New PDF release: A Course of Higher Mathematics. Volume IV

By V. I. Smirnov and A. J. Lohwater (Auth.)

ISBN-10: 1483167232

ISBN-13: 9781483167237

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Example text

The number of terms would thus be diminished. Let us take an equation with such a kernel and the adjoint equation: φ{8)=ί{8) + λ$Κ{8,ΐ)φ{ί)Μ; a V(8) = g{8) + (86) X^K{t,8)y>{t)ât. >(*) di; a & yk= $Qk (t) y){t) at. a Thus every solution of equations (87) must have the form (88), and the entire problem reduces t o finding numbers xk and yk instead of functions. On substituting expressions (88) in equations (87) and equating coefficients for t h e linearly independent functions gk(s) and 0k(s), we get two systems of equations for xk and yk: /C=l n (89,) yi-λJΣakiyk=:gl· /c=i where b b *ik = ^i(s)Qk(s)ds; b g{ = jg(s) Q i {s)ds.

The kernel 14] 47 GENERALIZATION OF THE RESULTS OBTAINED K(M, N) is taken to be a continuous function of the pair of points. (My N)y with each varying in the closed domain B. , m). We have here, instead of a kernel, a matrix of functions Kifc(s,t) The above system is readily reduced to a single integral equation with a single required function. To avoid unnecessary complexity in the notation we shall put m = 2: Ψι («) = h (*) + l [ # n («. ί) 9Ί (0 + #12 («. *) 9>2 (03 di, (95) 9>2 («) = / 2 («) + J [#21 («.

T/c—1,5, t £ + i , . · ·> tq V) <*-l,2,.. MÎ ). J 12. Degenerate equations. We shall now mention a class of integral equations, the solutions of which reduce to algebraic equations of the first degree. The kernel K(s, t) is said to be degenerate if it consists of a finite sum of products of functions of s only and of t only: *(*,<)=jb*(«)Mi)- (8s) The functions Qk(s), like the functions ^k(t), can be assumed to be linearly independent. For, if some gp(s) could be expressed linearly in terms of the remaining Qk(s), we could substitute this expression for QP(S) in (85).

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A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)


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