Integral equation of convolution type

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An integral equation containing the unknown function under the integral sign of a convolution transform (see Integral operator). The peculiarity of an integral equation of convolution type is that the kernel of such an equation depends on the difference of the arguments. The simplest example is the equation

 (1)

where and are given functions and is the unknown function. Suppose that and that one seeks a solution in the same class. In order that (1) is solvable it is necessary and sufficient that the following condition holds:

 (2)

where is the Fourier transform of . When (2) holds, equation (1) has a unique solution in the class , representable by the formula

 (3)

where is uniquely determined by its Fourier transform

An equation of convolution type on the half-line (a Wiener–Hopf equation)

 (4)

arises in the study of various questions of theoretical and applied character (see [1], [4]).

Suppose that the right-hand side and the unknown function belong to , , that the kernel and that

 (5)

The function is called the symbol of equation (4). The index of equation (4) is the number

 (6)

If , then the functions defined by the equations

 (7)

are the Fourier transforms of functions , respectively, such that for . Under the above conditions, equation (4) has a unique solution, which can be expressed by the formula

 (8)

where

If , all solutions of equation (4) are given by the formula

 (9)

where are arbitrary constants,

 (10)

and the functions are uniquely determined by their Fourier transforms:

 (11)

When , the homogeneous equation corresponding to (4) has exactly linearly independent solutions , which are absolutely-continuous functions on any bounded interval; these solutions can be chosen so that , for , and .

If , the equation is solvable only if the following condition holds:

 (12)

where is a system of linearly independent solutions of the transposed homogeneous equation of (4):

 (13)

Under these conditions, the (unique) solution is given by the formula

where

while the Fourier transforms and of the functions are defined by the equation

and the equations (11). Noether's theorem holds for equation (4) (see Singular integral equation).

The first significant results in the theory of equations (4) were obtained in [11], where an effective method (the so-called Wiener–Hopf method) for solving the homogeneous equation corresponding to (4) was given under the hypothesis that the kernel and the required solution satisfy the conditions: For some both

The main point of the Wiener–Hopf method is the idea of factorization of a function which is holomorphic in the strip , that is, the possibility to represent it as a product , where are functions holomorphic in the half-planes and , respectively, and satisfy certain additional requirements. These results have been developed and amplified (see [4]).

A method has been developed for reducing equation (4) to a boundary value problem of linear identification. In this way, equation (4) has been solved under the following hypotheses: , , , , , as , and , .

In addition to this, the role of the number in solving (4) has been clarified. In earlier articles an analogous role was played by the number of zeros of the analytic function in a strip (see ).

Condition (5) is both necessary and sufficient in order that Noether's theorem holds for equation (4). The solution of equation (4) given above simplifies in a number of particular cases important in practice. The asymptotics of the solution have been obtained for special right-hand sides (see [4]).

Equation (4) has also been studied in the case when and the Fourier transform of the kernel has discontinuities of the first kind (see [5]) or is an almost-periodic function (see [2]). In these cases, condition (5) turns out to be insufficient for Noether's theorem to hold.

The validity of the majority of results listed above has also been established for systems of equations of type (4); however, in contrast to the case of a single equation, a system of integral equations of convolution type in the general case cannot be solved explicitly by quadratures (see [6]).

Also related to integral equations of convolution type are paired equations (or dual equations)

 (15)

and their transposed equations, so-called integral equations of convolution type with two kernels:

 (16)

Equations (15) have been explicitly solved by quadratures in , and equation (16) in [10].

An integral equation of convolution type on a bounded interval,

 (17)

where , is called a Fredholm equation (see [7], [9]).

Integral equations of convolution type whose symbol vanishes at a finite number of points and the orders of whose zeros are integers, lend themselves to an explicit solution by quadratures (see [8], [12]).

References

 [1] B. Noble, "Methods based on the Wiener–Hopf technique for the solution of partial differential equations" , Pergamon (1958) [2] I.C. [I.Ts. Gokhberg] Gohberg, I.A. Feld'man, "Convolution equations and projection methods for their solution" , Transl. Math. Monogr. , 41 , Amer. Math. Soc. (1974) (Translated from Russian) [3a] I.M. Rapoport, "On a class of singular integral equations" Dokl. Akad. Nauk. SSSR , 59 : 8 (1948) pp. 1403–1406 (In Russian) [3b] I.M. Rapoport, Trudy Mat. Inst. Steklov. , 12 (1949) pp. 102–117 [4] M.G. Krein, "Integral equations on the half-line with kernel depending on the difference of the arguments" Uspekhi Mat. Nauk , 13 : 5 (1958) pp. 3–120 (In Russian) [5] R.V. Duduchava, "Wiener–Hopf integral operators" Math. Nachr. , 65 : 1 (1975) pp. 59–82 (In Russian) [6] I.Ts. Gokhberg, M.G. Krein, "Fundamental aspects of deficiency numbers, root numbers and indexes of linear operators" Uspekhi Mat. Nauk , 12 : 2 (1957) pp. 44–118 (In Russian) [7] M.P. Ganin, "On a Fredholm integral equation whose kernel depends on the difference of two arguments" Izv. Vuzov. Mat. : 2 (1963) pp. 31–43 (In Russian) [8] F.D. Gakhov, V.I. Smagina, "Exceptional cases of integral equations of convolution type and equations of the first kind" Izv. Akad. Nauk SSSR Ser. Mat. , 26 : 3 (1962) pp. 361–390 (In Russian) [9] I.B. Simonenko, "On some integro-differential equations of convolution type" Izv. Vuzov. Mat. : 2 (1959) pp. 213–226 (In Russian) [10] Yu.I. Cherskii, "On some special integral equations" Uchen. Zap. Kazansk. Univ. , 113 : 10 (1953) pp. 43–56 (In Russian) [11] N. Wiener, E. Hopf, "Ueber eine Klasse singulärer Integralgleichungen" Sitz. Ber. Akad. Wiss. Berlin 30/32 (1931) pp. 696–706 [12] S. Prössdorf, "Einige Klassen singulärer Gleichungen" , Birkhäuser (1974)

In general, systems of equations of type (4) cannot be solved explicitly. An exception occurs when the symbol is a rational matrix function. In that case can be written in the form , where is an identity matrix, is a square matrix of order , say, without real eigen values, and and are (possibly non-square) matrices of appropriate sizes. Such a representation, which comes from mathematical systems theory, is called a realization of . The system of Wiener–Hopf equations can now be analyzed in terms of , and . The analysis yields explicit formulas for solutions, as well as explicit formulas for Wiener–Hopf factorization. Interpreting (4) as a system of equations involving a matrix kernel and -vector functions and , one of the major results obtained through this approach reads as follows. For each in , equation (4) has a unique solution in if and only if has no real eigen values and is the direct sum of and , where and are the spectral subspaces corresponding to the eigen values of and located in the upper and lower half-planes, respectively. Furthermore, in that case the symbol admits a (canonical Wiener–Hopf) factorization with

and the resolvent kernel of (4) is given by

where is the projection of along onto . Of particular interest is the situation where the symbol is self-adjoint (cf. [a11]). For further details and additional results (also on non-canonical Wiener–Hopf factorization) see [a1], [a2], [a7], and [a9]. Generalizations are possible to certain classes of non-rational symbols. For these classes, the realizations involve infinite-dimensional, possibly unbounded, operators (cf. [a3], [a4] and [a9]). For applications to the transport equation and abstract kinetic theory, see [a2], [a8] and [a10]; for applications to control theory, cf. [a6].