HANDBOOK OFINTEGRAL EQUATIONS phần 6 docx

... special case of equation 6. 8.33 with f(t, y)=A coth(βy). 36. y(x)+A  b a sin(λx + µt) coth[βy(t)] dt = h(x). This is a special case of equation 6. 8.34 with f(t, y)=A coth(βy). 6. 5-5. Other Integrands 37. ... the roots of the quadratic equation Aβ 2 + λβ – Bλ =0. 6. 2. Equations With Quadratic Nonlinearity That Contain Arbitrary Functions 6. 2-1. Equations of the Form  b a G(··...

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... =  1 6 x(1 – t)(2t – x 2 – t 2 ) for x ≤ t, 1 6 t(1 – x)(2x – x 2 – t 2 ) for x ≥ t. We choose equidistant points x i and t j and take n = 5. This implies x 1 = t 1 = 1 6 , x 2 = t 2 = 2 6 , x 3 = ... 108, ˜ λ 4 = 2 16, ˜ λ 5 = 355.2. The exact values of the characteristic values of the equation under consideration are known: λ 1 = π 2 = 9. 869 , λ 2 =(2π) 2 = 39.478 , λ 3 =(3π) 2 =...

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... the asymmetric form). • Reference: F. D. Gakhov (1977). 3 .6. Equations Whose Kernels Contain Combinations of Elementary Functions 3 .6- 1. Kernels Containing Hyperbolic and Logarithmic Functions 1.  b a ln   cosh(λx) ... kind, respectively. • Reference: I. C. Gohberg and M. G. Krein (1 967 ). 3 .6- 2. Kernels Containing Logarithmic and Trigonometric Functions 6.  b a ln   cos(λ...

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... cot x. 65 .  tan xdx= – ln |cos x|. 66 .  tan 2 xdx= tan x – x. 67 .  tan 3 xdx= 1 2 tan 2 x +ln|cos x|. 68 .  tan 2n xdx=(–1) n x – n  k=1 (–1) k (tan x) 2n–2k+1 2n – 2k +1 , n =1,2, 69 .  tan 2n+1 xdx=(–1) n+1 ln ... a 2 exp  –b √ u 2 + a 2  • References for Supplement 6: G. Doetsch (1950, 19 56, 1958), H. Bateman and A. Erd ´ elyi (1954), V. A. Ditkin and A. P. Prudnikov (1...

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... m. Page 62 1 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 13.3. Complete Singular Integral Equations Solvable in a Closed Form In contrast with characteristic equations and their transposed equations, ... ϕ(x)=– 1 π  ∞ –∞ F (t) t – x dt. – ∞ < x < ∞. (6) The two formulas (6) are called the Hilbert transform pair (see Subsection 7 .6- 3). Page 62 5 © 1998 by CRC Press LLC © 1...

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... Verlan’ and V. S. Sizikov (19 86) , A. D. Polyanin and A. V. Manzhirov (1998). Page 463 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 9.3 -6. Systems of Volterra Integral Equations The Laplace transform ... solutions. • References for Section 9 .6: A. D. Polyanin and A. V. Manzhirov (1997, 1998). 9.7. Method of Differentiation for Integral Equations In some cases, the differentiation...

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... λ 2 k )I k = f  xx (x)+λ 2 n f(x). (6) Differentiating (6) with respect to x twice and eliminating I n–1 from the resulting equation with the aid of (6) , we obtain a similar equation whose ... coefficients. Page 329 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 4.9-3. Other Equations of the Form y(x)+  b a K(x, t)y(t) dt = F (x) 26. y(x) –  ∞ –∞ K(x + t)y(t) dt = f (x). The Fourier...

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... second-order linear ordinary differential equations (3) with vari- ous g(x), see E. Kamke (1977), G. M. Murphy (1 960 ), and A. D. Polyanin and V. F. Zaitsev (1995, 19 96) . 2 ◦ . Let y 1 = y 1 (x) and y 2 = ... λ 2 k )I k = f  xx (x)+λ 2 n f(x). (6) Differentiating (6) with respect to x twice followed by eliminating I n–1 from the resulting expression with the aid of (6) yields a simila...

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... B d dx   I ν  λ √ x  – A A+B  x a  I ν  λ √ t  – B A+B f  t (t) dt  . 61 .  x a  AI ν  λ √ x  + BI µ  β √ t  y(t) dt = f(x). This is a special case of equation 1.9 .6 with g(x)=AI ν  λ √ x  and h(t)=BI µ  β √ t  . 62 .  x a √ x – tI 1  λ √ x ... f(x). Solution: y(x)= 2 3/2 √ πλ 3/2 d 3 dx 3  x a cos  λ √ x – t  √ x – t f(t) dt. 65 .  x a (x – t) n/2 I n  λ...

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... Integrands 6. 6. Equations With Logarithmic Nonlinearity 6. 6-1. Integrands With Nonlinearity of the Form ln[βy(t)] 6. 6-2. Other Integrands 6. 7. Equations With Trigonometric Nonlinearity 6. 7-1. Integrands ... y(x)+  b a G(···) dt = F(x) 6. 3. Equations With Power-Law Nonlinearity 6. 3-1. Equations of the Form  b a G(···) dt = F(x) 6. 3-2. Equations of the Form y(x)+  b...

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