Get Smithies Integral Equations in PDF for Free: A Practical and Theoretical Approach to Integral Eq

An equation containing the unknown function under the integral sign. Integral equations can be divided into two main classes: linear and non-linear integral equations (cf. alsoLinear integral equation;Non-linear integral equation).

$$A(x)\phi(x) + \int_DK(x,s)\phi(s)ds = d(x),\quad x\in D \label1$$where $A$, $K$, $f$ are given functions, $A$ being called the coefficient, $K$ the kernel (cf.Kernel of an integral operator) and $f$ the free term (or right-hand side) of the integral equation, $D$ is a bounded or unbounded domain in a one- or higher-dimensional Euclidean space, $x,s$ are points of this space, $ds$ is the volume element, and $\phi$ is the unknown function. It is required to determine $\phi$ such that (1) holds for all (or almost all, if the integral is taken in the sense of Lebesgue) $x$ in $D$. In (1), if $A$, $K$ are matrices and $f$, $\phi$ are vector functions, then (1) is called a system of linear integral equations. If $f=0$, then the integral equation is said to be homogeneous, otherwise it is called inhomogeneous.

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There are three distinct types of linear integral equations, depending on the coefficient $A$. If $A(x)=0$ for all $x\in D$, then (1) is called an equation of the first kind; if $A(x)\ne 0$ for all $x\in D$, an equation of the second kind; and if $A(x)$ vanishes on some non-empty proper subset of $D$, an equation of the third kind.

For simplicity, only integral equations in the one-dimensional case will be considered, when $D$ is a finite interval $[a,b]$. In this case, linear equations of the first and second kind can be represented in the following form:

$$\phi(x)-\def\l\lambda\l\int_a^b K(x,s)\phi(s)ds = f(x), \quad x\in [a,b],\label3$$respectively. The constant $\l$ is called the parameter of the integral equation. Equations of the second kind are most frequently encountered in mathematical physics. If $K$ is aFredholm kernel, that is, if the integral operator in equations (2), (3) is completely continuous (also called compact, seeCompletely-continuous operator), then the integral equation (2) (respectively, (3)) is called a Fredholm equation of the first (second) kind. An important example of a Fredholm equation is one in which the kernel satisfies the condition

$$\phi(x)-\int_a^x K(x,s)\phi(s)ds = f(x), \quad a\le s\le x\le b,\label8$$These equations are called Volterra equations of the first and second kind, respectively (cf.Volterra equation). Special cases of integral equations began to appear in the first half of the 19th century. Integral equations became the object of special attention of mathematicians after the solution of theDirichlet problem for theLaplace equation had been reduced to the study of a linear integral equation of the second kind. The construction of a general theory of linear integral equations was begun at the end of the 19th century. The founders of this theory are considered to be V. Volterra (1896), E. Fredholm (1903,[Fr]), D. Hilbert (1912,[Hi]), and E. Schmidt (1907, ). Even before these investigations, the method of successive approximation for the construction of a solution of an integral equation was proposed (cf. alsoSequential approximation, method of). This method was initially applied to the solution of non-linear equations of Volterra type (in modern terminology) in connection with studies of ordinary differential equations in the work of J. Liouville (1838), L. Fuchs (1870), G. Peano (1888), and others; as well as by C. Neumann (1877) in constructing a solution of an integral equation of the second kind. The general form of the method of successive approximation is due to E. Picard (1893).

gave a simpler and somewhat more general version of the investigations of Hilbert. He constructed a theory of linear integral equations with real symmetric kernel (cf.Integral equation with symmetric kernel) independently of the Fredholm theory by representing the kernel as the sum of a degenerate and a "small" kernel. T. Carleman achieved a substantial weakening of the restrictions imposed on the data and the unknown elements in the theory of integral equations of the second kind for the case of real symmetric kernels. He extended the method of Fredholm (see[Ca]) to the case when the kernel of (3) satisfies condition (4). In papers of F. Riesz (1918) and J. Schauder (1930), Fredholm's theorems were generalized to a certain class of linear operator equations in Banach spaces.

The basic method for studying integral equations of the first kind is the so-calledregularization method (see alsoIll-posed problems). Integral equations of the third kind were the object of special investigations by H. Bateman (1907), Picard (1910), G. Fubini (1912), and Ch. Platrier (1912).

If a linear integral equation is not a Fredholm equation, then it is called a singular equation (cf.Singular integral equation). Hilbert's general theory of quadratic forms in an infinite number of variables provides the possibility in this case also of obtaining a number of important results. For certain specific classes of singular integral equations, special methods for solving them have been developed, taking into account the characteristic properties of these equations. For example, for singular integral equations and integral equations of convolution type (cf.Integral equation of convolution type), Fredholm's theorem that two transposed homogeneous integral equations have the same number of linearly independent solutions, is false. 2ff7e9595c