16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 843 On applying the Mellin transform to equation (16.4.6.15) and taking into account the fact that the integral with such a kernel is transformed into the product by the rule (see Subsection 11.3.2) M ∞ 0 1 t Q x t y(t) dt = ˆ Q(s)ˆy(s), we obtain the following equation for the transform ˆy(s): ˆy(s)– ˆ Q(s)ˆy(s)= ˆ f(s). The solution of this equation is given by the formula ˆy(s)= ˆ f(s) 1 – ˆ Q(s) .(16.4.6.18) On applying the Mellin inversion formula to equation (16.4.6.18) we obtain the solution of the original integral equation y(x)= 1 2πi c+i∞ c–i∞ ˆ f(s) 1 – ˆ Q(s) x –s ds.(16.4.6.19) This solution can also be represented via the resolvent in the form y(x)=f (x)+ ∞ 0 1 t N x t f(t) dt,(16.4.6.20) where we have used the notation N(x)=M –1 { ˆ N(s)}, ˆ N(s)= ˆ Q(s) 1 – ˆ Q(s) .(16.4.6.21) Under the application of this analytical method of solution, the following technical difficulties can occur: (a) in the calculation of the transform for a given kernel K(x)and (b) in the calculation of the solution for the known transform ˆy(s). To find the corresponding integrals, tables of direct and inverse Mellin transforms are applied (e.g., see Sections T3.5 and T3.6). In many cases, the relationship between the Mellin transform and the Fourier and Laplace transforms is first used: M{f (x), s} = F{f(e x ), is} = L{f(e x ), –s} + L{f (e –x ), s},(16.4.6.22) and then tables of direct and inverse Fourier transforms and Laplace transforms are applied (see Sections T3.1–T3.4). 16.4.6-3. Equation with the kernel K(x, t)=t β Q(xt)onthesemiaxis. Consider the following equation on the semiaxis: y(x)– ∞ 0 t β Q(xt)y(t) dt = f(x). (16.4.6.23) 844 INTEGRAL EQUATIONS To solve this equation, we apply the Mellin transform. On multiplying equation (16.4.6.23) by x s–1 and integrating with respect to x from zero to infinity, we obtain ∞ 0 y(x)x s–1 dx – ∞ 0 y(t)t β dt ∞ 0 Q(xt)x s–1 dx = ∞ 0 f(x)x s–1 dx.(16.4.6.24) Let us make the change of variables z = xt.Wefinally obtain ˆy(s)– ˆ Q(s) ∞ 0 y(t)t β–s dt = ˆ f(s). (16.4.6.25) Taking into account the relation ∞ 0 y(t)t β–s dt =ˆy(1 + β – s), we rewrite equation (16.4.6.25) in the form ˆy(s)– ˆ Q(s)ˆy(1 + β – s)= ˆ f(s). (16.4.6.26) On replacing s by 1 + β – s in equation (16.4.6.26), we obtain ˆy(1 + β – s)– ˆ Q(1 + β – s)ˆy(s)= ˆ f(1 + β – s). (16.4.6.27) Let us eliminate ˆy(1 + β – s) from (16.4.6.26) and (16.4.6.27), and then solve the resulting equation for ˆy(s). We thus fi nd the transform of the solution: ˆy(s)= ˆ f(s)+ ˆ Q(s) ˆ f(1 + β – s) 1 – ˆ Q(s) ˆ Q(1 + β – s) . On applying the Mellin inversion formula, we obtain the solution of the integral equa- tion (16.4.6.23) in the form y(x)= 1 2πi c+i∞ c–i∞ ˆ f(s)+ ˆ Q(s) ˆ f(1 + β – s) 1 – ˆ Q(s) ˆ Q(1 + β – s) x –s ds. 16.4.7. Method of Approximating a Kernel by a Degenerate One 16.4.7-1. Approximation of the kernel. For the approximate solution of the Fredholm integral equation of the second kind y(x)– b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b,(16.4.7.1) where, for simplicity, the functions f(x)andK(x, t) are assumed to be continuous, it is useful to replace the kernel K(x, t) by a close degenerate kernel K (n) (x, t)= n k=0 g k (x)h k (t). (16.4.7.2) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 845 Let us indicate several ways to perform such a change. If the kernel K(x, t)isdif- ferentiable with respect to x on [a, b]sufficiently many times, then, for a degenerate kernel K (n) (x, t), we can take a finite segment of the Taylor series: K (n) (x, t)= n m=0 (x – x 0 ) m m! K (m) x (x 0 , t), (16.4.7.3) where x 0 [a, b]. A similar trick can be applied for the case in which K(x, t) is differentiable with respect to t on [a, b]sufficiently many times. To construct a degenerate kernel, a finite segment of the double Fourier series can be used: K (n) (x, t)= n p=0 n q=0 a pq (x – x 0 ) p (t – t 0 ) q ,(16.4.7.4) where a pq = 1 p! q! ∂ p+q ∂x p ∂t q K(x, t) x=x 0 t=t 0 , a ≤ x 0 ≤ b, a ≤ t 0 ≤ b. A continuous kernel K(x, t) admits an approximation by a trigonometric polynomial of period 2l,wherel = b – a. For instance, we can set K (n) (x, t)= 1 2 a 0 (t)+ n k=1 a k (t)cos kπx l ,(16.4.7.5) where the a k (t)(k = 0, 1, 2, ) are the Fourier coefficients a k (t)= 2 l b a K(x, t)cos pπx l dx.(16.4.7.6) A similar decomposition can be obtained by interchanging the roles of the variables x and t.Afinite segment of the double Fourier series can also be applied by setting, for instance, a k (t) ≈ 1 2 a k0 + n m=1 a km cos mπt l , k = 0, 1, , n,(16.4.7.7) and it follows from formulas (16.4.7.5)–(16.4.7.7) that K (n) (x, t)= 1 4 a 00 + 1 2 n k=1 a k0 cos kπx l + 1 2 n m=1 a 0m cos mπt l + n k=1 n m=1 a km cos kπx l cos mπt l , where a km = 4 l 2 b a b a K(x, t)cos kπx l cos mπt l dx dt.(16.4.7.8) One can also use other methods of interpolating and approximating the kernel K(x, t). 846 INTEGRAL EQUATIONS 16.4.7-2. Approximate solution. If K (n) (x, t) is an approximate degenerate kernel for a given exact kernel K(x, t)andifa function f n (x) is close to f(x), then the solution y n (x) of the integral equation y n (x)– b a K (n) (x, t)y n (t) dt = f n (x)(16.4.7.9) can be regarded as an approximation to the solution y(x) of equation (16.4.7.1). Assume that the following error estimates hold: b a |K(x, t)–K (n) (x, t)| dt ≤ ε, |f(x)–f n (x)| ≤ δ. Next, let the resolvent R n (x, t) of equation (16.4.7.9) satisfy the relation b a |R n (x, t)| dt ≤ M n for a ≤ x ≤ b. Finally, assume that the following inequality holds: q = ε(1 + M n )<1. In this case, equation (16.4.7.1) has a unique solution y(x)and |y(x)–y n (x)| ≤ ε N(1 + M n ) 2 1 – q + δ, N =max a≤x≤b |f(x)|.(16.4.7.10) Example. Let us find an approximate solution of the equation y(x)– 1/2 0 e –x 2 t 2 y(t) dt = 1. (16.4.7.11) Applying the expansion in a double Taylor series, we replace the kernel K(x, t)=e –x 2 t 2 with the degenerate kernel K (2) (x, t)=1 – x 2 t 2 + 1 2 x 4 t 4 . Hence, instead of equation (16.4.7.11) we obtain y 2 (x)=1 + 1/2 0 1 – x 2 t 2 + 1 2 x 4 t 4 y 2 (t) dt. (16.4.7.12) Therefore, y 2 (x)=1 + A 1 + A 2 x 2 + A 3 x 4 , (16.4.7.13) where A 1 = 1/2 0 y 2 (x) dx, A 2 =– 1/2 0 x 2 y 2 (x) dx, A 3 = 1 2 1/2 0 x 4 y 2 (x) dx. (16.4.7.14) From (16.4.7.13) and (16.4.7.14) we obtain a system of three equations with three unknowns; to the fourth decimal place, the solution is A 1 = 0.9930, A 2 =–0.0833, A 3 = 0.0007. Hence, y(x) ≈ y 2 (x)=1.9930 – 0.0833 x 2 + 0.0007 x 4 , 0 ≤ x ≤ 1 2 . (16.4.7.15) An error estimate for the approximate solution (16.4.7.15) can be performed by formula (16.4.7.10). 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 847 16.4.8. Collocation Method 16.4.8-1. General remarks. Let us rewrite the Fredholm integral equation of the second kind in the form ε[y(x)] ≡ y(x)–λ b a K(x, t)y(t) dt – f(x)=0.(16.4.8.1) Let us seek an approximate solution of equation (16.4.8.1) in the special form Y n (x)=Φ(x, A 1 , , A n )(16.4.8.2) with free parameters A 1 , , A n (undetermined coefficients). On substituting the expres- sion (16.4.8.2) into equation (16.4.8.1), we obtain the residual ε[Y n (x)] = Y n (x)–λ b a K(x, t)Y n (t) dt – f (x). (16.4.8.3) If y(x) is an exact solution, then, clearly, the residual ε[y(x)] is zero. Therefore, one tries to choose the parameters A 1 , , A n so that, in a sense, the residual ε[Y n (x)] is as small as possible. The residual ε[Y n (x)] can be minimized in several ways. Usually, to simplify the calculations, a function Y n (x) linearly depending on the parameters A 1 , , A n is taken. On finding the parameters A 1 , , A n , we obtain an approximate solution (16.4.8.2). If lim n→∞ Y n (x)=y(x), (16.4.8.4) then, by taking a sufficiently large number of parameters A 1 , , A n ,wefind that the solution y(x) can be found with an arbitrary prescribed precision. Now let us go to the description of a concrete method of construction of an approximate solution Y n (x). 16.4.8-2. Approximate solution. We set Y n (x)=ϕ 0 (x)+ n i=1 A i ϕ i (x), (16.4.8.5) where ϕ 0 (x), ϕ 1 (x), , ϕ n (x) are given functions (coordinate functions)andA 1 , , A n are indeterminate coefficients, and assume that the functions ϕ i (x)(i = 1, , n) are linearly independent. Note that, in particular, we can take ϕ 0 (x)=f(x)orϕ 0 (x) ≡ 0. On substituting the expression (16.4.8.5) into the left-hand side of equation (16.4.8.1), we obtain the residual ε[Y n (x)] = ϕ 0 (x)+ n i=1 A i ϕ i (x)–f(x)–λ b a K(x, t) ϕ 0 (t)+ n i=1 A i ϕ i (t) dt, or ε[Y n (x)] = ψ 0 (x, λ)+ n i=1 A i ψ i (x, λ), (16.4.8.6) 848 INTEGRAL EQUATIONS where ψ 0 (x, λ)=ϕ 0 (x)–f(x)–λ b a K(x, t)ϕ 0 (t) dt, ψ i (x, λ)=ϕ i (x)–λ b a K(x, t)ϕ i (t) dt, i = 1, , n. (16.4.8.7) According to the collocation method, we require that the residual ε[Y n (x)] be zero at the given system of the collocation points x 1 , , x n on the interval [a, b], i.e., we set ε[Y n (x j )] = 0, j = 1, , n, where a ≤ x 1 < x 2 < ···< x n–1 < x n ≤ b. It is common practice to set x 1 = a and x n = b. This, together with formula (16.4.8.6), implies the linear algebraic system n i=1 A i ψ i (x j , λ)=–ψ 0 (x j , λ), j = 1, , n,(16.4.8.8) for the coefficients A 1 , , A n . If the determinant of system (16.4.8.8) is nonzero, det[ψ i (x j , λ)] ≠ 0, then system (16.4.8.8) uniquely determines the numbers A 1 , , A n , and hence makes it possible to find the approximate solution Y n (x) by formula (16.4.8.5). 16.4.8-3. Eigenfunctions of the equation. On equating the determinant with zero, we obtain the relation det[ψ i (x j , λ)] = 0, which enables us to find approximate values λ k (k = 1, , n) for the characteristic values of the kernel K(x, t). If we set f(x) ≡ 0, ϕ 0 (x) ≡ 0, λ = λ k , then, instead of system (16.4.8.8), we obtain the homogeneous system n i=1 A (k) i ψ i (x j , λ k )=0, j = 1, , n.(16.4.8.9) On finding nonzero solutions A (k) i (i = 1, , n) of system (16.4.8.9), we obtain approx- imate eigenfunctions for the kernel K(x, t): Y (k) n (x)= n i=1 A (k) i ϕ i (x), that correspond to its characteristic value λ k ≈ λ k . 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 849 Example. Let us solve the equation y(x)– 1 0 t 2 y(t) x 2 + t 2 dt = x arctan 1 x (16.4.8.10) by the collocation method. We set Y 2 (x)=A 1 + A 2 x. On substituting this expression into equation (16.4.8.10), we obtain the residual ε[Y 2 (x)] = –A 1 x arctan 1 x + A 2 x – 1 2 + x 2 2 ln 1 + 1 x 2 – x arctan 1 x . On choosing the collocation points x 1 = 0 and x 2 = 1 and taking into account the relations lim x→0 x arctan 1 x = 0, lim x→0 x 2 ln 1 + 1 x 2 = 0, we obtain the following system for the coefficients A 1 and A 2 : 0×A 1 – 1 2 A 2 = 0, – π 4 A 1 + 1 2 (1 +ln2)A 2 = π 4 . This implies A 2 = 0 and A 1 =–1. Thus, Y 2 (x)=–1. (16.4.8.11) We can readily verify that the approximate solution (16.4.8.11) thus obtained is exact. 16.4.9. Method of Least Squares 16.4.9-1. Description of the method. By analogy with the collocation method, for the equation ε[y(x)] ≡ y(x)–λ b a K(x, t)y(t) dt – f(x)=0 (16.4.9.1) we set Y n (x)=ϕ 0 (x)+ n i=1 A i ϕ i (x), (16.4.9.2) where ϕ 0 (x), ϕ 1 (x), , ϕ n (x) are given functions, A 1 , , A n are indeterminate coeffi- cients, and ϕ i (x)(i = 1, , n) are linearly independent. On substituting (16.4.9.2) into the left-hand side of equation (16.4.9.1), we obtain the residual ε[Y n (x)] = ψ 0 (x, λ)+ n i=1 A i ψ i (x, λ), (16.4.9.3) where ψ 0 (x, λ)andψ i (x, λ)(i = 1, , n)aredefined by formulas (16.4.8.7). According to the method of least squares, the coefficients A i (i = 1, , n) can be found from the condition for the minimum of the integral I = b a {ε[Y n (x)]} 2 dx = b a ψ 0 (x, λ)+ n i=1 A i ψ i (x, λ) 2 dx.(16.4.9.4) . method of solution, the following technical difficulties can occur: (a) in the calculation of the transform for a given kernel K(x )and (b) in the calculation of the solution for the known transform. tables of direct and inverse Mellin transforms are applied (e.g., see Sections T3.5 and T3.6). In many cases, the relationship between the Mellin transform and the Fourier and Laplace transforms. F{f(e x ), is} = L{f(e x ), –s} + L{f (e –x ), s},(16.4.6.22) and then tables of direct and inverse Fourier transforms and Laplace transforms are applied (see Sections T3.1–T3.4). 16.4.6-3. Equation