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By Guri I. Marchuk

New statements of difficulties arose lately hard thorough ana­ lysis. detect, to begin with, the statements of difficulties utilizing adjoint equations which steadily turned a part of our lifestyles. Adjoint equations are able to convey clean principles to numerous difficulties of recent know-how in keeping with linear and nonlinear tactics. They turned a part of golden fund of technological know-how via quantum mechanics, thought of nuclear reactors, optimum regulate, and eventually helped in fixing many difficulties at the foundation of perturbation strategy and sensitivity concept. to stress the real position of adjoint difficulties in technological know-how one should still point out 4-dimensional research challenge and resolution of inverse difficulties. This variety of difficulties comprises to start with difficulties of worldwide weather adjustments on our planet, nation of atmosphere and safeguard of environ­ ment opposed to pollutants, upkeep of the biosphere in stipulations of full of life progress of inhabitants, extensive improvement of undefined, and so forth. All this required advanced examine of enormous structures: interac­ tion among the ambience and oceans and continents within the conception of weather, cenoses within the biosphere stricken by pollutants of typical and anthropogenic foundation. difficulties of neighborhood and international perturbations and versions sensitivity to enter facts sign up for into universal advanced system.

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Meanwhile, let us see that the biorthogonality condition is satisfied. 5) and Wn from the inner product of the second and Vk and obtain (AVk,Wn) - (vk,A*wn) = (Ak - An)(Vk,W n) n=1,2, ... , k=1,2, ... 8) are referred to as biorthogonality condition. It is instrumental in expansion of the function into a Fourier series in eigenfunctions of the main or adjoint problems. 9) n=1 where In are Fourier coefficients to be found. 9) by the function W n , term by term. e Hilbert-Schmidt operators. MAIN AND ADJOINT EQUATIONS.

24) 27 MAIN AND ADJOINT EQUATIONS. A2 e x/2' SIn 21[X 1/4+41[2 4 ex / 2 sin 21[x 1 + 161[2 (1 2 25) .. 21). 26) where cp* E D(A*), P E H. The Fourier method offers a solution to this one, too. 14)). 26), which is done in every particular case on the basis of the functional analysis' theorems, proceeding from properties of the operator A *. As is known, if An happens to be a complex figure, a complex-adjoint figure 'xn will replace An in an adjoint spectral problem for W n. 3. Adjoint Equations and Functionals.

15) 35 MAIN AND ADJOINT EQUATIONS. 13). 15) by cp, subtract from each other and integrate in x on [0,1]. 1 to find the adjoint operator A*. 16) 1 1 Jfcp*dx = Jpcpdx o 0 i. 17) As a numerical example, let us examine a main problem in the form cp(O) = cp(l) = 0 where f(x) = { ~: x E [1/2, 2/3] x fj. [1/2, 2/3] Let us suppose we are seeking the value of the functional 1/3 J = J cp(x)dx. I Fig. 01. Fig. 6 shows the diagrams of the functions ~ and ~*. 10- 2 We see, that the difference in the values is of the order of O(h2).

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