Readable and systematic, this volume offers coherent presentations of not only the general theory of linear equations with a single integration, but also of applications to differential equations, the calculus of variations, and special areas in mathematical physics. Topics include the solution of Fredholm’s equation expressed as a ratio of two integral series in lambda, free and constrained vibrations of an elastic string, and auxiliary theorems on harmonic functions. Discussion of the Hilbert-Schmidt theory covers boundary problems for ordinary linear differential equations, vibration problems, and flow of heat in a bar. 1924 edition.
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1. Linear Integral Equation of the First Kind.—An equation of the form
is said to be a linear integral equation of the first kind. The functions K(x, t) and f(x) and the limits a and b are known. It is proposed so to determine the unknown function u that (1) is satisfied for all values of x in the closed interval a
x
b. K(x, t) is called the kernel of this equation.
Instead of equation (1), we have often to deal with equations of exactly the same form in which the upper limit of integration is the variable x. Such an equation is seen to be a special case of (1) in which the kernel K(x, t) vanishes when t > x, since it then makes no difference whether x or b is used as the upper limit of integration.
The characteristic feature of this equation is that the unknown function u occurs under a definite integral. Hence equation (1) is called an integral equation and, since u occurs linearly, equation (1) is called a linear integral equation.
2. Abel’s Problem.—As an illustration of the way in which integral equations arise, we give here a statement of Abel’s problem.
FIG. 1.
Given a smooth curve situated in a vertical plane. A particle starts from rest at any point P. Let us find, under the action of gravity, the time T of descent to the lowest point O. Choose O as the origin of coordinates, the x-axis vertically upward, and the y-axis horizontal. Let the coordinates of P be (x, y), of Q be (ξ, η), and s the arc OQ.
The velocity of the particle at Q is
Hence
The whole time of descent is, then,
If the shape of the curve is given, then s can be expressed in terms of ξ and hence ds can be expressed in terms of ξ. Let
Then
Abel set himself the problem1 of finding that curve for which the time T of descent is a given function of x, say f(x). Our problem, then, is to find the unknown function u from the equation
This is a linear integral equation of the first kind for the determination of u.
3. Linear Integral Equation of the Second Kind.—An equation of the form
is said to be a linear integral equation of the second kind.
K(x, t) is called the kernel of this equation. The functions K(x, t) and f(x) and the limits a and b are known. The function u is unknown.
The equation
is known as Volterra’s linear integral equation of the second kind.
If f(x) ≡ 0, then
This equation is said to be a, homogeneous linear int...
Table of contents
Cover Page
Title Page
Copyright Page
Preface
Contents
Chapter I: Introductory
Chapter II: Solution of Integral Equation of Second Kind by Successive Substitutions
Chapter III: Solution of Fredholm’s Equation Expressed as Ratio of Two Integral Series in λ
Chapter IV: Applications of the Fredholm Theory
Chapter V: Hilbeht-Schmidt Theory of Integral Equations with Symmetric Kernels
Chapter VI: Applications of the Hilbert-Schmidt Theory
