HANDBOOK OFINTEGRAL EQUATIONS phần 8 docx

Example 1. Let us solve the integral equation y(x) – λ  1 0 xty(t) dt = f(x), 0 ≤ x ≤ 1, by the method of successive approximations. Here we have K(x, t)=xt, a = 0, and b = 1. We successively define K 1 (x, t)=xt, K 2 (x, t)=  1 0 (xz)(zt) dz = xt 3 , K 3 (x, t)= 1 3  1 0 (xz)(zt) dz = xt 3 2 , , K n (x, t)= xt 3 n–1 . According to formula (5) for the resolvent, we obtain R(x, t; λ)= ∞  n=1 λ n–1 K n (x, t)=xt ∞  n=1  λ 3  n–1 = 3xt 3 – λ , where |λ| < 3, and it follows from formula (7) that the solution of the integral equation can be rewritten in the form y(x)=f(x)+λ  1 0 3xt 3 – λ f(t) dt,0≤ x ≤ 1, λ ≠ 3. In particular, for f (x)=x we obtain y(x)= 3x 3 – λ ,0≤ x ≤ 1, λ ≠ 3. 11.3-4. Orthogonal Kernels For some Fredholm equations, the Neumann series (5) for the resolvent is convergent for all values of λ. Let us establish this fact. Assume that two kernels K(x, t) and L(x, t) are given. These kernels are said to be orthogonal if the following two conditions hold:  b a K(x, z)L(z, t) dz =0,  b a L(x, z)K(z, t) dz = 0 (8) for all admissible values of x and t. There exist kernels that are orthogonal to themselves. For these kernels we have K 2 (x, t) ≡ 0, where K 2 (x, t) is the second iterated kernel. It is clear that in this case all the subsequent iterated kernels also vanish, and the resolvent coincides with the kernel K(x, t). Example 2. Let us find the resolvent of the kernel K(x, t) = sin(x – 2t), 0 ≤ x ≤ 2π,0≤ t ≤ 2π. We have  2π 0 sin(x – 2z) sin(z – 2t) dz = 1 2  2π 0 [cos(x +2t – 3z) – cos(x – 2t – z)] dz = = 1 2  – 1 3 sin(x +2t – 3z) + sin(x – 2t – z)  z=2π z=0 =0. Thus, in this case the resolvent of the kernel is equal to the kernel itself: R(x, t; λ) ≡ sin(x – 2t), so that the Neumann series (6) consists of a single term and clearly converges for any λ. Remark 2. If the kernels M (1) (x, t), , M (n) (x, t) are pairwise orthogonal, then the resolvent corresponding to the sum K(x, t)= n  m=1 M (m) (x, t) is equal to the sum of the resolvents corresponding to each of the summands. • References for Section 11.3: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V. I. Smirnov (1974), A. J. Jerry (1985). Page 535 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 11.4. Method of Fredholm Determinants 11.4-1. A Formula for the Resolvent A solution of the Fredholm equation of the second kind y(x) – λ  b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (1) is given by the formula y(x)=f(x)+λ  b a R(x, t; λ)f(t) dt, a ≤ x ≤ b, (2) where the resolvent R(x, t; λ)isdefined by the relation R(x, t; λ)= D(x, t; λ) D(λ) , D(λ) ≠ 0. (3) Here D(x, t; λ) and D(λ) are power series in λ, D(x, t; λ)= ∞  n=0 (–1) n n! A n (x, t)λ n , D(λ)= ∞  n=0 (–1) n n! B n λ n , (4) with coefficients defined by the formulas A 0 (x, t)=K(x, t), A n (x, t)=  b a ···  b a    n         K(x, t) K(x, t 1 ) ··· K(x, t n ) K(t 1 , t) K(t 1 , t 1 ) ··· K(t 1 , t n ) . . . . . . . . . . . . K(t n , t) K(t n , t 1 ) ··· K(t n , t n )         dt 1 dt n , (5) B 0 =1, B n =  b a ···  b a    n         K(t 1 , t 1 ) K(t 1 , t 2 ) ··· K(t 1 , t n ) K(t 2 , t 1 ) K(t 2 , t 2 ) ··· K(t 2 , t n ) . . . . . . . . . . . . K(t n , t 1 ) K(t n , t 2 ) ··· K(t n , t n )         dt 1 dt n ; n =0,1,2, (6) The function D(x, t; λ) is called the Fredholm minor and D(λ) the Fredholm determinant. The series (4) converge for all values of λ and hence define entire analytic functions of λ. The resolvent R(x, t; λ) is an analytic function of λ everywhere except for the values of λ that are roots of D(λ). These roots coincide with the characteristic values of the equation and are poles of the resolvent R(x, t; λ). Example 1. Consider the integral equation y(x) – λ  1 0 xe t y(t) dt = f(x), 0 ≤ x ≤ 1, λ ≠ 1. We have A 0 (x, t)=xe t , A 1 (x, t)=  1 0     xe t xe t 1 t 1 e t t 1 e t 1     dt 1 =0, A 2 (x, t)=  1 0  1 0       xe t xe t 1 xe t 2 t 1 e t t 1 e t 1 t 1 e t 2 t 2 e t t 2 e t 1 t 2 e t 2       dt 1 dt 2 =0, since the determinants in the integrand are zero. It is clear that the relation A n (x, t) = 0 holds for the subsequent coefficients. Let us find the coefficients B n : B 1 =  1 0 K(t 1 , t 1 ) dt 1 =  1 0 t 1 e t 1 dt 1 =1, B 2 =  1 0  1 0     t 1 e t 1 t 1 e t 2 t 2 e t 1 t 2 e t 2     dt 1 dt 2 =0. Page 536 © 1998 by CRC Press LLC © 1998 by CRC Press LLC It is clear that B n = 0 for all subsequent coefficients as well. According to formulas (4), we have D(x, t; λ)=K(x, t)=xe t ; D(λ)=1– λ. Thus, R(x, t; λ)= D(x, t; λ) D(λ) = xe t 1 – λ , and the solution of the equation can be represented in the form y(x)=f(x)+λ  1 0 xe t 1 – λ f(t) dt,0≤ x ≤ 1, λ ≠ 1. In particular, for f (x)=e –x we obtain y(x)=e –x + λ 1 – λ x,0≤ x ≤ 1, λ ≠ 1. 11.4-2. Recurrent Relations In practice, the calculation of the coefficients A n (x, t) and B n of the series (4) by means of formulas (5) and (6) is seldom possible. However, formulas (5) and (6) imply the following recurrent relations: A n (x, t)=B n K(x, t) – n  b a K(x, s )A n–1 (s, t) ds, (7) B n =  b a A n–1 (s, s) ds. (8) Example 2. Let us use formulas (7) and (8) to find the resolvent of the kernel K(x, t)=x – 2t, where 0 ≤ x ≤ 1 and 0 ≤ t ≤ 1. Indeed, we have B 0 = 1 and A 0 (x, t)=x – 2t. Applying formula (8), we see that B 1 =  1 0 (–s) ds = – 1 2 . Formula (7) implies the relation A 1 (x, t)=– x – 2t 2 –  1 0 (x – 2s)(s – 2t) ds = –x – t +2xt + 2 3 . Furthermore, we have B 2 =  1 0  –2s +2s 2 + 2 3  ds = 1 3 , A 2 (x, t)= x – 2t 3 – 2  1 0 (x – 2s)  –s – t +2st + 2 3  ds =0, B 3 = B 4 = ···=0, A 3 (x, t)=A 4 (x, t)=···=0. Hence, D(λ)=1+ 1 2 λ + 1 6 λ 2 ; D(x, t; λ)=x – 2t + λ  x + t – 2xt – 2 3  . The resolvent has the form R(x, t; λ)= x – 2t + λ  x + t – 2xt – 2 3  1+ 1 2 λ + 1 6 λ 2 . • References for Section 11.4: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), V. I. Smirnov (1974). Page 537 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 11.5. Fredholm Theorems and the Fredholm Alternative 11.5-1. Fredholm Theorems THEOREM 1. If λ is a regular value, then both the Fredholm integral equation of the second kind and the transposed equation are solvable for any right-hand side, and both the equations have unique solutions. The corresponding homogeneous equations have only the trivial solutions. THEOREM 2. For the nonhomogeneous integral equation to be solvable, it is necessary and sufficient that the right-hand side f(x) satisfies the conditions  b a f(x)ψ k (x) dx =0, k =1, , n, where ψ k (x) is a complete set of linearly independent solutions of the corresponding transposed homogeneous equation. THEOREM 3. If λ is a characteristic value, then both the homogeneous integral equation and the transposed homogeneous equation have nontrivial solutions. The number of linearly independent solutions of the homogeneous integral equation is finite and is equal to the number of linearly independent solutions of the transposed homogeneous equation. THEOREM 4. A Fredholm equation of the second kind has at most countably many characteristic values, whose only possible accumulation point is the point at infinity. 11.5-2. The Fredholm Alternative The Fredholm theorems imply the so-called Fredholm alternative, which is most frequently used in the investigation of integral equations. T HE FREDHOLM ALTERNATIVE. Either the nonhomogeneous equation is solvable for any right- hand side or the corresponding homogeneous equation has nontrivial solutions. The first part of the alternative holds if the given value of the parameter is regular and the second if it is characteristic. Remark. The Fredholm theory is also valid for integral equations of the second kind with weak singularity. • References for Section 11.5: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V. I. Smirnov (1974), A. J. Jerry (1985), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), W. Hackbusch (1995), R. P. Kanwal (1997). 11.6. Fredholm Integral Equations of the Second Kind With Symmetric Kernel 11.6-1. Characteristic Values and Eigenfunctions Integral equations whose kernels are symmetric, that is, satisfy the condition K(x, t)=K(t, x), are called symmetric integral equations. Each symmetric kernel that is not identically zero has at least one characteristic value. For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth powers of the characteristic values of the first kernel. The eigenfunctions of a symmetric kernel corresponding to distinct characteristic values are orthogonal, i.e., if ϕ 1 (x)=λ 1  b a K(x, t)ϕ 1 (t) dt, ϕ 2 (x)=λ 2  b a K(x, t)ϕ 2 (t) dt, λ 1 ≠ λ 2 , Page 538 © 1998 by CRC Press LLC © 1998 by CRC Press LLC then (ϕ 1 , ϕ 2 )=0, (ϕ, ψ) ≡  b a ϕ(x)ψ(x) dx. The characteristic values of a symmetric kernel are real. The eigenfunctions can be normalized; namely, we can divide each characteristic function by its norm. If several linearly independent eigenfunctions correspond to the same characteristic value, say, ϕ 1 (x), , ϕ n (x), then each linear combination of these functions is an eigenfunction as well, and these linear combinations can be chosen so that the corresponding eigenfunctions are orthonormal. Indeed, the function ψ 1 (x)= ϕ 1 (x) ϕ 1  , ϕ 1  =  (ϕ 1 , ϕ 1 ), has the norm equal to one, i.e., ψ 1  = 1. Let us form a linear combination αψ 1 + ϕ 2 and choose α so that (αψ 1 + ϕ 2 , ψ 1 )=0, i.e., α = – (ϕ 2 , ψ 1 ) (ψ 1 , ψ 1 ) = –(ϕ 2 , ψ 1 ). The function ψ 2 (x)= αψ 1 + ϕ 2 αψ 1 + ϕ 2  is orthogonal to ψ 1 (x) and has the unit norm. Next, we choose a linear combination αψ 1 + βψ 2 + ϕ 3 , where the constants α and β can be found from the orthogonality relations (αψ 1 + βϕ 2 + ϕ 3 , ψ 1 )=0, (αψ 1 + βψ 2 + ϕ 3 , ψ 2 )=0. For the coefficients α and β thus defined, the function ψ 3 = αψ 1 + βψ 2 + ϕ 2 αψ 1 + βϕ 2 + ϕ 3  is orthogonal to ψ 1 and ψ 2 and has the unit norm, and so on. As was noted above, the eigenfunctions corresponding to distinct characteristic values are orthogonal. Hence, the sequence of eigenfunctions of a symmetric kernel can be made orthonormal. In what follows we assume that the sequence of eigenfunctions of a symmetric kernel is or- thonormal. We also assume that the characteristic values are always numbered in the increasing order of their absolute values. Thus, if λ 1 , λ 2 , , λ n , (1) is the sequence of characteristic values of a symmetric kernel, and if a sequence of eigenfunctions ϕ 1 , ϕ 2 , , ϕ n , (2) corresponds to the sequence (1) so that ϕ n (x) – λ n  b a K(x, t)ϕ n (t) dt = 0, (3) then  b a ϕ i (x)ϕ j (x) dx =  1 for i = j, 0 for i ≠ j, (4) Page 539 © 1998 by CRC Press LLC © 1998 by CRC Press LLC and |λ 1 |≤|λ 2 |≤···≤|λ n |≤··· . (5) If there are infinitely many characteristic values, then it follows from the fourth Fredholm theorem that their only accumulation point is the point at infinity, and hence λ n →∞as n →∞. The set of all characteristic values and the corresponding normalized eigenfunctions of a sym- metric kernel is called the system of characteristic values and eigenfunctions of the kernel. The system of eigenfunctions is said to be incomplete if there exists a nonzero square integrable function that is orthogonal to all functions of the system. Otherwise, the system of eigenfunctions is said to be complete. 11.6-2. Bilinear Series Assume that a kernel K(x, t) admits an expansion in a uniformly convergent series with respect to the orthonormal system of its eigenfunctions: K(x, t)= ∞  k=1 a k (x)ϕ k (t) (6) for all x in the case of a continuous kernel or for almost all x in the case of a square integrable kernel. We have a k (x)=  b a K(x, t)ϕ k (t) dt = ϕ k (x) λ k , (7) and hence K(x, t)= ∞  k=1 ϕ k (x)ϕ k (t) λ k . (8) Conversely, if the series ∞  k=1 ϕ k (x)ϕ k (t) λ k (9) is uniformly convergent, then K(x, t)= ∞  k=1 ϕ k (x)ϕ k (t) λ k . The following assertion holds: the bilinear series (9) converges in mean-square to the ker- nel K(x, t). If a symmetric kernel K(x, t) has finitely many characteristic values, then it is degenerate, because in this case we have K(x, t)= n  k=1 ϕ k (x)ϕ k (t) λ k . (10) A kernel K(x, t) is said to be positive definite if for all functions ϕ(x) that are not identically zero we have  b a  b a K(x, t)ϕ(x)ϕ(t) dx dt >0, and the above quadratic functional vanishes for ϕ(x) = 0 only. Such a kernel has positive characteristic values only. A negative definite kernel is defined similarly. Page 540 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Each symmetric positive definite (or negative definite) continuous kernel can be decomposed in a bilinear series in eigenfunctions that is absolutely and uniformly convergent with respect to the variables x, t. The assertion remains valid if we assume that the kernel has finitely many negative (positive, respectively) characteristic values. If a kernel K(x, t) is symmetric, continuous on the square S = {a ≤ x ≤ b, a ≤ t ≤ b}, and has uniformly bounded partial derivatives on this square, then this kernel can be expanded in a uniformly convergent bilinear series in eigenfunctions. 11.6-3. The Hilbert–Schmidt Theorem If a function f(x) can be represented in the form f(x)=  b a K(x, t)g(t) dt, (11) where the symmetric kernel K(x, t) is square integrable and g(t) is a square integrable function, then f(x) can be represented by its Fourier series with respect to the orthonormal system of eigenfunctions of the kernel K(x, t): f(x)= ∞  k=1 a k ϕ k (x), (12) where a k =  b a f(x)ϕ k (x) dx, k =1,2, Moreover, if  b a K 2 (x, t) dt ≤ A < ∞, (13) then the series (12) is absolutely and uniformly convergent for any function f(x) of the form (11). Remark 1. In the Hilbert–Schmidt theorem, the completeness of the system of eigenfunctions is not assumed. 11.6-4. Bilinear Series of Iterated Kernels By the definition of the iterated kernels, we have K m (x, t)=  b a K(x, z)K m–1 (z, t) dz, m =2,3, (14) The Fourier coefficients a k (t) of the kernel K m (x, t), regarded as a function of the variable x, with respect to the orthonormal system of eigenfunctions of the kernel K(x, t) are equal to a k (t)=  b a K m (x, t)ϕ k (x) dx = ϕ k (t) λ m k . (15) On applying the Hilbert–Schmidt theorem to (14), we obtain K m (x, t)= ∞  k=1 ϕ k (x)ϕ k (t) λ m k , m =2,3, (16) In formula (16), the sum of the series is understood as the limit in mean-square. If in addition to the above assumptions, inequality (13) is satisfied, then the series in (16) is uniformly convergent. Page 541 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 11.6-5. Solution of the Nonhomogeneous Equation Let us represent an integral equation y(x) – λ  b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (17) where the parameter λ is not a characteristic value, in the form y(x) – f(x)=λ  b a K(x, t)y(t) dt (18) and apply the Hilbert–Schmidt theorem to the function y(x) – f(x): y(x) – f(x)= ∞  k=1 A k ϕ k (x), A k =  b a [y(x) – f(x)]ϕ k (x) dx =  b a y(x)ϕ k (x) dx –  b a f(x)ϕ k (x) dx = y k – f k . Taking into account the expansion (8), we obtain λ  b a K(x, t)y(t) dt = λ ∞  k=1 y k λ k ϕ k (x), and thus λ y k λ k = y k – f k , y k = λ k f k λ k – λ , A k = λf k λ k – λ . (19) Hence, y(x)=f(x)+λ ∞  k=1 f k λ k – λ ϕ k (x). (20) However, if λ is a characteristic value, i.e., λ = λ p = λ p+1 = ···= λ q , (21) then, for k ≠ p, p +1, , q, the terms (20) preserve their form. For k = p, p +1, , q, formula (19) implies the relation f k = A k (λ – λ k )/λ, and by (21) we obtain f p = f p+1 = ··· = f q = 0. The last relation means that  b a f(x)ϕ k (x) dx =0 for k = p, p+1, , q, i.e., the right-hand side of the equation must be orthogonal to the eigenfunctions that correspond to the characteristic value λ. In this case, the solutions of Eqs. (17) have the form y(x)=f(x)+λ ∞  k=1 f k λ k – λ ϕ k (x)+ q  k=p C k ϕ k (x), (22) where the terms in the first of the sums (22) with indices k = p, p +1, , q must be omitted (for these indices, f k and λ – λ k vanish in this sum simultaneously). The coefficients C k in the second sum are arbitrary constants. Page 542 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Remark 2. On the basis of the bilinear expansion (8) and the Hilbert–Schmidt theorem, the solution of the symmetric Fredholm integral equation of the first kind  b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b, can be constructed in a similar way in the form y(x)= ∞  k=1 f k λ k ϕ k (x), and the necessary and sufficient condition for the existence and uniqueness of such a solution in L 2 (a, b) is the completeness of the system of the eigenfunctions ϕ k (x) of the kernel K(x, t) together with the convergence of the series ∞  k=1 f 2 k λ 2 k , where the λ k are the corresponding characteristic values. It should be noted that the verification of the last condition for specific equations is quite complicated. In the solution of Fredholm equations of the first kind, the methods presented in Chapter 10 are usually applied. 11.6-6. The Fredholm Alternative for Symmetric Equations The above results can be unified in the following alternative form. A symmetric integral equation y(x) – λ  b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (23) for a given λ, either has a unique square integrable solution for an arbitrarily given function f(x) ∈ L 2 (a, b), in particular, y = 0 for f = 0, or the corresponding homogeneous equation has finitely many linearly independent solutions Y 1 (x), , Y r (x), r >0. For the second case, the nonhomogeneous equation has a solution if and only if the right-hand side f(x) is orthogonal to all the functions Y 1 (x), , Y r (x) on the interval [a, b]. Here the solution is defined only up to an arbitrary additive linear combination A 1 Y 1 (x)+···+ A r Y r (x). 11.6-7. The Resolvent of a Symmetric Kernel The solution of a Fredholm equation of the second kind (23) can be written in the form y(x)=f(x)+λ  b a R(x, t; λ)f(t) dt, (24) where the resolvent R(x, t; λ) is given by the series R(x, t; λ)= ∞  k=1 ϕ k (x)ϕ k (t) λ k – λ . (25) Here the collections ϕ k (x) and λ k form the system of eigenfunctions and characteristic values of Eqs. (23). It follows from formula (25) that the resolvent of a symmetric kernel has only simple poles. Page 543 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 11.6-8. Extremal Properties of Characteristic Values and Eigenfunctions Let us introduce the notation (u, w)=  b a u(x)w(x) dx, u 2 =(u, u), (Ku, u)=  b a  b a K(x, t)u(x)u(t) dx dt, where (u, w) is the inner product of functions u(x) and w(x), u is the norm of a function u(x), and (Ku, u)isthequadratic form generated by the kernel K(x, t). Let λ 1 be the characteristic value of the symmetric kernel K(x, t) with minimum absolute value and let y 1 (x) be the eigenfunction corresponding to this value. Then 1 |λ 1 | = max y /≡0 |(Ky, y)| y 2 ; (26) in particular, the maximum is attained, and y = y 1 is a maximum point. Let λ 1 , , λ n be the first n characteristic values of a symmetric kernel K(x, t) (in the ascending order of their absolute values) and let y 1 (x), , y n (x) be orthonormal eigenfunctions corresponding to λ 1 , , λ n , respectively. Then the formula 1 |λ n+1 | = max |(Ky, y)| y 2 (27) is valid for the characteristic value λ n+1 following λ n . The maximum is taken over the set of functions y which are orthogonal to all y 1 , , y n and are not identically zero, that is, y ≠ 0 (y, y j )=0, j =1, , n; (28) in particular, the maximum in (27) is attained, and y = y n+1 is a maximum point, where y n+1 is any eigenfunction corresponding to the characteristic value λ n+1 which is orthogonal to y 1 , , y n . Remark 3. For a positive definite kernel 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 K(x, t)ρ(t)y(t) dt = f(x), (29) where K(s, t) is a symmetric kernel and ρ(t) > 0 is a continuous function on [a, b], can be reduced to a symmetric equation. Indeed, on multiplying Eq. (29) by √ ρ(x) and introducing the new unknown function z(x)= √ ρ(x) y(x), we arrive at the integral equation z(x) – λ  b a L(x, t)z(t) dt = f (x)  ρ(x), L(x, t)=K(x, t)  ρ(x)ρ(t), (30) where L(x, t) is a symmetric kernel. Page 544 © 1998 by CRC Press LLC © 1998 by CRC Press LLC [...]... (1974), P P Zabreyko, A I Koshelev, et al (1975), F D Gakhov and Yu I Cherskii (19 78) , A D Polyanin and A V Manzhirov (1997, 19 98) © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 550 11.9 The Carleman Method for Integral Equations of Convolution Type of the Second Kind 11.9-1 The Wiener–Hopf Equation of the Second Kind Equations of convolution type of the second kind of the form* 1 y(x) + √ 2π ∞ K(x... 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 569 Remark 2 For functions with nontrivial growth at infinity, the complete solution of Wiener– Hopf equations of the second kind is presented in the cited book by F D Gakhov and Yu I Cherskii (19 78) Remark 3 The Wiener–Hopf method can be applied to solve Wiener–Hopf integral equations of the first kind under the assumption that the kernels of these equations. .. sides of the above relation are equal to C2 C1 + , u – α – iβ u + α – iβ © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 552 Fig 4 Scheme of solving the Wiener–Hopf integral equations For β = 0, we have the equation of the first kind, and for β = 1, we have the equation of the second kind © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 553 where the constants C1 and C2 must be defined Hence,... Hankel transform: ∞ tJν (xt)y(t) dt L [y] = (8) 0 and acts by the rule L2 = 1 (see Subsection 7.6-1) We obtain the solution by formula (3), for k = 1, taking into account Eq (8) : y(x) = 1 f (x) + λ 1 – λ2 ∞ λ ≠ ±1 tJν (xt)f (t) dt , (9) 0 • Reference for Section 11.7: A D Polyanin and A V Manzhirov (19 98) 11 .8 Methods of Integral Transforms and Model Solutions 11 .8- 1 Equation With Difference Kernel on the... Remark 2 The equation ∞ y(x) – H(xt)xp tq y(t) dt = f (x) 0 can be rewritten in the form of Eq (25) under the notation Q(z) = z p H(z), where β = q – p © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 549 11 .8- 4 The Method of Model Solutions for Equations on the Entire Axis Let us illustrate the capability of a generalized modification of the method of model solutions (see Subsection 9.6) by an example... ( 48) where W(z) is subjected to condition (20), as well as in the case of a homogeneous equation We now note that Eq ( 48) is a special case of Eq (34) In the strip v– < Im z < v+ , the function W(z) is analytic and uniformly tends to 1 as |z| → ∞ because |K(z)| → 0 as |z| → ∞ In this case, this function has the representation (see (43)–(45)) W(z) = W+ (z) , W– (z) (49) © 19 98 by CRC Press LLC © 19 98. .. (see Supplements 4–7) Remark 1 The equation ∞ x α –α–1 y(t) dt = f (x) x t t 0 can be rewritten in the form of Eq (16) under the notation K(z) = z α H(z) y(x) – H (24) © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 5 48 11 .8- 3 Equation With the Kernel K(x, t) = tβ Q(xt) on the Semiaxis Consider the following equation on the semiaxis: ∞ y(x) – tβ Q(xt)y(t) dt = f (x) (25) 0 To solve this equation,... described 11.7-2 Solution of Equations of the Second Kind on the Semiaxis 1◦ Consider the equation ∞ y(x) – λ cos(xt)y(t) dt = f (x) (4) 0 In this case, the operator L coincides, up to a constant factor, with the Fourier cosine transform: ∞ cos(xt)y(t) dt = L [y] = 0 and acts by the rule L2 = k, where k = π 2 π Fc [y] 2 (5) (see Subsection 7.5-1) © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 545... 1)2 ]2 4α2 β + ρ∗ 4α2 ∞ 0 ∞ e–β(x+t) cos[α(x – t)]f (t) dt 0 e–β(x+t) cos[ψ + α(x + t)]f (t) dt, ρ∗ eiψ = (β – 1 – iα)4 8 2 (β – iα) (15) Since Y + (u) = Y1 (u) + Y2 (u), it follows that the desired solution is the sum of the functions (14) and (15) © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 554 11.9-2 An Integral Equation of the Second Kind With Two Kernels Consider an integral equation... (u)]–1 of the problem is a function that has an analytic continuation to the upper half-plane, possibly except for finitely many poles that are zeros of the function 1 + K+ (z) © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC Page 5 58 (we assume that 1 + K+ (z) ≠ 0 on the real axis) Therefore, the index ν of the problem is always nonpositive, ν ≤ 0 On rewriting the problem in the form [1 + K+ (u)]Y + (u) = . rewritten in the form of Eq. (16) under the notation K(z)=z α H(z). Page 5 48 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 11 .8- 3. Equation With the Kernel K(x, t)=t β Q(xt) on the Semiaxis Consider. notation Q(z)=z p H(z), where β = q – p. Page 549 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 11 .8- 4. The Method of Model Solutions for Equations on the Entire Axis Let us illustrate the capability. A. D. Polyanin and A. V. Manzhirov (1997, 19 98) . Page 550 © 19 98 by CRC Press LLC © 19 98 by CRC Press LLC 11.9. The Carleman Method for Integral Equations of Convolution Type of the Second Kind 11.9-1.