complete differential is transformed into a complete differential (i = 1, 2, . . . ,n), or

Such a transformation is called a contact transformation; a contact transformation necessarily preserves every strip condition and will thus preserve contact (osculation) of regular surface elements for n = 2 (see also Sec. 10.2-1).

A contact transformation (23) transforms the given partial differential equation (14) into a new partial differential equation

with solutions It may happen that the new equation (25) does not contain the ₁ and is thus an ordinary equation.

EXAMPLE: n-dimensional Legendre transformation (see also Secs. 9.2-3b and 11.5-6).

10.2-6. Canonical Equations and Canonical Transformations. (a) Canonical Equations. For a first-order partial differential equation

which does not contain the dependent variable z explicitly, the characteristic equations (16) take the especially simple form

NOTE: The solution of every given first-order partial differential equation (14) can be reduced to the solution of a partial differential equation of the simpler form (27) with n + 1 independent variables x₁, x₂, . . . , x_n, z; for every solution

of the partial differential equation

yields a corresponding solution z = z(x₁, x₂, . . . , x_n) of the given partial differential equation (14) such that u(x₁, x₂, . . . , x_n; z) = 0.

(b) Canonical Transformations (see also Sec. 11.6-8). A twice continuously differentiable transformation

is a canonical transformation if and only if it transforms the canonical equations (29) into a new set of canonical equations

for an arbitrary twice-differentiable function G(x₁, x₂, . . . , x_n; p₁, p₂, . . . , p_n). This is true if and only if

i.e., if and only if is the complete differential dΩ of a “generating function” Ω = Ω(x₁, x₂, . . . , x_n; p₁, p₂, . . . , p_n). Note that Eq. (33) implies Δ ≡ 1.

For every canonical transformation (31) the function appearing in the transformed canonical equations (32) is simply given by

Given a first-order partial differential equation (27) with the canonical equations (29), the new canonical equations (32) are those associated with the partial differential equation

The solution of the transformed partial differential equation (34) is related to the solution z = z(x₁ x₂, . . . , x_n) of the original equation (27) by

Equations (31) and (35) together constitute a contact transformation (Sec. 10.2-5).

A canonical transformation can be specified in terms of its generating function Ω(x₁, x₂, . . . , x_n; p₁, p₂, . . . , p_n); the latter is often given indirectly as a function of the x_i and _i, or of the p_i and _i. In particular, every twice-differentiable function Ω = Ψ(x₁, x₂, . . . , x_n; ₁ ₂, . . . , _n) defines a canonical transformation (31) such that

The canonical transformations (31) constitute a group (see also Sec. 12.2-8).

(c) Poisson Brackets. Given any pair of twice continuously differentiable functions g(x₁, x₂, . . . , x_n, p₁, p₂, . . . , p_n), h(x₁, x₂, . . . , x_n; p₁, p₂, . . . , p_n) one defines the Poisson bracket

so that

Note that [ƒ, g] = 0, [ƒ, h] = 0 implies [g, h] = 0. Given a transformation (31), let

A given transformation (31) is a canonical transformation if and only if it preserves Poisson brackets, i.e., if and only if for all twice continuously differentiable functions g, h.

Now let the variables x_i and p_i be functions of a parameter t such that a set of canonical equations (29) holds. Then

and for every suitably differentiable function ƒ(t; x₁, x₂, . . . , x_n; p₁, p₂, . . . , p_n)

In particular, ∂ƒ/∂t = 0, [ƒ, G] = 0 imply ƒ = constant. Two functions g, h of the x_i(t) and p_i(t) are canonically conjugate if and only if [g, h] = 1; this is true whenever g and h satisfy a pair of canonical equations (e.g., x_i, p_i, and t, G). A given transformation (31) is a canonical transformation if and only if it preserves the relations (42), i.e., if and only if

10.2-7. The Hamilton-Jacobi Equation. Solution of the Canonical Equations, (a) An important application of the theory of first-order partial differential equations is the solution of systems of ordinary differential equations which can be written as canonical equations associated with a partial differential equation of the special form

Note that n + 1 independent variables x and x_i are involved. Since Eq. (29) yields dx/dt = 1, one can write x ≡ t (assuming x = 0 for t = 0); the 2n canonical equations (29) for the x_i and p_i become*

Systems of ordinary differential equations having the precise form (46) are of importance in the calculus of variations (Sec. 11.6-8) and in analytical dynamics and optics.

If it is possible to find an n-parameter solution

of the Hamilton-Jacobi equation (45) with det [∂²Φ/∂x_i∂α_k] ≠ 0, then the solution x_i = x_i(t), p_i = p_i(t) (i = 1, 2, . . . , n) of the system of 2n ordinary differential equations (46) is given by

where the α_k and β_i are 2n constants of integration. One first solves the n equations (48) for the x_i = x_i(t); the p_i = p_i(t) are obtained by substitution of x_i = x_i(t) into p_i = ∂Φ/∂x_i (i = 1, 2, . . . , n) (see also Sec. 10.2-4b).

(b) Use of Canonical Transformations (see also Sec. 10.2-6b). If a complete integral (47) solving the given equations (46) is not known, one may try to introduce a canonical transformation relating the 2n + 2 variables x = t, p, x_i, p_i to 2n + 2 new variables = t, , _i, _i so that p + H ≡ and

In this case,

must be a complete differential of a “generating function” Ω = Ω(t, x₁, x₂, . . . , x_n; p, p₁, p₂, . . . , p_n). In particular, let t = t; then every twice continuously differ-entiable function Ω = Ψ(t, x₁, x₂, . . . , x_n; ₁ ₂, . . . , _n) with det [∂²Ψ/∂x_i ∂_k] ≠ 0 defines a canonical transformation such that

* The remaining canonical equation is

It may be possible to choose this transformation so that does not depend explicitly on the _i (transformation to cyclic variables _i).

defines a canonical transformation (51) yielding constant transformed variables:

As shown in Sec. 10.2-7a, the 2n equations (52) yield the solution x_i = x_i(t), p_i = p_i(t) of the canonical system (46).

Given such a solution of the “unperturbed” canonical system (46), one often desires to solve the canonical equations

where ∊Hi is a small correction term (perturbation, e.g., the effect of a small disturbing field in celestial mechanics). Using the known solution (47) of the "unperturbed" Hamilton-Jacobi equation (45), introduce new variables _i, _i by a canonical transformation (51) with the generating function

Since Eq. (46) has been replaced by Eq. (53), the _i and _i are no longer constants but satisfy transformed canonical equations

which may be easier to solve than the given system (53). If one writes

the corrections ∊X_i(t), ∊P_i(t) to the constants (52) may yield corresponding corrections to the solutions of the unperturbed system (46) by an approximately linear transformation.

10.3. HYPERBOLIC, PARABOLIC, AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS. CHARACTERISTICS

10.3-1. Quasilinear Partial Differential Equations of Order 2 with Two Independent Variables. Characteristics, (a) A partial differential equation of order r is quasilinear if and only if it is linear in the r^th-order derivatives of the unknown function Φ. Thus a real quasi-linear second-order equation with two independent variables x, y has the form

where α₁₁, α₁₂, α₂₂ and B are suitably differentiable real functions of x, y, Φ, ∂Φ/∂x, and ∂Φ/∂y.

(b) Characteristics. Given a real boundary curve C₀ described by

a set of Cauchy-type boundary conditions (called initial conditions in Sec. 10.2-2) specifies boundary values*

A given set of suitably differentiable functions (2) uniquely defines the values of ∂²Φ/∂x² = u(τ), ∂²Φ/∂x ∂y = (τ), ∂²Φ/∂y² = ω(τ) (and also the values of higher-order derivatives of Φ) on the curve (2a) at every point P₀ where the functions (2a) do not satisfy the ordinary differential equation

This is true because the derivatives u, ϑ, ω of Φ on C₀ must satisfy Eq. (1) and the “second-order strip conditions”

so that, for instance,

Equation (4) holds at P₀ if C₀ is a segment of a characteristic base curve (often called a characteristic) y = y(x) satisfying Eq. (4), or if C₀ touches such a curve at P₀.

Properly speaking, the characteristics associated with the given partial differential equation (1) are curves x = x(τ), y = y(τ), z = z(τ) on the solution surface z = Φ(x, y) such that y = y(x) satisfies Eq. (4). Since the expression (6) must be

* If one is given the boundary values of the normal derivative ∂Φ/∂n (Sec. 5.6-1), say

solve Eq. (3) together with dz/dτ = p(dx/dτ) + q(dy/dτ) = 0 to obtain q(τ) and finite on the solution surface, p = ∂Φ/∂x and q = dΦ/∂y must satisfy the ordinary differential equation

on every characteristic defined by Eq. (4), with corresponding plus and minus signs.

NOTE: The second-order derivatives of Φ may be discontinuous (though finite) on a characteristic, so that different solutions can be “patched together” along characteristics.

(c) Hyperbolic, Parabolic, and Elliptic Partial Differential Equations. The given partial differential equation (1) is

Hyperbolic if α₁₁α₂₂ – α₁₂² < 0 in the region of points (x, y) under consideration, so that Eq. (4) describes two distinct families of real characteristic base curves

Parabolic if α₁₁α₂₂ – α₁₂² = 0, so that there exists a single family of real characteristic base curves

Elliptic if α₁₁α₂₂ – α₁₂² > 0, so that no real characteristics exist

10.3-2. Solution of Hyperbolic Partial Differential Equations by the Method of Characteristics (see also Sec. 10.3-4). In the hyperbolic case (α₁₁α₂₂ – α₁₂² < 0), simultaneous solution of the four ordinary differential equations (4) and (7) yields p = ∂Φ/∂x and q = ∂Φ/∂y on the solution surface as functions of x and y, so that Φ = Φ(x, y) can be obtained by further integration. In many applications, ∂Φ/dx and ∂Φ/∂y rather than Φ(x, y) are of paramount interest (velocity components); the method forms the basis for many analytical and numerical solution procedures in the theory of compressible flow.

Computations are considerably simplified in special cases. If B ≡ 0, one has

where the subscripts refer to the characteristics derived, respectively, with a plus sign and a minus sign in Eqs. (4) and (6). If, in addition, α₁₁, α₁₂, and α₂₂ depend only on ∂Φ/∂x, ∂Φ/∂y one need only solve Eq. (7) to obtain the characteristics (e.g., two-dimensional steady supersonic flow). Again, if α₁₁, α₁₂, and α₂₂ depend only on x, y one need only solve Eq. (4).

10.3-3. Transformation of Hyperbolic, Parabolic, and Elliptic Differential Equations to Canonical Form. For convenience, let α₁₁, α₁₂, and α₂₂ be functions of x and y alone, so that the ordinary differential equation (4) separates into two linear first-order equations

where α₁ α₂ are arbitrary constants. Depending on the sign of the function α₁₁α₂₂ – α₁₂² in the region of points (x, y) under consideration, * three cases arise.

1. Hyperbolic Partial Differential Equation (α₁₁α₂₂ – α₁₂² < 0). λ₁(x, y) and λ₂(x, y) are real and distict. There exist two one-param-eter families of real characteristics (9a) and (9b); a curve of each family passes through every point (x, y) under consideration. Introduce new coordinates

to transform the given partial differential equation (1) to the canonical form

The alternative coordinate system

yields the second canonical form

2. Parabolic Partial Differential Equation (α₁₁α₁₂ – α₁₂² = 0). λ₁(x, y) and λ₂(x, y) are real and identical. There exists a single one-parameter family of real characteristics (9); one characteristic passes through each point (x, y) under consideration. Introduce

where h₀(x, y) is an arbitrary suitably differentiable function such that Equation (1) is transformed to the canonical form

3. Elliptic Partial Differential Equation (α₁₁α₂₂ – α₁₂² > 0). λ₁(x, y) and λ₂(x, y) and hence also h₁(x, y) and h₂(x, y) are complex conjugates; no real characteristics exist. Introduce

* In the usual applications, the discriminant α₁₁α₂₂ – α₁₂² does not change sign in the region under consideration. Note also that the sign of α₁₁α₂₂ – α₁₂² is invariant with respect to any real continuously differentiable coordinate transformation with nonvanishing Jacobian.

to transform Eq. (1) to the canonical form

These three types of partial differential equations differ significantly with respect to the types of boundary conditions yielding valid and unique integrals (Secs. 10.3-4 and 15.6-2).

10.3-4. Typical Boundary-value Problems for Second-order Equations. (a) Hyperbolic Differential Equations. The Cauchy initial-value problem of Sec. 10.3-16 requires one to solve the hyperbolic differential equation (1), given Φ, ∂Φ/∂x, and ∂Φ/∂y on an arc C₀ of a regular

FIG. 10.3-1.Boundary-value problems for hyperbolic differential equations.

curve which is neither a characteristic (6) nor touches any characteristic. Such a curve intersects each characteristic at most once; the given initial values determine Φ in a triangular region D₀ bounded by C₀ and a characteristic of each family (Fig. 10.3-la). More specifically, the value of Φ at each point P of D₀ is determined by the values of Φ and its derivatives on the portion C_P of C₀ which is hounded by the characteristics through P.

A second type of boundary-value problem prescribes only a linear relation α(∂Φ/∂n) + βΦ = b(x, y) on an arc C₀ specified as above; in addition, Φ is given on a characteristic arc C_c through one end point of C₀(Fig. 10.3-16).

A third type of boundary-value problem prescribes Φ on two intersecting characteristic arcs C_c and C′_c (>Fig. 10.3-lc).

Combinations of these three types of problems will indicate admissible boundary conditions for more complicated boundaries. Thus in Fig. 10.3-ld, Φ, ∂Φ/∂x, and ∂Φ/∂y may be given on C₀, but only one relation of the type α(∂Φ/∂n) + βΦ = b(x, y) can be prescribed on each of C′_oand C′′₀. The solutions in the various regions indicated in 10.3-5 are “patched” together along characteristics (“patching curves”) so that Φ is continuous, while ∂Φ/∂x and ∂Φ/∂y may be discontinuous. Note that closed boundaries are not admissible.

EXAMPLE: Initial-value problems for the hyperbolic one-dimensional wave equation, Sec. 10.3-5.

(b)Parabolic Differential Equations, There exists one and only one family of characteristics. Although the Cauchy problem can again be solved for a suitable arc C₀, one is usually given Φ on a characteristic (= t) = 0 and α(∂Φ/∂n) + βΦ on two curves which do not intersect or touch each other or any characteristic. Closed boundaries in the xy plane are not admissible.

EXAMPLE: An admissible boundary-value problem for the parabolic diffusion equation ∂²Φ/∂x² + (I/γ²) ∂Φ/∂t = 0 specifies Φ(x, t) = Φ₀(x) on the characteristic t = 0 (initial conditions) and α(x, t) ∂Φ/∂t + β(x, t)Φ on the curves x = a and x = b (boundary conditions).

(c) Elliptic Differential Equations (see also Secs. 10.4-1 and 15.6-2). No real characteristics exist; Cauchy-type boundary conditions are not admissible. Typical problems specify α(x, y)(∂Φ/∂n) + β(x, y)Φ on a curve C enclosing a bounded or unbounded solution region (“true” boundary-value problems).

10.3-5. The One-dimensional Wave Equation (see also Secs. 10.3-4a, 10.4-8a,and 10.4-9b). The hyperbolic differential equation

has the general solution

which represents a pair of arbitrarily-shaped waves respectively propagated in the +x and –x directions with phase velocity c. The characteristics x ± ct = constant are loci of constant phase (Fig. 10.3-2). Sections 10.3-5a, b, and c list solutions for three types of initial-value problems (e.g., waves in a string); in practice, the Fourier-expansion

Fig. 10.3-2. Characteristics for the one-dimensional wave equation

method of Sec. 10.4-9b may be preferable, since it applies also to non-homogeneous differential equations (forced vibrations).

(a) The initial conditions

specify a true Cauchy initial-value problem (Secs. 10.3-1 and 10.3-4a; see also Figs. 10.3-la and 10.3-2). The solution is

Disturbances initially restricted to any given interval a < x < b affect the solution in the interval a – ct < x < a + ct. Discontinuities in Φo(x) are propagated in both directions.

(b) The initial conditions

and the boundary condition

specify a combination-type boundary-value problem (see also Sec. 10.3-4a and Figs. 10.3-ld and 10.3.2). The solution is

which corresponds to superposition of incoming and reflected waves.

and the boundary conditions

define another combination-type problem (see also Figs. 10.3-ld and 10.3-2). The solution is given by Eq. (24) if one interprets P(x) and Q(x) as odd periodic functions with period 2L and respectively equal to Φ₀(x) and ϑ₀(x) for 0 ≤ x ≤ L.

10.3-6. The Riemann-Volterra Method for Linear Hyperbolic Equations (see also Secs. 10.3-4a and 15.5-1). It is desired to solve the Cauchy problem (Sec. 10.3-1) for the real hyperbolic differential equation

with Φ and ∂Φ/∂x, ∂Φ/∂y given on a boundary curve C₀ satisfying the conditions of Sec. 10.3-1. Referring to Fig. 10.3-la, the solution at each point P with the coordinates x = ξ, y = η is given in terms of the “initial values” of Φ(x, y), ∂Φ/∂x, and ∂Φ/∂y on the boundary-curve segment C_p cut off by the characteristics through P:*

* See also the footnote to Sec. 10.3-1b.

where the so-called Riemann-Green function G_R(x, y; ξ, η) is continuously differentiable on and inside the region D_P bounded by C_Pand the characteristics through P and satisfies the conditions of the simpler boundary-value problem

EXAMPLES: For a ≡ b ≡ c ≡ 0. G_R ≡ 1; and for a ≡ b ≡ 0, c ≡ constant, one has where J₀(z) is the zero-order Bessel function of the first kind (Sec. 21.8-1). For many practical applications involving linear hyperbolic differential equations with constant coefficients, the integral-transform methods (Sec. 10.5-1) are preferable.

10.3-7. Equations with Three or More Independent Variables. A real partial differential equation of the form

where Φ = Φ(x₁, x₂, . . . , x_n), is an elliptic partial differential equation if and only if the matrix [α_ik] is positive definite (Sec. 13.5-2) throughout the region of interest.

In many problems involving nonelliptic partial differential equations, the unknown function Φ depends on n “space coordinates” x₁, x₂, . . . , x_n and a “time coordinate” t; the partial differential equation takes either of the forms

where the matrix [c_ik] ≡ [c_ik(x₁, x₂, . . . , x_n; t)] is real and positive definite and B is a function of the x_i, t, Φ, and the first-order derivatives. The two types of partial differential equations are respectively referred to as hyperbolic and parabolic differential equations.

Characteristics of the more general partial differential equations (31) and (32) are surfaces or hypersurfaces on which Cauchy-type boundary conditions cannot determine higher-order derivatives of the solution (Ref. 10.5). Elliptic partial differential equations have no real characteristics. The concept of characteristics has also been extended to certain partial differential equations of order higher than two, and to some systems of partial differential equations (Ref. 10.5).

10.4. LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS. PARTICULAR SOLUTIONS

10.4-1. Physical Background and Survey. (a) Many problems of classical physics require one to find a solution Φ(x, t) or Φ(r, t) of a linear partial differential equation on a given space interval or region V (Table 10.4-1). The unknown function Φ and/or its derivatives must, in addition, satisfy given initial conditions for t = 0 and linear boundary conditions on the boundary S of V. Related problems arise in quantum mechanics.

Each partial differential equation listed in Table 10.4-1 is homogeneous (Sec. 10.1-2b) if ƒ ≡ 0. A given boundary condition is, again, homogeneous if and only if it holds for every multiple αΦ of any function which satisfies the condition. Inhomogeneities represent the action of external influences (forces, heat sources, electric charges or currents) on the physical system under consideration. Typically, an elliptic differential equation describes an equilibrium situation (steady-state heat flow, elastic deformation, electrostatic field). Parabolic and hyperbolic differential equations describe transients (free vibrations, return to equilibrium after a given initial disturbance) or, if there are time-dependent inhomogeneities (“forcing functions” in the differential equation or boundary conditions), such equations describe the propagation of disturbances (forced vibrations, radiation).

One can relate each problem of the type discussed here to an approximating system of ordinary differential equations by replacing each space derivative by a difference coefficient in the manner of Sec. 20.8-3 (method of difference-differential equations). This method is not only useful for numerical calculations; the analogy to a discrete-variable problem of the type described in Secs. 9.4-1 to 9.4-8 may give interesting physical insight.

(b) Construction of Solutions by Superposition. The most important methods for the solution of linear differential equations are based on the fundamental superposition theorems stated explicitly in Secs. 9.3-1 and 15.4-2. The most important solution methods superimpose a judiciously chosen set of trial functions to construct solutions which match given forcing functions, given boundary conditions, and/or given initial conditions. Eigenfunction expansions (Secs. 10.4-2 and 15.4-12) and integral-transform methods (Secs. 9.3-7, 9.4-5, and 10.5-1 to 10.5-3) are systematic schemes for constructing such solutions. Green's-function methods (Secs. 9.3-3, 9.4-3, 15.5-1 to 15.5-4, 15.6-6, 15.6-9, and 15.6-10)

Table 10.4-1. The Most Important Linear Partial Differential Equations of Classical Physics

are superposition schemes which reduce the solution of suitable problems to that of problems with simpler forcing functions or boundary conditions.

The general theory of linear boundary-value problems is treated in Chap. 15; Secs. 10.4-3 to 10.4-8 present useful particular solutions from the heuristic point of view of an elementary course.

(c) Choice of Coordinate System. The system of coordinates x¹, x², or x^l, x², x³ used to specify the point (r) is usually chosen so that (1) separation of variables is possible (Sec. 10.1-3) and/or (2) the given boundary S becomes a coordinate line or surface, or a pair of coordinate lines or surfaces.

10.4-2. Linear Boundary-value Problems (see also Secs. 15.4-1 and 15.4-2). (a) Let V be a given three- or two-dimensional region of points (r), and let S be the boundary surface or boundary curve of V. One desires to solve the partial differential equation

subject to a set of boundary conditions

where LΦ and B_iΦ are linear homogeneous functions of the unknown function Φ(r) and its derivatives. Every solution of this linear boundary-value problem can be written as the sum Φ = Φ_A + Φ_B of solutions of the simpler boundary-value problems

and

Note that Eq. (2) involves a homogeneous differential equation, whereas Eq. (3) has homogeneous boundary conditions.

(b) Homogeneous Differential Equation with Nonhomogeneous Boundary Conditions. The particular solutions listed in Secs. 10.4-3 to 10.4-6 often permit one to expand Φ_A(r) as an infinite series or definite integral

over suitably chosen solutions Φ_μ(r) of Eq. (2a); the coefficients α_μ or α(μ) are chosen so as to satisfy the boundary conditions (2b). Frequently the approximation functions Φ_μ(r) are complete orthonormal sets (Sec. 15.2-4; e.g., Fourier series, Fourier integrals); one may then expand the

given functions b_i(r) in the form (4) and obtain the unknown coefficients α_μ or α(μ) by comparison of coefficients (Sec. 10.4-9).

(c) Nonhomogeneous Differential Equation with Homogeneous Boundary Conditions: Eigenfunction Expansions (see also Secs. 15.4-5 to 15.4-12). For an important class of partial differential equations (1), the solution Φ_B(r) of Eq. (3) can be constructed by a similar superposition of solutions Ψ(r) of the related homogeneous differential equation

for the different possible values of λ permitting Ψ(r) to satisfy the homogeneous boundary conditions

In general, such solutions exist only for specific values of the parameter λ (eigenvalues); the solutions Ψ = Ψ_λ(r) corresponding to each eigenvalue are called eigenfunctions of the boundary-value problem (5).

Sections 10.4-3 to 10.4-8 list particular solutions Ψ(r) for a number of partial differential equations of the form (5a). These functions may be superimposed to form solutions of the corresponding nonhomogeneous problem (3). Eigenfunctions corresponding to discrete sets of eigenvalues often form convenient orthonormal sets (Sec. 15.2-4) for expansion of forcing functions and solutions in the form

(refer also to Table 8.7-1), whereas continuous sets (continuous spectra) S_λ of eigenvalues λ yield integral-transform expansions.

Substitution of Eq. (6a) or (6b) into Eq. (3) yields the unknown coefficients β_λ or β(λ). Refer to Secs. 10.5-1 and 15.4-12 for the general theory of this solution method and its range of applicability. An important alternative method of solving Eq. (3), the so-called Green's-function method, is treated in Secs. 15.5-1 to 15.5-4.

(d) Problems Involving the Time Variable. In problems involving the time as well as the space coordinates, one desires to solve a linear differential equation

subject to a set of linear initial conditions

as well as the linear boundary conditions

Since the initial conditions (7b) are simply boundary conditions on the “coordinate surface” t = 0, the methods of Secs. 10.4-2a and b apply (Sec. 10.4-9). The following procedures may simplify the treatment of initial conditions:

1. Φ_B = Φ_B(r, t) can be further split into a sum of functions respectively satisfying homogeneous initial conditions and homogeneous boundary conditions.

2. Separation of variables (Secs. 10.1-3, 10.4-7b, and 10.4-8).

3. Laplace transformation of the time variable (Secs. 10.5-2 and 10.5-3a).

4. Duhamel's method (Sec. 10.5-4).

10.4-3. Particular Solutions of Laplace's Differential Equation: Three-dimensional Case (see also Table 6.5-1, Sec. 10.1-3, and Secs. 15.6-1 to 15.6-9; see Ref. 10.5 for solutions employing other coordinate systems)

(a) Rectangular Cartesian Coordinates x, y, z.

admits the particular solutions

which combine into various products of real linear, exponential, trigonometric, and/or hyperbolic functions.

(b)Cylindrical Coordinates r′, φ, z. Let Φ = u(φ)ϑ(z)ω(r′) (Sec. 10.1-3). Then

separates into

where uniqueness requires u(φ + 2π) = u(φ) or m = 0, ±1, ±2, . . . , and K is an arbitrary constant (separation constant, Sec. 10.1-3) to be determined by the given boundary conditions. Equation (10) admits particular solutions (cylindrical harmonics) of the form

where Z_m(ζ) is a cylinder function (Sec. 21.8-1); in particular, if a given problem requires to be analytic for r = 0, then Z_m(ζ) must be a Bessel function of the first kind (Sec. 21.8-1). Note that complex-conjugate solutions (14) combine into real particular solutions like

for real K. Such solutions can be superimposed to form real Fourier series. Note that m = 0 in cases of axial symmetry.

separates into

where regularity for ϑ = 0, ϑ = π and uniqueness require that m = 0, ±1, ±2, . . . , ±j and,j = 0, 1, 2, ... . Equation (15) admits particular solutions of the form

where the P_j^m(ζ)are associated Legendre functions of the first kind of degree j (Sec. 21.8-10). Combination of such solutions yields more general particular solutions

with

The functions (21) satisfy Eq. (15) for r = constant and are called spherical surface harmonics of degree,j (see also Sec. 21.8-12).

There are 2j + 1 linearly independent spherical surface harmonics of degree j. For orthogonal-series expansion of solutions, note that the functions

(j = 0, 1, 2, . . . ; m = 0, 1, 2, . . . , j) or the sometimes more convenient functions

constitute orthonormal sets in the sense of Sec. 21.8-2. These functions are called tesseral spherical harmonics (sectorial spherical harmonics for m = j; see also Secs. 10.4-9a, 15.2-6, and 21.8-11). The orthonormal functions

are known as zonal spherical harmonics.

If one admits solutions with singularities for ϑ = 0, ϑ = π, one must add analogous solutions involving the associated Legendre functions of the second kind (Ref. 21.3).

10.4-4. Particular Solutions for the Space Form of the Three-dimensional Wave Equation (see also Secs. 10.3-5 and 15.6-10). The differential equation

is obtained, for example, by separation of the time variable t in the three-dimensional wave equation (44). The coefficient k² may be negative (k = i_k, space form of the Klein-Gordon equation).

For suitably given homogeneous linear boundary conditions (e.g., Φ = 0 on the boundary of S of a bounded region V), Eq. (22) admits solutions only for a corresponding discrete set of values of k² (eigenvalue problem, Sec. 15.4-5; see Sec. 10.4-9b for an example).

(a) Rectangular Cartesian Coordinates x, y, z. Equation (19) has the particular solutions

which may be combined into various products of real linear, exponential, trigonometric, and hyperbolic functions.

(b) Cylindrical Coordinates r′, φ, z (see also Sec. 10.4-3b). Let φ = u(φ)ϑ(z)w(r′). Then Eq. (22) separates into Eq. (11), Eq. (12), and

where uniqueness requires m = 0, ±1, ±2, . . . , and K is an arbitrary separation constant to be determined by the boundary conditions. Equation (22) admits solutions of the form

Note that m = 0 for axial symmetry.

(c) Spherical Coordinates r, ϑ, φ (see also Sec. 10.4-3c). Let Φ = u(Φ)ϑ(cos ϑ)ω(r). Then Eq. (22) separates into Eq. (16), Eq. (17), and

where regularity for ϑ = 0, ϑ = π and uniqueness require that m = 0, ±1, ±2, . . . , ±j, and j = 0, 1, 2, . . . . Equation (22) admits particular solutions of the form

where Y_j(ϑ, φ) is a spherical surface harmonic (21). In particular, if a given problem requires Φ to be analytic for r = 0, the are spherical Bessel functions of the first kind (Sec. 21.8-8).

10.4-5. Particular Solutions for Two-dimensional Problems (see also Secs. 15.6-7 and 15.6-10b). (a) Laplace's equation

admits the particular solutions

where K, like a, b, α, β, A, B, is an arbitrary parameter to be determined by the boundary conditions.

(b) Space Form of the Wave Equation. The two-dimensional space form (22) of the wave equation admits the particular solutions

(c) Complex-conjugate solutions (29) and (31) combine into various products of real linear, exponential, trigonometric, and/or hyperbolic functions.

10.4-6. The Static Schrödinger Equation for Hydrogenlike Wave Functions.

The three-dimensional partial differential equation

admits the particular solutions

where the L_n^k(ζ) are Laguerre functions (Sec. 21.7-5). If Φ is to be normalizable (Sec. 15.2-1b), one is restricted to the first type of solution (34). Normalizable solutions exist only for eigenvalues (Sec. 15.4-5) λ² such that c/λ = n = 0, 1, 2, . . . . In this case the Laguerre functions reduce to associated Laguerre polynomials, and the solutions (34) constitute an orthogonal set in the sense of Secs. 21.7-5 and 21.8-12 (see also Ref. 10.8).

10.4-7. Particular Solutions for the Diffusion Equation (see also Secs. 10.3-4b, 10.4-9, 10.5-3, 10.5-4, and 15.5-3). (a) The one-dimensional diffusion equation

admits the particular solutions

where k is an arbitrary separation constant to be determined by the boundary conditions.

(b) The two- or three-dimensional diffusion equation

admits the particular solutions

where Φ_k(r) is any particular solution of the corresponding Helmholtz equation (22) (Secs. 10.4-4 and 10.4-5b); k is an arbitrary separation constant to be determined by the boundary conditions. Equation (37) also admits the particular solution

10.4-8. Particular Solutions for the Wave Equation. Sinusoidal Waves (see also Secs. 4.11-4b, 10.3-5, 10.4-9, 10.5-2, and 15.6-10). (a) The one-dimensional wave equation

admits particular solutions of the form

where k is an arbitrary constant to be determined by the boundary conditions. The functions (41) and the corresponding values of k² are usually the eigenfunctions and eigenvalues of an eigenvalue problem, Sees. 10.4-2 and 15.4-5.

Solutions of the form (41) combine into real solutions

The circular frequency ω, the frequency v = ω/2π, the wave number k, the wavelength λ = 2π/k, and the phase velocity c of the sinusoidal waves are related by

Sinusoidal waves (42) may be superimposed to form Fourier series or Fourier integrals for more general waves.

(b) The two- or three-dimensional wave equation

admits particular solutions of the form

The cylindrical waves (47) are propagated in the ±z direction with phase velocity c^′ = ω/K = kc/K, which is seen to depend on ω and K (dispersion). One defines the group velocity in the z direction as dω/dK = Kc/k.

admits particular solutions of the form

where s = σ + iω is a root of the quadratic equation

Complex-conjugate solutions (53) combine into damped sinusoidal waves in the manner of Sec. 9.4-1b; again, double roots of Eq. (54) are treated as in Sec. 9.4-1b. Equation (52) includes Eqs. (35) and (40) as special cases. An analogous generalization applies to the multidimensional case.

10.4-9. Solution of Boundary-value Problems by Orthogonal-series Expansions: Examples (see also Sec. 10.5-3). The following examples illustrate the use of the particular solutions listed in Secs. 10.4-3 to 10.4-8; see also Table 8.7-1.

(a) Dirichlet Problem for a Sphere (see also Secs. 10.4-3c, 15.6-2a, and 15.6-6c). One desires to find the function Φ(r) which satisfies Laplace's equation ∇²Φ = 0 throughout a given sphere r < R and assumes suitably given boundary values Φ(R, ϑ, φ) = b(ϑ, 6) on the sphere. The use of spherical coordinates r, ϑ, φ is clearly indicated (Sec. 10.4-lc). One attempts to write the unknown function as a sum of solutions (19) which remain finite for r = 0 (A = 0):

The unknown coefficients α_jm, β_jm are obtained from the given boundary conditions

with the aid of the orthogonality conditions of Sec. 21.8-12:

(b) Free Vibrations of an Elastic String (see also Sec. 10.3-5). The lateral displacement Φ(x, t) of an elastic string satisfies the one-dimensional wave equation

In addition, let

Consider the special case Φ_a(t) ≡ Φ_b(t) ≡ 0. The standing-wave solutions (42) of the one-dimensional wave equation fit these given boundary conditions whenever γ = –π/2 and the half wavelength λ/2 ≡ π/k is an integral fraction of the string length L, or

One next attempts to write the solution as an infinite sum of these particular solutions, i.e.,

This is seen to be a Fourier series (Sec. 4.11-4b) whose coefficients α_n, b_n fit the given initial conditions if

The solution is seen to be the sum of harmonic standing waves (modes of vibration) “excited” by the given initial conditions.

(c) Free Oscillations of a Circular Membrane. The transverse displacement Φ(r) of a membrane clamped along its boundary circle r = 1 satisfies the wave equation together with the boundary condition Φ = 0 (r = 1) and a given set of initial conditions

One uses polar coordinates r, φ. Since Φ is to be regular for r = 0, one attempts to superimpose solutions (51) involving only Bessel functions of the first kind, i.e., Z_m(kr) = J_m(kr) (Sec. 21.8-1). Such solutions represent characteristic oscillations satisfying the given boundary condition Φ = 0 (r = 1) if

where k_mn is the n^th real positive zero of J_m(ζ); the problem is an eigenvalue problem. The solution appears as a sum of characteristic oscillations

where the unknown coefficients are determined as ordinary Fourier coefficients upon substitution of the given initial conditions for t = 0.

10.5. INTEGRAL-TRANSFORM METHODS

10.5-1. General Theory (see also Secs. 8.1-1, 8.6-1, 9.3-7, 9.4-5, 10.4-2c, and 15.4-12). It is desired to find the solution Φ = Φ(x₁, x₂, . . . , x_n), subject to suitable boundary conditions, of the real linear partial differential equation

where L2 is a linear homogeneous differential operator whose derivatives and coefficients involve x₁, x₂, . . . , x_n-1 but not x_n. For simplicity, assume x_n to range over a fixed bounded or unbounded interval (a, b) for all x₁, x₂, . . . , x_n-1, so that the boundary S of the solution region consists of two x_n-coordinate surfaces (see also Sec. 10.4-lc). One can often simplify Eq. (1) by applying a linear integral transformation

to both sides. If the transformation kernel K(x_n, s) is chosen so that

then partial integration (generalized Green's formula, Sec. 15.4-3) yields a possibly simpler differential equation

to be satisfied by the unknown integral transformΦ(x₁,x₂ , . . . ,x_n-1;s). Note that the new differential equation introduces the boundary values of Φ and ∂Φ/∂x_n given for x_n = α, x_n = b and variable x₁, x₂, . . . , x_n-1. The integral-transform method assumes the existence of the relevant integrals (2) and a convergent inversion formula

which usually involves complex integration (see also Sec. 15.3-7). Table 8.6-1 lists a number of useful integral transformations and inversion formulas.

Note that constants of integration in solutions of the transformed differential equation (4) must be regarded as arbitrary functions of s; similarly, arbitrary functions of x₁, x₂, . . . , x_n-1 must also be arbitrary functions of s.

The integral-transform method may be generalized to apply to differential equations involving higher-order derivatives of x_n; again, the integral transformation may operate on two variables x_n, x_n-1 simultaneously (Ref. 8.6). If 5-function terms in H(x_n, s) are admitted, Eq, (5) can yield series expansions for Φ over finite intervals (a, b) (finite integral transforms, Sec. 8.7-1 and Ref. 10.21).

10.5-2. Laplace Transformation of the Time Variable (see also Secs. 8.2-1, 8.3-1, 9.3-7, and 9.4-5). Initial-value problems involving hyperbolic or parabolic differential equations are often simplified by the unilateral Laplace transformation

which transforms every suitable linear partial differential equation with constant coefficients

into a new and possibly simpler differential equation

for the unknown integral transform Φ(r, s) of Φ(r, t). The boundary conditions are similarly transformed; note that Eq. (7) includes the effect of given initial conditions.

10.5-3. Solution of Boundary-value Problems by Integral-transform Methods: Examples (see also Sec. 10.4-9). The following examples illustrate the simplest applications of integral transformations to the solution of boundary-value problems (see also Refs. 10.3, 10.18, and 10.19).

(a) One-dimensional Heat Conduction in a Wall with Fixed Boundary Temperatures: Use of Laplace Transforms. Duhamel's theorem (Sec. 10.5-4) reduces an important class of one-dimensional heat-conduction problems to the form

The differential equation transforms into

If, in particular, Φ₀(x) = Φ₀ = constant, one has

where the functions C₁(s) and C₂(s) must be chosen so as to match the transformed boundary conditions

Φ(x, t) is obtained as an inverse Laplace transform by one of the methods of Secs. 8.4-2 to 8.4-9. In particular, for a = 0, Φ₀ = 0, Φ₀ = 0, one has

Note that this problem can also be solved by the method of Sec. 10.4-9.

(b) Heat Conduction into a Wall of Infinite Thickness: Use of Fourier Sine and Cosine Transforms. The method of Sec. 10.5-3a still applies if a = 0, b = ∞, and one is given

One may, instead, apply the Fourier sine transformation

(Sec. 4.11-3) with the aid of Sec. 4.11-5c to obtain the transformed differential equation

and the transformed initial condition Φ(s, 0 + 0) = Φ₀(s). The transformed problem has the solution

and If, in particular, one is given Φ₀(t) = con-stant = Φ₀ and Φ₀(x) ≡ 0, one has Φ₀(s) = 0, and

If the given boundary condition specifies, instead,

(zero heat flux through boundary), one uses the Fourier cosine transformation

and proceeds as before.

10.5-4. Duhamel's Formulas (see also Sec. 9.4-3). (a) Let L be a homogeneous linear operator whose coefficients and derivatives do not involve the time variable t. Let Φ(x, t) be the solution of the initial-value problem

with as many homogeneous linear boundary conditions for x = 0 and/or x = L as needed, and let Ψ(x, t) be the solution of the same problem for b(t) ≡ 1 (t > 0). Then

(b) The solution Φ(r, t) of the generalized diffusion problem

with time-dependent forcing functions ƒ(r, t) and b(r, t) is related to the solution Ψ(r, t; λ) of the simpler problem

where ƒ(r, λ) and b(r, λ) depend on a fixed parameter λ rather than on the variable t. One obtains Φ(r, t) from

Fourier series and Fourier integrals Chap. 4

Curvilinear coordinates Chap. 6

The Laplace transformation and other integral transformations Chap. 8

Calculus of variations Chap. 11

Boundary-value problems and eigenvalue problems Chap. 15

Numerical solutions Chap. 20

Special transcendental functions Chap. 21

10.6-2. References and Bibliography (see also Sec. 15.7-2).

10.1. Bers, L., F. John, and M. Schechter: Partial Differential Equations, Wiley, New York, 1963.

10.2. Bieberbach, L.: Differentialgleichungen, 2d ed., Springer, Berlin, 1965.

10.3. Churchill, R. V.: Fourier Series and Boundary-value Problems, 2d ed., McGraw-Hill, New York, 1963.

10.4. ———: Operational Mathematics, 2d ed., McGraw-Hill, New York, 1958.

10.5. Courant, R., and D. Hilbert: Methoden der mathematischen Physik, 3 vols., Wiley, New York, 1953/67.

10.6. Duff, G. F., and D. Naylor: Differential Equations of Applied Mathematics, Wiley, New York, 1965.

10.7. Epstein, B.: Partial Differential Equations, McGraw-Hill, New York, 1962.

10.8. Feshbach, H., and P. Morse: Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

10.9. Frank, P., and R. Von Mises: Die Differentialgleichungen der Mechanik und Physik, 2d ed., M. S. Rosenberg, New York, 1943.

10.10. Friedman, B.: Principles and Techniques of Applied Mathematics, Wiley, New York, 1956.

10.11. Garabedian, P. R.: Partial Differential Equations, Wiley, New York, 1964.

10.12. Hopf, L.: Introduction to the Partial Differential Equations of Physics, Dover, New York, 1948.

10.13. Kamke, E.: Differentialgleichungen, Losungsmethoden und Losungen, vol. II, Chelsea, New York, 1948.

10.14. Lebedev, N. N., et al.: Problems of Mathematical Physics, Prentice-Hall, Englewood Cliffs, N.J., 1965.

10.15. Petrovsky, E. G.: Lectures on Partial Differential Equations, Interscience, New York, 1955.

10.16. Sagan, H.: Boundary and Eigenvalue Problems in Mathematical Physics, Wiley, New York, 1961.

10.17. Shapiro, A. H.: The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, Ronald, New York, 1953.

10.18. Sneddon, I. N.: Elements of Partial Differential Equations, McGraw-Hill, New York, 1957.

10.19. ———: Fourier Transforms, McGraw-Hill, New York, 1951.

10.20. Sommerfeld, A.: Partial Differential Equations in Physics, Academic Press, New York, 1949.

10.21. Tranter, C. J.: Integral Transforms in Mathematical Physics, 2d ed., Wiley, New York, 1956.

10.22. Tychonov, A. N., and A. A. Samarski: Partial Differential Equations in Mathematical Physics, Holden-Day, San Francisco, 1964.

10.23. Margenau, H., and G. M. Murphy: The Mathematics of Physics and Chemistry, 2d ed., Van Nostrand, Princeton, N.J., 1952.