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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 92', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14.4. Method of Separation of Variables Fourier Method 605 where eij A si pj X kppf . For system 14.4.1.10 to have nontrivial solutions its determinant must be zero we have 11 A E22 A - E12 A E21 A 0. 14.4.1.11 Solving the transcendental equation 14.4.1.11 for A one obtains the eigenvalues A An where n 1 2 . For these values of A equation 14.4.1.6 has nontrivial solutions n x en An P1 x An - 11 An 2 x An 14.4.1.12 which are called eigenfunctions these functions are defined up to a constant multiplier . To facilitate the further analysis we represent equation 14.4.1.6 in the form p x X Ap x - q x p 0 14.4.1.13 where x b x 1 c x r f b x 1 1 f b x p x exp ---------dx q x --------exp ----------dx p x ------exp ----------dx . J a x J a x J a x J a x J a x 14.4.1.14 It follows from the adopted assumptions see the end of Paragraph 14.4.1-1 that p x p x x q x and p x are continuous functions with p x 0 and p x 0. The eigenvalue problem 14.4.1.13 14.4.1.8 is known to possess the following properties 1. All eigenvalues A1 A2 . are real and An œ as n 00 consequently the number of negative eigenvalues is finite. 2. The system of eigenfunctions p1 x 2 x . is orthogonal on the interval x1 x x2 with weight p x i.e. r x2 p x pn x pm x dx 0 for n m. 14.4.1.15 Jx1 3. If q x 0 s1k1 0 s2k2 0 14.4.1.16 there are no negative eigenvalues. If q 0 and k1 k2 0 the least eigenvalue is A1 0 and the corresponding eigenfunction is p1 const. Otherwise all eigenvalues are positive provided that conditions 14.4.1.16 are satisfied the first inequality in 14.4.1.16 is satisfied if c x 0. Subsection 12.2.5 presents some estimates for the eigenvalues An and eigenfunctions pn x . The procedure of constructing solutions to nonstationary boundary value problems is further different for parabolic and hyperbolic equations see Subsections 14.4.2 and 14.4.3 below forresults elliptic equations are treated in Subsection 14.4.4 . 14.4.2. Problems for Parabolic Equations Final Stage of Solution 14.4.2-1. Series
