850 INTEGRAL EQUATIONS This requirement leads to the algebraic system of equations ∂I ∂A j = 0, j = 1, , n,(16.4.9.5) and hence, on the basis of (16.4.9.4), by differentiating with respect to the parameters A 1 , , A n under the integral sign, we obtain 1 2 ∂I ∂A j = b a ψ j (x, λ) ψ 0 (x, λ)+ n i=1 A i ψ i (x, λ) dx = 0, j = 1, , n.(16.4.9.6) Using the notation c ij (λ)= b a ψ i (x, λ)ψ j (x, λ) dx,(16.4.9.7) we can rewrite system (16.4.9.6) in the form of the normal system of the method of least squares: c 11 (λ)A 1 + c 12 (λ)A 2 + ···+ c 1n (λ)A n =–c 10 (λ), c 21 (λ)A 1 + c 22 (λ)A 2 + ···+ c 2n (λ)A n =–c 20 (λ), ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ c n1 (λ)A 1 + c n2 (λ)A 2 + ···+ c nn (λ)A n =–c n0 (λ). (16.4.9.8) Note that if ϕ 0 (x) ≡ 0,thenψ 0 (x)=–f(x). Moreover, since c ij (λ)=c ji (λ), the matrix of system (16.4.9.8) is symmetric. 16.4.9-2. Construction of eigenfunctions. The method of least squares can also be applied for the approximate construction of char- acteristic values and eigenfunctions of the kernel K(x, s),similarlytothewayinwhichit can be done in the collocation method. Namely, by setting f (x) ≡ 0 and ϕ 0 (x) ≡ 0,which implies ψ 0 (x) ≡ 0, we determine approximate values of the characteristic values from the algebraic equation det[c ij (λ)] = 0. After this, approximate eigenfunctions can be found from the homogeneous system of the form (16.4.9.8), where, instead of λ, the corresponding approximate value is substituted. 16.4.10. Bubnov–Galerkin Method 16.4.10-1. Description of the method. Let ε[y(x)] ≡ y(x)–λ b a K(x, t)y(t) dt – f(x)=0.(16.4.10.1) Similarly to the above reasoning, we seek an approximate solution of equation (16.4.10.1) in the form of a finite sum Y n (x)=f(x)+ n i=1 A i ϕ i (x), i = 1, , n,(16.4.10.2) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 851 where the ϕ i (x)(i = 1, , n) are some given linearly independent functions (coordinate functions)andA 1 , , A n are indeterminate coefficients. On substituting the expres- sion (16.4.10.2) into the left-hand side of equation (16.4.10.1), we obtain the residual ε[Y n (x)] = n j=1 A j ϕ j (x)–λ b a K(x, t)ϕ j (t) dt – λ b a K(x, t)f(t) dt.(16.4.10.3) According to the Bubnov–Galerkin method, the coefficients A i (i = 1, , n)are defined from the condition that the residual is orthogonal to all coordinate functions ϕ 1 (x), , ϕ n (x). This gives the system of equations b a ε[Y n (x)]ϕ i (x) dx = 0, i = 1, , n, or, by virtue of (16.4.10.3), n j=1 (α ij – λβ ij )A j = λγ i , i = 1, , n,(16.4.10.4) where α ij = b a ϕ i (x)ϕ j (x) dx, β ij = b a b a K(x, t)ϕ i (x)ϕ j (t) dt dx, γ i = b a b a K(x, t)ϕ i (x)f(t) dt dx, i, j = 1, , n. If the determinant of system (16.4.10.4) D(λ)=det[α ij – λβ ij ] is nonzero, then this system uniquely determines the coefficients A 1 , , A n . In this case, formula (16.4.10.2) gives an approximate solution of the integral equation (16.4.10.1). 16.4.10-2. Characteristic values. The equation D(λ)=0 gives approximate characteristic values λ 1 , , λ n of the integral equation. On finding nonzero solutions of the homogeneous linear system n j=1 (α ij – λ k β ij ) A (k) j = 0, i = 1, , n, we can construct approximate eigenfunctions Y (k) n (x) corresponding to characteristic val- ues λ k : Y (k) n (x)= n i=1 A (k) i ϕ(x). It can be shown that the Bubnov–Galerkin method is equivalent to the replacement of the kernel K(x, t) by some degenerate kernel K (n) (x, t). Therefore, for the approximate solution Y n (x) we have an error estimate similar to that presented in Subsection 16.4.7-2. 852 INTEGRAL EQUATIONS Example. Let us find the first two characteristic values of the integral equation ε[y(x)] ≡ y(x)–λ 1 0 K(x, t)y(t) dt = 0, where K(x, t)= t for t ≤ x, x for t > x. (16.4.10.5) On the basis of (16.4.10.5), we have ε[y(x)] = y(x)–λ x 0 ty(t) dt + 1 x xy(t) dt . We set Y 2 (x)=A 1 x + A 2 x 2 . In this case ε[Y 2 (x)] = A 1 x + A 2 x 2 – λ 1 3 A 1 x 3 + 1 4 A 2 x 4 + x 1 2 A 1 + 1 3 A 2 – 1 2 A 1 x 3 + 1 3 A 2 x 4 = = A 1 1 – 1 2 λ x + 1 6 λx 3 + A 2 – 1 3 λx + x 2 + 1 12 λx 4 . On orthogonalizing the residual ε[Y 2 (x)], we obtain the system 1 0 ε[Y 2 (x)]xdx= 0, 1 0 ε[Y 2 (x)]x 2 dx = 0, or the following homogeneous system of two algebraic equations with two unknowns: A 1 (120 – 48λ)+A 2 (90 – 35λ)=0, A 1 (630 – 245λ)+A 2 (504 – 180λ)=0. (16.4.10.6) On equating the determinant of system (16.4.10.6) with zero, we obtain the following equation for the characteristic values: D(λ) ≡ 120 – 48λ 90 – 35λ 630 – 245λ 504 – 180λ = 0. Hence, λ 2 – 26.03λ + 58.15 = 0. (16.4.10.7) Equations (16.4.10.7) imply λ 1 = 2.462 and λ 2 = 23.568 For comparison we present the exact characteristic values: λ 1 = 1 4 π 2 = 2.467 and λ 2 = 9 4 π 2 = 22.206 , which can be obtained from the solution of the following boundary value problem equivalent to the original equation: y xx (x)+λy(x)=0; y(0)=0, y x (1)=0. Thus, the error of λ 1 is approximately equal to 0.2% and that of λ 2 ,to6%. 16.4.11. Quadrature Method 16.4.11-1. General scheme for Fredholm equations of the second kind. In the solution of an integral equation, the reduction to the solution of systems of algebraic equations obtained by replacing the integrals with finite sums is one of the most effective tools. The method of quadratures is related to the approximation methods. It is widespread in practice because it is rather universal with respect to the principle of constructing algo- rithms for solving both linear and nonlinear equations. 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 853 Just as in the case of Volterra equations, the method is based on a quadrature formula (see Subsection 16.1.5): b a ϕ(x) dx = n j=1 A j ϕ(x j )+ε n [ϕ], (16.4.11.1) where the x j are the nodes of the quadrature formula, A j are given coefficients that do not depend on the function ϕ(x), and ε n [ϕ] is the error of replacement of the integral by the sum (the truncation error). If in the Fredholm integral equation of the second kind y(x)–λ b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b,(16.4.11.2) we set x = x i (i = 1, , n), then we obtain the following relation that is the basic formula for the method under consideration: y(x i )–λ b a K(x i , t)y(t) dt = f(x i ), i = 1, , n.(16.4.11.3) Applying the quadrature formula (16.4.11.1) to the integral in (16.4.11.3), we arrive at the following system of equations: y(x i )–λ n j=1 A j K(x i , x j )y(x j )=f(x i )+λε n [y]. (16.4.11.4) By neglecting the small term λε n [y] in this formula, we obtain the system of linear algebraic equations for approximate values y i of the solution y(x) at the nodes x 1 , , x n : y i – λ n j=1 A j K ij y j = f i , i = 1, , n,(16.4.11.5) where K ij = K(x i , x j ), f i = f(x i ). The solution of system (16.4.11.5) gives the values y 1 , , y n , which determine an approximate solution of the integral equation (16.4.11.2) on the entire interval [a, b]by interpolation. Here for the approximate solution we can take the function obtained by linear interpolation, i.e., the function that coincides with y i at the points x i and is linear on each of the intervals [x i , x i+1 ]. Moreover, for an analytic expression of the approximate solution to the equation, a function y(x)=f(x)+λ n j=1 A j K(x, x j )y j (16.4.11.6) can be chosen, which also takes the values y 1 , , y n at the points x 1 , , x n . Example. Consider the equation y(x)– 1 2 1 0 xty(t) dt = 5 6 x. 854 INTEGRAL EQUATIONS Let us choose the nodes x 1 = 0, x 2 = 1 2 , x 3 = 1 and calculate the values of the right-hand side f(x)= 5 6 x and of the kernel K(x, t)=xt at these nodes: f(0)=0, f 1 2 = 5 12 , f(1)= 5 6 , K(0, 0)=0, K 0, 1 2 = 0, K(0, 1)=0, K 1 2 , 0 = 0, K 1 2 , 1 2 = 1 4 , K 1 2 , 1 = 1 2 , K(1, 0)=0, K 1, 1 2 = 1 2 , K(1, 1)=1. On applying Simpson’s rule (see Subsection 16.1.5) 1 0 F (x) dx ≈ 1 6 F (0)+4F 1 2 + F(1) to determine the approximate values y i (i = 1, 2, 3) of the solution y(x) at the nodes x i we obtain the system y 1 = 0, 11 12 y 2 – 1 24 y 3 = 5 12 , – 2 12 y 2 + 11 12 y 3 = 5 6 , whose solution is y 1 = 0, y 2 = 1 2 , y 3 = 1. In accordance with the expression (16.4.11.6), the approximate solution can be presented in the form y(x)= 5 6 x + 1 2 × 1 6 0 + 4× 1 2 × 1 2 x + 1×1×x = x. We can readily verify that it coincides with the exact solution. 16.4.11-2. Construction of the eigenfunctions. The method of quadratures can also be applied for solutions of homogeneous Fredholm equations of the second kind. In this case, system (16.4.11.5) becomes homogeneous (f i = 0) and has a nontrivial solution only if its determinant D(λ) is equal to zero. The algebraic equation D(λ)=0 of degree n for λ makes it possible to find the roots λ 1 , , λ n , which are approximate values of n characteristic values of the equation. The substitution of each value λ k (k = 1, , n) into (16.4.11.5) for f i ≡ 0 leads to the system of equations y (k) i – λ k n j=1 A j K ij y (k) j = 0, i = 1, , n, whose nonzero solutions y (k) i make it possible to obtain approximate expressions for the eigenfunctions of the integral equation: y k (x)= λ k n j=1 A j K(x, x j )y (k) j . If λ differs from each of the roots λ k , then the nonhomogeneous system of linear algebraic equations (16.4.11.5) has a unique solution. In the same case, the homogeneous system of equations (16.4.11.5) has only the trivial solution. 16.4.12. Systems of Fredholm Integral Equations of the Second Kind 16.4.12-1. Some remarks. A system of Fredholm integral equations of the second kind has the form y i (x)–λ n j=1 b a K ij (x, t)y j (t) dt = f i (x), a ≤ x ≤ b, i = 1, , n.(16.4.12.1) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 855 Assume that the kernels K ij (x, t) are continuous or square integrable on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} and the right-hand sides f i (x) are continuous or square integrable on [a, b]. We also assume that the functions y i (x)tobedefined are continuous or square integrable on [a, b] as well. The theory developed above for Fredholm equations of the second kind can be completely extended to such systems. In particular, it can be shown that for systems (16.4.12.1), the successive approximations converge in mean-square to the solution of the system if λ satisfies the inequality |λ| < 1 B ∗ ,(16.4.12.2) where n i=1 n j=1 b a b a |K ij (x, t)| 2 dx dt = B 2 ∗ < ∞ .(16.4.12.3) If the kernel K ij (x, t) satisfies the additional condition b a K 2 ij (x, t) dt ≤ A ij , a ≤ x ≤ b,(16.4.12.4) where A ij are some constants, then the successive approximations converge absolutely and uniformly. If all kernels K ij (x, t) are degenerate, then system (16.4.12.1) can be reduced to a linear algebraic system. It can be established that for a system of Fredholm integral equations, all Fredholm theorems are satisfied. 16.4.12-2. Method of reducing a system of equations to a single equation. System (16.4.12.1) can be transformed into a single Fredholm integral equation of the second kind. Indeed, let us introduce the functions Y (x)andF (x)on[a, nb –(n – 1)a]by setting Y (x)=y i x –(i – 1)(b – a) , F (x)=f i x –(i – 1)(b – a) , for (i – 1)b –(i – 2)a ≤ x ≤ ib –(i – 1)a. Let us define a kernel K(x, t) on the square {a ≤ x ≤ nb –(n –1)a, a ≤ t ≤ nb –(n–1)a} as follows: K(x, t)=K ij x –(i – 1)(b – a), t –(j – 1)(b – a) for (i – 1)b –(i – 2)a ≤ x ≤ ib –(i – 1)a,(j – 1)b –(j – 2)a ≤ t ≤ jb –(j – 1)a. Now system (16.4.12.1) can be rewritten as the single Fredholm equation Y (x)–λ nb–(n–1)a a K(x, t)Y (t) dt = F (x), a ≤ x ≤ nb –(n – 1)a. If the kernels K ij (x, t) are square integrable on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} and the right-hand sides f i (x) are square integrable on [a, b], then the kernel K(x, t) is square integrable on the new square S n = {a < x < nb –(n – 1)a, a < t < nb –(n – 1)a}, and the right-hand side F(x) is square integrable on [a, nb –(n – 1)a]. 856 INTEGRAL EQUATIONS If condition (16.4.12.4) is satisfied, then the kernel K(x, t) satisfies the inequality b a K 2 (x, t) dt ≤ A ∗ , a < x < nb –(n – 1)a, where A ∗ is a constant. 16.5. Nonlinear Integral Equations 16.5.1. Nonlinear Volterra and Urysohn Integral Equations 16.5.1-1. Nonlinear integral equations with variable integration limit. Nonlinear Volterra integral equations have the form x a K x, t, y(t) dt = F x, y(x) ,(16.5.1.1) where K x, t, y(t) is the kernel of the integral equation and y(x) is the unknown function (a ≤ x ≤ b). All functions in (16.5.1.1) are usually assumed to be continuous. 16.5.1-2. Nonlinear integral equations with constant integration limits. Nonlinear Urysohn integral equations have the form b a K x, t, y(t) dt = F (x, y(x)), α ≤ x ≤ β,(16.5.1.2) where K x, t, y(t) is the kernel of the integral equation and y(x) is the unknown function. Usually, all functions in (16.5.1.2) are assumed to be continuous and the case of α = a and β = b is considered. Conditions for existence and uniqueness of the solution of an Urysohn equation are discussed below in Paragraphs 16.5.3-4 and 16.5.3-5. Remark. A feature of nonlinear equations is that it frequently has several solutions. 16.5.2. Nonlinear Volterra Integral Equations 16.5.2-1. Method of integral transforms. Consider a Volterra integral equation with quadratic nonlinearity μy(x)–λ x 0 y(x – t)y(t) dt = f(x). (16.5.2.1) To solve this equation, the Laplace transform can be applied, which, with regard to the convolution theorem (see Section 11.2), leads to a quadratic equation for the transform y(p)=L{y(x)}: μy(p)–λy 2 (p)= f(p). This implies y(p)= μ μ 2 – 4λ f(p) 2λ .(16.5.2.2) The inverse Laplace transform y(x)=L –1 {y(p)} (if it exists) is a solution to equa- tion (16.5.2.1). Note that for the two different signs in formula (16.5.2.2), there are two corresponding solutions of the original equation. . continuous and the case of α = a and β = b is considered. Conditions for existence and uniqueness of the solution of an Urysohn equation are discussed below in Paragraphs 16.5.3-4 and 16.5.3-5. Remark the error of λ 1 is approximately equal to 0.2% and that of λ 2 ,to6%. 16.4.11. Quadrature Method 16.4.11-1. General scheme for Fredholm equations of the second kind. In the solution of an integral. principle of constructing algo- rithms for solving both linear and nonlinear equations. 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 853 Just as in the case of