HANDBOOK OFINTEGRAL EQUATIONS phần 7 pps

... and G. I. Makarenko (1 971 ), V. I. Smirnov (1 974 ), K. B. Oldham and J. Spanier (1 974 ), P. P. Zabreyko, A. I. Koshelev, et al. (1 975 ), F. D. Gakhov and Yu. I. Cherskii (1 978 ), Yu. I. Babenko (1986), ... known. Below we list some classes of integral equations that can be reduced to ordinary differential equations with constant coefficients. 9 .7- 1. Equations With Kernel Containin...

Ngày tải lên: 23/07/2014, 16:20

... (x). This is a special case of equation 4.9. 17 with g(x)=K µ (x) and h(t)=K ν (t). 4.9. Equations Whose Kernels Contain Arbitrary Functions 4.9-1. Equations With Degenerate Kernel: K(x, t)=g 1 (x)h 1 (t)+···+ ... corresponding to the characteristic value λ 1 . Page 308 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 47. y(x) – λ  b a cot(βx) cot(βt) y(t) dt = f (x). This is a special...

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... 1) √ π 2 n , C = 0. 577 2 7 x ν–1 ln x, ν >0 Γ(ν)p –ν  ψ(ν) – ln p  , ψ(ν) is the logarithmic derivative of the gamma function 8 (ln x) 2 1 p  (ln x + C) 2 + 1 6 π 2  , C = 0. 577 2 9 e –ax ln ... C), C = 0. 577 2 is the Euler constant 2 ln(1 + ax) – 1 p e p/a Ei(–p/a) 3 ln(x + a) 1 p  ln a – e ap Ei(–ap)  4 x n ln x, n =1,2, n! p n+1  1+ 1 2 + 1 3 + ···+ 1 n – ln p – C  , C =...

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... Complete Singular Integral Equations Solvable in a Closed Form In contrast with characteristic equations and their transposed equations, complete singular integral equations cannot be solved ... dτ  , λ ≠ ±1. • Reference for Section 13.3: F. D. Gakhov (1 977 ). 13.4. The Regularization Method for Complete Singular Integral Equations 13.4-1. Certain Properties of Singular Operators...

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... Root (1958), I. C. Gohberg and M. G. Krein (19 67) , P. P. Zabreyko, A. I. Koshelev, et al. (1 975 ), A. D. Polyanin and A. V. Manzhirov (1998). Page 577 © 1998 by CRC Press LLC © 1998 by CRC Press ... K(x, t), the symbol of modulus on the right-hand sides of ( 27) and (28) can be omitted. 11.6-9. Integral Equations Reducible to Symmetric Equations An equation of the form y(x) – λ  b a...

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... arbitrary positive constant. Page 377 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 6.8-3. Equations of the Form y(x)+  b a K(x, t)G  t, y(t)  dt = F (x) 27. y(x)+  b a f  t, y(t)  dt ... ke λt  e –λt dt. 57. y(x)+  b a y(x – t)f  t, y(t)  dt = A sinh λx. A solution: y(x)=p sinh λx + q cosh λx, (1) where p and q are roots of the following system of algebraic (or transcenden...

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... cos x – 2 cos ξ . • Reference: I. C. Gohberg and M. G. Krein (19 67) . 3 .7. Equations Whose Kernels Contain Special Functions 3 .7- 1. Kernels Containing Bessel Functions 1.  ∞ 0 tJ ν (xt)y(t) ... solution form a Hilbert transform pair (in the asymmetric form). • Reference: F. D. Gakhov (1 977 ). 3.6. Equations Whose Kernels Contain Combinations of Elementary Functions 3.6-1. Kernels...

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... form y(x)= m  k=1 A k B k exp(–λ k x), B k =1– a  π λ k  ln(4λ k )+C  , where C = 0. 577 2 is the Euler constant. 2.5. Equations Whose Kernels Contain Trigonometric Functions 2.5-1. Kernels Containing ... (4) For exact solutions of second-order linear ordinary differential equations (3) with vari- ous g(x), see E. Kamke (1 977 ), G. M. Murphy (1960), and A. D. Polyanin and V. F. Zaits...

Ngày tải lên: 23/07/2014, 16:20

... 1.9.6 with g(x)=A arccot β (λx) and h(t)=B arccot γ (µt)+C. 1 .7. Equations Whose Kernels Contain Combinations of Elementary Functions 1 .7- 1. Kernels Containing Exponential and Hyperbolic Functions 1.  x a e µ(x–t)  A 1 cosh[λ 1 (x ... 1.9.6 with g(x)=AI k ν (λx) and h(t)=BI s µ (βt). 76 .  x a [K 0 (λx) – K 0 (λt)]y(t) dt = f(x). Solution: y(x)=– d dx  f  x (x) λK 1 (λx)  . 77...

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... Formulas 7. 1-1. Some Definitions 7. 1-2. The Structure of Solutions to Linear Integral Equations 7. 1-3. Integral Transforms 7. 1-4. Residues. Calculation Formulas 7. 1-5. The Jordan Lemma 7. 2. The ... Right-Hand Side 9.6 -7. Some Generalizations 9 .7. Method of Differentiation for Integral Equations 9 .7- 1. Equations With Kernel Containing a Sum of Exponential Functions 9 .7- 2...

Ngày tải lên: 23/07/2014, 16:20