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Handbook of mathematics for engineers and scienteists part 127

Handbook of mathematics for engineers and scienteists part 127. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 850 Integral Equations This requirement leads to the algebraic system of equations dI n Q A 0 j 1 n dAj 16.4.9.5 and hence on the basis of 16.4.9.4 by differentiating with respect to the parameters Ai . An under the integral sign we obtain 1 dL- r 2 dAj -Ja n fj x X f0 x X Aifi x X dx 0 j 1 . n. 16.4.9.6 a i 1 using the notation cij X fi x X fj x X dx a 16.4.9.7 we can rewrite system 16.4.9.6 in the form of the normal system of the method of least squares C11 X A1 c12 X A2 C1n X An -C10 X C21 X A1 C22 X A2 C2n X An -C20 X 16.4.9.8 Cn1 X A1 Cn2 X A2 Cnn X An -Cn0 X . Note that if q0 x 0 then f0 x -f x . Moreover since cij X Cji X the matrix of system 16.4.9.8 is symmetric. 16.4.9-2. Construction of eigenfunctions. The method of least squares can also be applied for the approximate construction of characteristic values and eigenfunctions of the kernel K x s similarly to the way in which it can be done in the collocation method. Namely by setting f x 0 and Q 0 x 0 which implies f0 x 0 we determine approximate values of the characteristic values from the algebraic equation det Cij X 0. After this approximate eigenfunctions can be found from the homogeneous system of the form 16.4.9.8 where instead of X the corresponding approximate value is substituted. 16.4.10. Bubnov-Galerkin Method 16.4.10-1. Description of the method. Let y x y x - X i K x t y t dt - f x 0. X a 16.4.10.1 Similarly to the above reasoning we seek an approximate solution of equation 16.4.10.1 in the form of a finite sum n Yn x f x 2 Ai i x i 1 . n 16.4.10.2 i 1 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration 851 where the i x i 1 . n are some given linearly independent functions coordinate functions and A1 . An are indeterminate coefficients. On substituting the expression 16.4.10.2 into the left-hand side of equation 16.4.10.1 we obtain the residual n r r b 1 b e Yn x Aj i j x - X K x t j t dt - X K x t f t dt. 16.4.10.3 According to the Bubnov-Galerkin method .