INTEGRAL EQUATIONS

An integral equation is a mathematical expression that includes a required function under an integration sign. Such equations often describe an elementary or a complex physical process wherein the characteristics at a given point depend on values in the whole domain and can't be defined only on the bases of the values near the given point (as in local-type problems described by differential equations). Most frequently, integral equations as well as integro-differential equations are found in such problems of heat and mass transfer as diffusion, the potential theory and radiation heat transfer. For instance, nondimensional radiation heat flux in a given point on diffusely reflecting inner surfaces of two parallel isothermal optical grey plates, (x) = Q(x)/σ , dependent on fluxes and reflected from other points, is defined by the equation

where K(x,s) =[(x − s)2 + γ2]3/2; γ = h/d; x = X/d; d and h are the width of the plates and the distance between them; and Tw is surface temperature. Using an integral equation for local-type boundary value problems leads to more compact expressions than when differential equations are employed, including boundary conditions in the governing equation. This approach allows the possibility of using alternative solution methods such as the perturbation method or the approximate semianalytic averaging method.

Linear integral equations are of the form

(1)

where function A is called a coefficient; K is a kernel; and f is a free term (right-hand side) of the equation while D is the domain of function y(s). If A and K are matrices while y and f are vectors, Eq. (1) is a set of integral equations. If f = 0, Eq. (1) is called homogeneous; otherwise it is a nonhomogeneous equation. Three kinds of linear integral equations may be distinguished depending on whether A ≡ 0 (first kind), A ≠ 0 for all x (second kind) and A = 0 for some x (third kind). In the one-dimensional case, the first and second kind of equations are of the form

(2)
(3)

If K(x,s) and f(x) are continuous functions, λ is a real number and a and b are constants, Eqs. (2) and (3) are called the Fredholm integral equation of the first and second kinds, respectively. The homogeneous integral equation always has a trivial (zero) solution. Those values of λ which provide nontrivial solutions of the integral equation are known as the eigen values of the equation (or the kernel) and the corresponding nontrivial solutions, the characteristic ones. In case of a symmetrical nondegenerate kernel, K(x,s) = K(s,x), the integral equation has at least one characteristic solution and all its eigen values are real. For the kernel of the form K(x,s) = K(x − s), the integral equation is of the convoluted type. If K(x,s) becomes zero for s > x. Eqs. (2) and (3) take the form

(4)
(5)

and are called the Volterra integral equations of the first and second kind. Linear equation (3), which is not the Fredholm equation (i.e., when the integral operator is not continuous) is referred to as a singular integral equation for which solutions can be found by special methods. The equations below provide an example of nonlinear integral equations:

For continuous K and f, analytic solutions of Volterra integral equations of the second kind, Eq. (5), can be obtained by means of operational calculus or the method of successive approximations. For the latter method, the number of iterations, which are necessary to obtain a solution with given accuracy, significantly depend on the choice of initial approximation y0(x). The solution for the Fredholm equation of the second kind (Eq. (3)) exists and can be constructed by means of the successive approximations method if parameter |λ| < M−1, where M is such, that

Projection methods can be developed for approximate solution of integral equations of the second kind with constant integration limits. The solution is often approached by summing a finite series

where yi are known linear independent functions and Ci are free parameters which have to be defined. Depending on the approach to the approximate solution and definition of the free parameters, this method is classified as being a least squares method, the Galerkin method or the collocation method. Analytical solutions of integral equations of the second kind can be computed also by means of a resolvent R(x, s; λ) in the form

where R is expressed as an infinite series of multiple integrals of kernel K. The Volterra equation of the first kind can be frequently reduced to the second kind of equation. The solution of the Fredholm equation of the first kind is difficult because small perturbations in K and f or errors in solving methods can give uncontrollable errors. Solving this equation involves the use of regularization algorithms (i.e., the Tikhonov method, statistical regularization methods, etc.).

Numerical methods for solution of integral equation are based either on a discretization of the calculation domain and then reducing the problem to solving a set of algebraic equations (linear for linear integral equation) for discrete values {yj}, or on the methods described above; results are usually impracticable without the use of a computer.

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