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# Integral Equations

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INTEGRAL EQUATIONS. An integral equation is an equation connecting two vari ables, an independent variable x, and a depend ent variable it, in whicli u occurs (in at least one term) under the sign of integration. An example of such an equation is (1) g(x)u(x) +f b K(x, t)u(t) dt, in which f(x) and q(x) are given functions of x, K(x, t) is a given function of x and t, and a are known functions of x, one or both of which may be constant. The above equa tion is called linear, since u enters to the first degree only. It is homogeneous when f(x) is identically zero. It is the most general form of linear equation. The fundamental problem connected with eq. (1) is to solve it for the unknown u; that is, to express u as a known function of x.

The theory of integral equations is the most recent development of importance in the domain of pure mathematics. The foundation for a general theory of such equations was laid by Volterra in several papers published in Italian scientific journals in 1896. Since then this new field has developed with great rapidity and already has an extensive literature. In spite of this rapid growth the magnitude and diffi culties of the problem in its general form are such that only the simplest cases have been worked out with any degree of completeness. These cases group themselves under the fol lowing four types: • I. Volterra's equation of the 1st kind: f o K(x, t)u(t)dt; II. Volterra's equation of the 2d kind: +Af K(x, t)u(t)dt; III. Fredholm's equation of the 1st kind: t)u(t)dt; IV. Fredholm's equation of the 2d kind: f(x)==u(x) K(x, t)u(t)dt.

The function K(x, t) is called the kernel of the given equation (It., nucleo; Fr., noyau; Ger., Kern) ; A is an arbitrary constant.

These types include the equations that natu rally present themselves in connection with other branches of mathematics or in the solu tion of physical problems. It was, in fact, from such sources that integral equations began to claim the attention of mathematicians and to press for a systematic investigation. The first attempt to solve an integral equation occurs in a memoir by Abel, published in 1823 in connec tion with the following mechanical problem: To determine the arc of a curve situated in a vertical plane such that a heavy particle start ing at rest from the highest point P of the arc and moving by the attraction of gravity along the curve to its lowest point 0, shall make the descent in a time f(h) which is re quired to be a given function of the vertical altitude Is between 0 and P. This problem leads to the integral equation \rig f (h) fh u (y) dy in which 2g is the acceleration due to gravity, approximately 32, and u(y) is the unknown d function (xx and y being the rectan dy gular coordinates of a variable point on the required curve. It will be observed that Abel's equation is of Type I, with K(h, y) = 1 ,— • V Abel found the solution of this equation to be u V 2g d fy f (h) dh dy 0 Vy—k The simplest case is that in which f(h) is con stant. This is the celebrated problem of the tautochrone, which may be stated as follows: To find an arc of a curve such that a heavy particle descending along it under the action of gravity shall arrive at the lowest point in the same time, from whatever point of the curve the point begins to fall. The above solution readily leads, in this case, to an in verted cycloid with its vertex at the lowest point Abel was the first to perceive the essential character of integral equations and to make some effort to develop a method for their solu tion. He applied his ideas to the more general

equation f (x) f u (t) dt , o < a < 1 , Ft o (xi)" • for which he obtained a solution by the aid of infinite series. This method is not regarded, however, as satisfactory and cannot therefore be considered as affording a basis for a gen eral theory. In the interval between Abel's first paper and Volterra's fundamental work, several writers had occasion to solve integral equations in connection with problems of mathe matical physics. In particular, Lionville (1832), who was the next after Abel to consider this subject, and who apparently was unaware of the work of his predecessor, was led to Abel's equa tion in connection with various physical prob lems. He introduced the method of successive substitutions which has recently been made ap much more difficult equations. The first attempt to solve an equation of the Fred holm type was made by Neumann (1877) in his celebrated method for the solution of Dirichlet's problem. This consists in develop ing u(x) in increasing powers of a. But Neu mann s development, while converging in the case of Dirichlet's problem, does not converge in the general case. Neumann's method was applied with success, however, by Volterra to the general equation of Type II. In a brilliant memoir published by the Swedish mathema tician, Fredholm, in 1903, a general method for solving equations of types III and IV was given, thus completing, at least in outline, the theory for linear equations.

While integral equations have found their greatest usefulness in the field of applied mathe matics, they are of importance also in connec tion with other lines of pure mathematics, such as Theory of Functions, Calculus of Varia tions and Differential Equations. In the first of these, mention has already been made of the fundamental problem of Dirichlet: To deter mine the values of a harmonic function within a region bounded by a closed curve, given the value of the function at every point of the boundary. It is with differential equations, however, that this subject has the closest con tact. In many cases a single integral equation is the equivalent of a differential equation to gether with certain auxiliary equations of con dition. Lionville, in 1837, solved a differential equation by means of an integral equation in order to express the desired solution in the form of a series which should converge with sufficient rapidity. In dealing with equations containing more than one independent variable, the integral equation has this advantage over the corresponding partial differential equation, in that the extension from one variable to several variables offers no appreciable increase in difficulty in the former case, while the diffi culty is seriously augmented in the latter case. There has also been developed a theory of integro-differential equations which involve not only the unknown function u(x) but its deriv atives as well. These are likewise of import ance in physical problems.

For a detailed exposition of the theory of integral equations, the reader is referred to the following admirable works: An Introduction to the Study of Integral Equations,' by Bucher (Cambridge 1909) ; sur les equations integrales et les equations integro-differen tie:Iles,' by Volterra (Paris 1913) • della teoria delle equazioni integ;ali lineari,' by Vivanti (Milan 1916).