HANDBOOK OFINTEGRAL EQUATIONS phần 8 docx

... 43.2, ˜ λ 3 = 1 08, ˜ λ 4 = 216, ˜ λ 5 = 355.2. The exact values of the characteristic values of the equation under consideration are known: λ 1 = π 2 = 9 .86 9 , λ 2 =(2π) 2 = 39.4 78 , λ 3 =(3π) 2 = 88 .82 6 ... manipulations, we obtain the following equation: 130µ 5 – 441µ 4 + 488 µ 3 – 206µ 2 +30µ – 1=0 ( ˜ λ = 216µ), Page 582 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC where...

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... second-order algebraic equations (this system can be obtained from the more general system presented in 6 .8. 42). Page 384 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 38. y(x)+  b a f(t)y(x ... equation 6 .8. 28 with f(t, y)=A cot(βy). 32. y(x)+  b a g(x) cot[βy(t)] dt = h(x). This is a special case of equation 6 .8. 29 with f(t, y) = cot(βy). Page 401 © 19 98 by CRC Press...

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... equation 3 .8. 3 with g(x)=ln β x. 8.  b a y(t) |ln(x/t)| β dt = f (x), 0 < β <1. This is a special case of equation 3 .8. 7 with g(x)=lnx + A, where A is an arbitrary number. Page 2 18 © 19 98 by ... s 2 . • Reference: P. P. Zabreyko, A. I. Koshelev, et al. (1975). Page 229 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 3 .8- 2. Equations Containing Modulus 3.  b a |g(x...

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... b) 2 – a a 2 +(x + b) 2 πe –au sin(bu) 8 x + b a 2 +(x + b) 2 – x – b a 2 +(x – b) 2 πe –au cos(bu) Page 739 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 5 .8. Expressions With Trigonometric Functions No ... Supplement 6: G. Doetsch (1950, 1956, 19 58) , H. Bateman and A. Erd ´ elyi (1954), V. A. Ditkin and A. P. Prudnikov (1965). Page 7 38 © 19 98 by CRC Press LLC © 19 98 by CRC...

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... 621 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 13.3. Complete Singular Integral Equations Solvable in a Closed Form In contrast with characteristic equations and their transposed equations, ... Tricomi (1 985 ), N. I. Muskhelishvili (1992). Page 645 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Relations (41) are the solvability conditions for the regularized equation ( 38...

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... Section9.10: V. I. Krylov, V.V. Bobkov, and P. I. Monastyrnyi (1 984 ), A. F. Verlan’ andV. S. Sizikov (1 986 ). Page 488 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Moreover, in the special case λ = –A,wehaveR(x)=A 2 x. ... Chapter 10. • References for Section 8. 8: F. D. Gakhov and Yu. I. Cherskii (19 78) , A. D. Polyanin and A. V. Manzhirov (1997). Page 4 58 © 19 98 by CRC Pre...

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... 3 08 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 47. y(x) – λ  b a cot(βx) cot(βt) y(t) dt = f (x). This is a special case of equation 4.9.1 with g(x) = cot(βx) and h(t)= 1 cot(βt) . 48. ... is a special case of equation 4.9. 18 with g 1 (x) = tan(αx), h 1 (t)=A cot(βt), g 2 (x) = tan(γx), and h 2 (t)=B cot(δt). Page 289 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 3 ◦ ....

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... sin(λt)   f(t) dt. Page 146 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 8. y(x)+A  x a t cosh[λ(x – t)]y(t) dt = f (x). This is a special case of equation 2.9. 28 with g(t)=At. 9. y(x)+A  x a t k cosh m (λx)y(t) ... I k (x). (3) Page 147 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 28. y(x)+  x a  A sinh(kt)+B + AB(x – t) sinh(kt)  y(t) dt = f(x). This is a special...

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... f(x). Solution: y(x)=  λ 2π d dx  x a cos  λ √ x – t  √ x – t f(t) dt. Page 88 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 60.  x a e µ(x–t) sin[λ(x – t)]y(t) dt = f(x), f(a)=f  x (a)=0. Solution: ... + C:  x a  A tanh λ x + B tanh β t + C  w(t) dt = e –µx f(x). Page 68 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 86 .  x s (x – t) c–1 F  a, b, c;1– x t  y(t)...

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... = 1 2 . See 1 .8. 86 for the solution of this equation. Page 14 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC HANDBOOK OF INTEGRAL EQUATIONS © 19 98 by CRC Press LLC © 19 98 by CRC Press ... Integrands 6 .8. Equations With Nonlinearity of General Form 6 .8- 1. Equations of the Form  b a G(···) dt = F(x) 6 .8- 2. Equations of the Form y(x)+  b a K(x, t)G  y(t)  dt...

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