We set a = e C and obtain L   ∞ 0 x ν e –Cν Γ(ν +1) dν  = 1 p (ln p + C) . (12) Let us proceed with relation (9). By (12), we have f  x (0) p (ln p + C) = L  f  x (0)  ∞ 0 x ν e –Cν Γ(ν +1) dν  . (13) Taking into account (10) and (12), we can regard the first summand on the right-hand side in (9) as a product of transforms. To find this summand itself we apply the convolution theorem: p 2 ˜ f(p) – f  x (0) p (ln p + C) = L   x 0 f  tt (t)  ∞ 0 (x – t) ν e –Cν Γ(ν +1) dν dt  . (14) On the basis of relations (9), (13), and (14) we obtain the solution of the integral equation (4) in the form y(x)=–  x 0 f  tt (t)  ∞ 0 (x – t) ν e –Cν Γ(ν +1) dν dt – f  x (0)  ∞ 0 x ν e –Cν Γ(ν +1) dν. (15) • References for Section 8.6: V. Volterra (1959), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971). 8.7. Method of Quadratures 8.7-1. Quadrature Formulas The method of quadratures is a method for constructing an approximate solution of an integral equation based on the replacement of integrals by finite sums according to some formula. Such formulas are called quadrature formulas and, in general, have the form  b a ψ(x) dx = n  i=1 A i ψ(x i )+ε n [ψ], (1) where x i (i =1, , n) are the abscissas of the partition points of the integration interval [a, b], or quadrature (interpolation) nodes, A i (i =1, , n) are numerical coefficients independent of the choice of the function ψ(x), and ε n [ψ] is the remainder (the truncation error) of formula (1). As a rule, A i ≥ 0 and n  i=1 A i = b – a. There are quite a few quadrature formulas of the form (1). The following formulas are the simplest and most frequently used in practice. Rectangle rule: A 1 = A 2 = ···= A n–1 = h, A n =0, h = b – a n – 1 , x i = a + h(i – 1) (i =1, , n). (2) Trapezoidal rule: A 1 = A n = 1 2 h, A 2 = A 3 = ···= A n–1 = h, h = b – a n – 1 , x i = a + h(i – 1) (i =1, , n). (3) Simpson’s rule (or prizmoidal formula): A 1 = A 2m+1 = 1 3 h, A 2 = ···= A 2m = 4 3 h, A 3 = ···= A 2m–1 = 2 3 h, h = b – a n – 1 , x i = a + h(i – 1) (n =2m +1, i =1, , n), (4) where m is a positive integer. In formulas (2)–(4), h is a constant integration step. The quadrature formulas due to Chebyshev and Gauss with various numbers of interpolation nodes are also widely applied. Let us illustrate these formulas by an example. Page 454 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Example. For the interval [–1, 1], the parameters in formula (1) acquire the following values: Chebyshev’s formula (n = 6): A 1 = A 2 = ···= 2 n = 1 3 , x 2 = –x 5 = –0.4225186538, x 1 = –x 6 = –0.8662468181, x 3 = –x 4 = –0.2666354015. (5) Gauss’ formula (n = 7): A 1 = A 7 = 0.1294849662, A 3 = A 5 = 0.3818300505, x 1 = –x 7 = –0.9491079123, x 3 = –x 5 = –0.4058451514, A 2 = A 6 = 0.2797053915, A 4 = 0.4179591837, x 2 = –x 6 = –0.7415311856, x 4 =0. (6) Note that a vast literature is devoted to quadrature formulas, and the reader can find books of interest (e.g., see G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), S. M. Nikol’skii (1979)). 8.7-2. The General Scheme of the Method Let us solve the Volterra integral equation of the first kind  x a K(x, t)y(t) dt = f(x), f(a)=0, (7) on an interval a ≤ x ≤ b by the method of quadratures. The procedure of constructing the solution involves two stages: 1 ◦ . First, we determine the initial value y(a). To this end, we differentiate Eq. (7) with respect to x, thus obtaining K(x, x)y(x)+  x a K  x (x, t)y(t) dt = f  x (x). By setting x = a,wefind that y 1 = y(a)= f  x (a) K(a, a) = f  x (a) K 11 . 2 ◦ . Let us choose a constant integration step h and consider the discrete set of points x i = a+h(i–1), i =1, , n.Forx = x i , Eq. (7) acquires the form  x i a K(x i , t)y(t) dt = f(x i ), i =2, , n, (8) Applying the quadrature formula (1) to the integral in (8) and choosing x j (j =1, , i)tobethe nodes in t, we arrive at the system of equations i  j=1 A ij K(x i , x j )y(x j )=f(x i )+ε i [y], i =2, , n, (9) where the A ij are the coefficients of the quadrature formula on the interval [a, x i ] and ε i [y]isthe truncation error. Assume that the ε i [y] are small and neglect them; then we obtain a system of linear algebraic equations in the form i  j=1 A ij K ij y j = f i , i =2, , n, (10) Page 455 © 1998 by CRC Press LLC © 1998 by CRC Press LLC where K ij = K(x i , x j )(j =1, , i), f i = f (x i ), and y j are approximate values of the unknown function at the nodes x i . Now system (10) permits one, provided that A ii K ii ≠ 0(i =2, , n), to successively find the desired approximate values by the formulas y 1 = f  x (a) K 11 , y 2 = f 2 – A 21 K 21 y 1 A 22 K 22 , , y n = f n – n–1  j=1 A nj K nj y j A nn K nn , whose specific form depends on the choice of the quadrature formula. 8.7-3. An Algorithm Based on the Trapezoidal Rule According to the trapezoidal rule (3), we have A i1 = A ii = 1 2 h, A i2 = ···= A i,i–1 = h, i =2, , n. The application of the trapezoidal rule in the general scheme leads to the following step algorithm: y 1 = f  x (a) K 11 , f  x (a)= –3f 1 +4f 2 – f 3 2h , y i = 2 K ii  f i h – i–1  j=1 β j K ij y j  , β j =  1 2 for j =1, 1 for j >1, i =2, , n, where the notation coincides with that introduced in Subsection 8.7-2. The trapezoidal rule is quite simple and effective and frequently used in practice for solving integral equations with variable limit of integration. On the basis of Subsections 8.7-1 and 8.7-2, one can write out similar expressions for other quadrature formulas. However, they must be used with care. For example, the application of Simpson’s rule must be alternated, for odd nodes, with some other rule, e.g., the rectangle rule or the trapezoidal rule. For equations with variable integration limit, the use of Chebyshev’s formula or Gauss’ formula also has some difficulties as well. 8.7-4. An Algorithm for an Equation With Degenerate Kernel A general property of the algorithms of the method of quadratures in the solution of the Volterra equations of the first kind with arbitrary kernel is that the amount of computational work at each step is proportional to the number of the step: all operations of the previous step are repeated with new data and another term in the sum is added. However, if the kernel in Eq. (7) is degenerate, i.e., K(x, t)= m  k=1 p k (x)q k (t), (11) or if the kernel under consideration can be approximated by a degenerate kernel, then an algorithm can be constructed for which the number of operations does not depend on the index of the digitalization node. With regard to (11), Eq. (7) becomes m  k=1 p k (x)  x a q k (t)y(t) dt = f(x). (12) Page 456 © 1998 by CRC Press LLC © 1998 by CRC Press LLC By applying the trapezoidal rule to (12), we obtain recurrent expressions for the solution of the equation (see formulas in Subsection 8.7-3): y(a)= f  x (a) m  k=1 p k (a)q k (a) , y i = 2 m  k=1 p ki q ki  f i h – m  k=1 p ki i–1  j=1 β j q kj y j  , where y i are approximate values of y(x)atx i , f i = f(x i ), p ki = p k (x i ), and q ki = q k (x i ). • References for Section 8.7: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), V. I. Krylov, V. V. Bobkov, and P. I. Monastyrnyi (1984), A. F. Verlan’ and V. S. Sizikov (1986). 8.8. Equations With Infinite Integration Limit Integral equations of the first kind with difference kernel in which one of the limits of integration is variable and the other is infinite are of interest. Sometimes the kernels and the functions of these equations do not belong to the classes described in the beginning of the chapter. The investigation of these equations can be performed by the method of model solutions (see Section 9.6) or by the method of reducing to equations of the convolution type. Let us consider these methods for an example of an equation of the first kind with variable lower limit of integration. 8.8-1. An Equation of the First Kind With Variable Lower Limit of Integration Consider the equation of the first kind with difference kernel  ∞ x K(x – t)y(t) dt = f(x). (1) Equation (1) cannot be solved by direct application of the Laplace transform, because the convolution theorem cannot be used here. According to the method of model solutions whose detailed exposition can be found in Section 9.6, we consider the auxiliary equation with exponential right-hand side  ∞ x K(x – t)y(t) dt = e px . (2) The solution of (2) has the form Y (x, p)= 1 ˜ K(–p) e px , ˜ K(–p)=  ∞ 0 K(–z)e pz dz. (3) On the basis of these formulas and formula (11) from Section 9.6, we obtain the solution of Eq. (1) for an arbitrary right-hand side f(x) in the form y(x)= 1 2πi  c+i∞ c–i∞ ˜ f(p) ˜ K(–p) e px dp, (4) where ˜ f(p) is the Laplace transform of the function f(x). Page 457 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Example. Consider the following integral equation of the first kind with variable lower limit of integration:  ∞ x e a(x–t) y(t) dt = A sin(bx), a >0. (5) According to (3) and (4), we can write out the expressions for ˜ f(p) (see Supplement 4) and ˜ K(–p), ˜ f(p)= Ab p 2 + b 2 , ˜ K(–p)=  ∞ 0 e (p–a)z dz = 1 a – p , (6) and the solution of Eq. (5) in the form y(x)= 1 2πi  c+i∞ c–i∞ Ab(a – p) p 2 + b 2 e px dp. (7) Now using the tables of inverse Laplace transforms (see Supplement 5), we obtain the exact solution y(x)=Aa sin(bx) – Ab cos(bx), a >0, (8) which can readily be verified by substituting (8) into (5) and using the tables of integrals in Supplement 2. 8.8-2. Reduction to a Wiener–Hopf Equation of the First Kind Equation (1) can be reduced to a first-kind one-sided equation  ∞ 0 K – (x – t)y(t) dt = –f(x), 0 < x < ∞, (9) where the kernel K – (x – t) has the following form: K – (s)=  0 for s >0, –K(s) for s <0. Methods for studying Eq. (9) are described in Chapter 10. • References for Section 8.8: F. D. Gakhov and Yu. I. Cherskii (1978), A. D. Polyanin and A. V. Manzhirov (1997). Page 458 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Chapter 9 Methods for Solving Linear Equations of the Form y(x)–  x a K(x, t)y(t) dt = f(x) 9.1. Volterra Integral Equations of the Second Kind 9.1-1. Preliminary Remarks. Equations for the Resolvent In this chapter we present methods for solving Volterra integral equations of the second kind, which have the form y(x)–  x a K(x, t)y(t) dt = f(x), (1) where y(x) is the unknown function (a ≤ x ≤ b), K(x, t) is the kernel of the integral equation, and f(x) is the right-hand side of the integral equation. The function classes to which y(x), f(x), and K(x, t) can belong are defined in Subsection 8.1-1. In these function classes, there exists a unique solution of the Volterra integral equation of the second kind. Equation (1) is said to be homogeneous if f(x) ≡ 0 and nonhomogeneous otherwise. The kernel K(x, t) is said to be degenerate if it can be represented in the form K(x, t)= g 1 (x)h 1 (t)+···+ g n (x)h n (t). The kernel K(x, t) of an integral equation is called difference kernel if it depends only on the difference of the arguments, K(x, t)=K(x – t). Remark 1. A homogeneous Volterra integral equation of the second kind has only the trivial solution. Remark 2. The existence and uniqueness of the solution of a Volterra integral equation of the second kind hold for a much wider class of kernels and functions. Remark 3. A Volterra equation of the second kind can be regarded as a Fredholm equation of the second kind whose kernel K(x, t) vanishes for t > x (see Chapter 11). Remark 4. The case in which a =–∞ and/or b = ∞ is not excluded, but in this case the square integrability of the kernel K(x, t) on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} is especially significant. The solution of Eq. (1) can be presented in the form y(x)=f(x)+  x a R(x, t)f(t) dt, (2) where the resolvent R(x, t) is independent of f (x) and the lower limit of integration a and is determined by the kernel of the integral equation alone. The resolvent of the Volterra equation (1) satisfies the following two integral equations: R(x, t)=K(x, t)+  x t K(x, s)R(s, t) ds , (3) Page 459 © 1998 by CRC Press LLC © 1998 by CRC Press LLC R(x, t)=K(x, t)+  x t K(s, t)R(x, s) ds , (4) in which the integration is performed with respect to different pairs of variables of the kernel and the resolvent. 9.1-2. A Relationship Between Solutions of Some Integral Equations Let us present two useful formulas that express the solution of one integral equation via the solutions of other integral equations. 1 ◦ . Assume that the Volterra equation of the second kind with kernel K(x, t) has a resolvent R(x, t). Then the Volterra equation of the second kind with kernel K ∗ (x, t)=–K(t, x) has the resolvent R ∗ (x, t)=–R(t, x). 2 ◦ . Assume that two Volterra equations of the second kind with kernels K 1 (x, t) and K 2 (x, t) are given and that resolvents R 1 (x, t) and R 2 (x, t) correspond to these equations. In this case the Volterra equation with kernel K(x, t)=K 1 (x, t)+K 2 (x, t) –  x t K 1 (x, s)K 2 (s, t) ds (5) has the resolvent R(x, t)=R 1 (x, t)+R 2 (x, t)+  x t R 1 (s, t)R 2 (x, s) ds. (6) Note that in formulas (5) and (6), the integration is performed with respect to different pairs of variables. • References for Section 9.1: E. Goursat (1923), H. M. M ¨ untz (1934), V. Volterra (1959), S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V. I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. J. Jerry (1985), F. G. Tricomi (1985), A. F. Verlan’ and V. S. Sizikov (1986), G. Gripenberg, S O. Londen, and O. Staffans (1990), C. Corduneanu (1991), R. Gorenflo and S. Vessella (1991), A. C. Pipkin (1991). 9.2. Equations With Degenerate Kernel: K(x, t)=g 1 (x)h 1 (t)+···+ g n (x)h n (t) 9.2-1. Equations With Kernel of the Form K(x, t)=ϕ(x)+ψ(x)(x – t) The solution of a Volterra equation (see Subsection 9.1-1) with kernel of this type can be expressed by the formula y = w  xx , (1) where w = w(x) is the solution of the second-order linear nonhomogeneous ordinary differential equation w  xx – ϕ(x)w  x – ψ(x)w = f (x), (2) with the initial conditions w(a)=w  x (a) = 0. (3) Let w 1 = w 1 (x) be a nontrivial particular solution of the corresponding homogeneous linear differ- ential equation (2) for f (x) ≡ 0. Assume that w 1 (a) ≠ 0. In this case, the other nontrivial particular solution w 2 = w 2 (x) of this homogeneous linear differential equation has the form w 2 (x)=w 1 (x)  x a Φ(t) [w 1 (t)] 2 dt, Φ(x)=exp   x a ϕ(s) ds  . Page 460 © 1998 by CRC Press LLC © 1998 by CRC Press LLC The solution of the nonhomogeneous equation (2) with the initial conditions (3) is given by the formula w(x)=w 2 (x)  x a w 1 (t) Φ(t) f(t) dt – w 1 (x)  x a w 2 (t) Φ(t) f(t) dt. (4) On substituting expression (4) into formula (1) we obtain the solution of the original integral equation in the form y(x)=f(x)+  x a R(x, t)f(t) dt, where R(x, t)=[w  2 (x)w 1 (t) – w  1 (x)w 2 (t)] 1 Φ(t) = ϕ(x) Φ(x) w 1 (x) w 1 (t) Φ(t) +[ϕ(x)w  1 (x)+ψ(x)w 1 (x)] w 1 (t) Φ(t)  x t Φ(s) [w 1 (s)] 2 ds. Here Φ(x)=exp   x a ϕ(s) ds  and the primes stand for x-derivatives. For a degenerate kernel of the above form, the resolvent can be defined by the formula R(x, t)=u  xx , where the auxiliary function u is the solution of the homogeneous linear second-order ordinary differential equation u  xx – ϕ(x)u  x – ψ(x)u = 0 (5) with the following initial conditions at x = t: u   x=t =0, u  x   x=t = 1. (6) The parameter t occurs only in the initial conditions (6), and Eq. (5) itself is independent of t. Remark 1. The kernel of the integral equation in question can be rewritten in the form K(x, t)= G 1 (x)+tG 2 (x), where G 1 (x)=ϕ(x)+xψ(x) and G 2 (x)=–ϕ(x). 9.2-2. Equations With Kernel of the Form K(x, t)=ϕ(t)+ψ(t)(t – x) For a degenerate kernel of the above form, the resolvent is determined by the expression R(x, t)=–v  tt , (7) where the auxiliary function v is the solution of the homogeneous linear second-order ordinary differential equation v  tt + ϕ(t)v  t + ψ(t)v = 0 (8) with the following initial conditions at t = x: v   t=x =0, v  t   t=x = 1. (9) The point x occurs only in the initial data (9) as a parameter, and Eq. (8) itself is independent of x. Assume that v 1 = v 1 (t) is a nontrivial particular solution of Eq. (8). In this case, the general solution of this differential equation is given by the formula v(t)=C 1 v 1 (t)+C 2 v 1 (t)  t a ds Φ(s)[v 1 (s)] 2 , Φ(t)=exp   t a ϕ(s) ds  . Page 461 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Taking into account the initial data (9), we find the dependence of the integration constants C 1 and C 2 on the parameter x. As a result, we obtain the solution of problem (8), (9): v = v 1 (x)Φ(x)  t x ds Φ(s)[v 1 (s)] 2 . (10) On substituting the expression (10) into formula (7) and eliminating the second derivative by means of Eq. (8) we find the resolvent: R(x, t)=ϕ(t) v 1 (x)Φ(x) v 1 (t)Φ(t) + v 1 (x)Φ(x)[ϕ(t)v  t (t)+ψ(t)v 1 (t)]  t x ds Φ(s)[v 1 (s)] 2 . Remark 2. The kernel of the integral equation under consideration can be rewritten in the form K(x, t)=G 1 (t)+xG 2 (t), where G 1 (t)=ϕ(t)+tψ(t) and G 2 (t)=–ϕ(t). 9.2-3. Equations With Kernel of the Form K(x, t)=  n m=1 ϕ m (x)(x – t) m–1 To find the resolvent, we introduce an auxiliary function as follows: u(x, t)= 1 (n – 1)!  x t R(s, t)(x – s) n–1 ds + (x – t) n–1 (n – 1)! ; at x = t, this function vanishes together with the first n – 2 derivatives with respect to x, and the (n – 1)st derivative at x = t is equal to 1. Moreover, R(x, t)=u (n) x (x, t), u (n) x = d n u(x, t) dx n . (11) On substituting relation (11) into the resolvent equation (3) of Subsection 9.1-1, we see that u (n) x (x, t)=K(x, t)+  x t K(x, s)u (n) s (s, t) ds. (12) Integrating by parts the right-hand side in (12), we obtain u (n) x (x, t)=K(x, t)+ n–1  m=0 (–1) m K (m) s (x, s)u (n–m–1) s (s, t)   s=x s=t . (13) On substituting the expressions for K(x, t) and u(x, t) into (13), we arrive at a linear homogeneous ordinary differential equation of order n for the function u(x, t). Thus, the resolvent R(x, t) of the Volterra integral equation with degenerate kernel of the above form can be obtained by means of (11), where u(x, t) satisfies the following differential equation and initial conditions: u (n) x – ϕ 1 (x)u (n–1) x – ϕ 2 (x)u (n–2) x – 2ϕ 3 (x)u (n–3) x – ···– (n – 1)! ϕ n (x)u =0, u   x=t = u  x   x=t = ···= u (n–2) x   x=t =0, u (n–1) x   x=t =1. The parameter t occurs only in the initial conditions, and the equation itself is independent of t explicitly. Remark 3. A kernel of the form K(x, t)= n  m=1 φ m (x)t m–1 can be reduced to a kernel of the above type by elementary transformations. Page 462 © 1998 by CRC Press LLC © 1998 by CRC Press LLC 9.2-4. Equations With Kernel of the Form K(x, t)=  n m=1 ϕ m (t)(t – x) m–1 Let us represent the resolvent of this degenerate kernel in the form R(x, t)=–v (n) t (x, t), v (n) t = d n v(x, t) dt n , where the auxiliary function v(x, t) vanishes at t = x together with n – 2 derivatives with respect to t, and the (n – 1)st derivative with respect to t at t = x is equal to 1. On substituting the expression for the resolvent into Eq. (3) of Subsection 9.1-1, we obtain v (n) t (x, t)=  x t K(s, t)v (n) s (x, s) ds – K(x, t). Let us apply integration by parts to the integral on the right-hand side. Taking into account the properties of the auxiliary function v(x, t), we arrive at the following Cauchy problem for an nth-order ordinary differential equation: v (n) t + ϕ 1 (t)v (n–1) t + ϕ 2 (t)v (n–2) t +2ϕ 3 (t)v (n–3) t + ···+(n – 1)! ϕ n (t)v =0, v   t=x = v  t   t=x = ···= v (n–2) t   t=x =0, v (n–1) t   t=x =1. The parameter x occurs only in the initial conditions, and the equation itself is independent of x explicitly. Remark 4. A kernel of the form K(x, t)= n  m=1 φ m (t)x m–1 can be reduced to a kernel of the above type by elementary transformations. 9.2-5. Equations With Degenerate Kernel of the General Form In this case, the Volterra equation of the second kind can be represented in the form y(x) – n  m=1 g m (x)  x a h m (t)y(t) dt = f(x). (14) Let us introduce the notation w j (x)=  x a h j (t)y(t) dt, j =1, , n, (15) and rewrite Eq. (14) as follows: y(x)= n  m=1 g m (x)w m (x)+f(x). (16) On differentiating the expressions (15) with regard to formula (16), we arrive at the following system of linear differential equations for the functions w j = w j (x): w  j = h j (x)  n  m=1 g m (x)w m + f(x)  , j =1, , n, with the initial conditions w j (a)=0, j =1, , n. Once the solution of this system is found, the solution of the original integral equation (14) is defined by formula(16) or any of the expressions y(x)= w  j (x) h j (x) , j =1, , n, which can be obtained from formula (15) by differentiation. • References for Section 9.2: E. Goursat (1923), H. M. M ¨ untz (1934), A. F. Verlan’ and V. S. Sizikov (1986), A. D. Polyanin and A. V. Manzhirov (1998). Page 463 © 1998 by CRC Press LLC © 1998 by CRC Press LLC [...]... construct a solution of systems of Volterra equations of the first kind and of integro-differential equations as well • References for Section 9.3: V A Ditkin and A P Prudnikov (1965), M L Krasnov, A I Kiselev, 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), R Gorenflo and S... nonhomogeneous ordinary differential equations with constant coefficients 9 .7- 3 Equations With Kernel Containing a Sum of Trigonometric Functions Equations with difference kernel of the form m x y(x) + K(x – t)y(t) dt = f (x), K(x) = a x y(x) + Ak cos(λk x), (4) Ak sin(λk x), (5) k=1 m K(x – t)y(t) dt = f (x), K(x) = a k=1 can also be reduced to linear nonhomogeneous ordinary differential equations of order 2m with... Sometimes by differentiating we can reduce a given equation to a simpler integral equation whose solution is 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 Containing a Sum of Exponential Functions Consider the equation x n Ak eλk (x–t) y(t) dt = f (x) y(x) + a (1) k=1 In the general case,... is arbitrary, this approach extends the abilities of the method of model solutions • References for Section 9.6: A D Polyanin and A V Manzhirov (19 97, 1998) 9 .7 Method of Differentiation for Integral Equations In some cases, the differentiation of integral equations (once, twice, and so on) with the subsequent elimination of integral terms by means of the original equation makes it possible to reduce... to linear nonhomogeneous ordinary differential equations with constant coefficients 9 .7- 4 Equations Whose Kernels Contain Combinations of Various Functions Any equation with difference kernel that contains a linear combination of summands of the form (x – t)m (m = 0, 1, 2, ), cosh β(x – t) , exp α(x – t) , sinh γ(x – t) , cos λ(x – t) , sin µ(x – t) , (7) can also be reduced by differentiation to a... we apply a slight modification of the above scheme, which corresponds to the case M ≡ const Let us rewrite Eq (6) as follows: d dx x f (x) – y(x) y(t) dt √ = (7) λ x–t Let us assume that the right-hand side of Eq (7) is known and treat Eq (7) as an Abel equation of the first kind Its solution can be written in the following form (see the example in Subsection 8.4-4): a or y(x) + 1 d πλ dx x 1 d π dx... instead of w(x), where y0 is the general solution of the homogeneous equation © 1998 by CRC Press LLC © 1998 by CRC Press LLC Page 472 9.4-4 Solution of Operator Equations of Polynomial Form The method described in Subsection 9.4-3 can be generalized to the case of operator equations of polynomial form Suppose that the solution of the linear nonhomogeneous equation (21) is given by formula (22) and that... (34) © 1998 by CRC Press LLC © 1998 by CRC Press LLC Page 473 can be represented in the form of a product n y(x) – Q [y] ≡ 1 – Lk [y], (35) k=1 where the Lk are linear operators Suppose that the solutions of the auxiliary equations y(x) – Lk [y] = f (x), k = 1, , n (36) are known and are given by the formulas y(x) = Yk f (x) , k = 1, , n ( 37) The solution of the auxiliary equation (36) for k = n,... of the auxiliary problem for Eq (1) whose right-hand side is the kernel of the inverse transform P–1 : L [Y (x, λ)] = ψ(x, λ) (7) * Before reading this section, it is useful to look over Section 9.5 © 1998 by CRC Press LLC © 1998 by CRC Press LLC Page 479 Let us multiply Eq (7) by F (λ) and integrate with respect to λ within the same limits that stand in the inverse transform (6) Taking into account... Section 9 .7 9.3-4 Reduction to a Wiener–Hopf Equation of the Second Kind A Volterra equation of the second kind with the difference kernel of the form x y(x) + K(x – t)y(t) dt = f (x), 0 < x < ∞, (16) 0 can be reduced to the Wiener–Hopf equation ∞ y(x) + K+ (x – t)y(t) dt = f (x), 0 < x < ∞, ( 17) 0 where the kernel K+ (x – t) is given by K+ (s) = K(s) for s > 0, 0 for s < 0 Methods for studying Eq ( 17) are . formula (n = 7) : A 1 = A 7 = 0.1294849662, A 3 = A 5 = 0.3818300505, x 1 = –x 7 = –0.9491 079 123, x 3 = –x 5 = –0.4058451514, A 2 = A 6 = 0. 279 7053915, A 4 = 0.4 179 5918 37, x 2 = –x 6 = –0 .74 15311856, x 4 =0. (6) Note. 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),. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1 971 ), J. A. Cochran (1 972 ), V. I. Smirnov (1 974 ), P. P. Zabreyko, A. I. Koshelev, et al. (1 975 ), A. J. Jerry (1985), F. G. Tricomi (1985), A.