Handbook of mathematics for engineers and scienteists part 121 docx

808 INTEGRAL EQUATIONS 16.1.5. Method of Quadratures 16.1.5-1. 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], (16.1.5.1) 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 (16.1.5.1). As a rule, A i ≥ 0 and n  i=1 A i = b – a. There are quite a few quadrature formulas of the form (16.1.5.1). 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, 2, , n). (16.1.5.2) 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, 2, , n). (16.1.5.3) Simpson’s rule (or prismoidal 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), (16.1.5.4) where m is a positive integer. In formulas (16.1.5.2)–(16.1.5.4), h is a constant integration step. The quadrature formulas due to Chebyshev and Gauss with various numbers of inter- polation nodes are also widely applied. Let us illustrate these formulas by an example. Example. For the interval [–1, 1], the parameters in formula (16.1.5.1) 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. (16.1.5.5) 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. (16.1.5.6) Note that a vast literature is devoted to quadrature formulas, and the reader can find books of interest [e.g., see Bakhvalov (1973), Korn and Korn (2000)]. 16.1. LINEAR INTEGRAL EQUATIONS OF THE FIRST KIND WITH VARIABLE INTEGRATION LIMIT 809 16.1.5-2. 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,(16.1.5.7) 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 equation (16.1.5.7) 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 , equation (16.1.5.7) acquires the form  x i a K(x i , t)y(t) dt = f (x i ), i = 2, , n.(16.1.5.8) Applying the quadrature formula (16.1.5.1) to the integral in (16.1.5.8) and choosing x j (j = 1, , i) to be the 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,(16.1.5.9) where A ij are the coefficients of the quadrature formula on the interval [a, x i ]andε i [y]is the truncation error. Assume that 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,(16.1.5.10) 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 (16.1.5.10) permits one, provided that A ii K ii ≠ 0 (i = 2, , n), to succes- sively 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. 810 INTEGRAL EQUATIONS 16.1.5-3. Algorithm based on the trapezoidal rule. According to the trapezoidal rule (16.1.5.3), 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 Paragraph 16.1.5-2. The trapezoidal rule is quite simple and effective and frequently used in practice for solving integral equa- tions with variable limit of integration. 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit 16.2.1. Volterra Equations of the Second Kind 16.2.1-1. Some definitions. Equations for the resolvent. A Volterra linear integral equation of the second kind has the general form y(x)–  x a K(x, t)y(t) dt = f (x), (16.2.1.1) where y(x) is the unknown function (a ≤ x ≤ b), K(x, t) is the kernel of the integral equation, and f(x)istheright-hand side of the integral equation. The function classes to which y(x), f(x), and K(x, t) can belong are defined in Paragraph 16.1.1-1. In these function classes, there exists a unique solution of the Volterra integral equation of the second kind. Equation (16.2.1.1) is said to be homogeneous if f (x) ≡ 0 and nonhomogeneous other- wise. The kernel K(x, t)issaidtobedegenerate 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. 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 Section 16.4). 16.2.1-2. Structure of the solution. The resolvent. The solution of equation (16.2.1.1) can be presented in the form y(x)=f(x)+  x a R(x, t)f (t) dt,(16.2.1.2) where the resolvent R(x, t) is independent of f (x) and the lower limit of integration a;itis determined by the kernel of the integral equation alone. 16.2. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH VARIABLE INTEGRATION LIMIT 811 16.2.1-3. 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 resol- vent 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)andK 2 (x, t) are given and that resolvents R 1 (x, t)andR 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 (16.2.1.3) 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.(16.2.1.4) Note that in formulas (16.2.1.3) and (16.2.1.4), the integration is performed with respect to different pairs of variables. 16.2.2. Equations with Degenerate Kernel: K(x, t) = g 1 (x)h 1 (t) + ···+ g n (x)h n (t) 16.2.2-1. Equations with kernel of the form K(x, t)=ϕ(x)+ψ(x)(x – t). 1 ◦ . The solution of a Volterra equation (see Paragraph 16.2.1-1) with kernel of this type can be expressed by the formula y = w  xx , 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), with the initial conditions w(a)=w  x (a)=0. 2 ◦ . For a degenerate kernel of the above form, the resolvent can be defined by the formula R(x, t)=u  xx ,(16.2.2.1) 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 (16.2.2.2) with the following initial conditions at x = t: u   x=t = 0, u  x   x=t = 1.(16.2.2.3) The parameter t occurs only in the initial conditions (16.2.2.3), and equation (16.2.2.2) itself is independent of t. 812 INTEGRAL EQUATIONS 16.2.2-2. 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 ,(16.2.2.4) 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 (16.2.2.5) with the following initial conditions at t = x: v   t=x = 0, v  t   t=x = 1.(16.2.2.6) The point x occurs only in the initial data (16.2.2.6) as a parameter, and equation (16.2.2.5) itself is independent of x. 16.2.2-3. 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). (16.2.2.7) Let us introduce the notation w j (x)=  x a h j (t)y(t) dt, j = 1, , n,(16.2.2.8) and rewrite equation (16.2.2.7) as follows: y(x)= n  m=1 g m (x)w m (x)+f(x). (16.2.2.9) On differentiating the expressions (16.2.2.8) with regard to formula (16.2.2.9), 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 equa- tion (16.2.2.7) is defined by formula (16.2.2.9) or any of the expressions y(x)= w  j (x) h j (x) , j = 1, , n, which can be obtained from formula (16.2.2.8) by differentiation. 16.2. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH VARIABLE INTEGRATION LIMIT 813 16.2.3. Equations with Difference Kernel: K(x, t) = K(x – t) 16.2.3-1. Solution method based on the Laplace transform. Volterra equations of the second kind with kernel depending on the difference of the arguments have the form y(x)–  x 0 K(x – t)y(t) dt = f(x). (16.2.3.1) Applying the Laplace transform L to equation (16.2.3.1) and taking into account the fact that by the convolution theorem (see Subsection 11.2.2) the integral with kernel depending on the difference of the arguments is transformed into the product  K(p)y(p), we arrive at the following equation for the transform of the unknown function: y(p)–  K(p)y(p)=  f(p). (16.2.3.2) The solution of equation (16.2.3.2) is given by the formula y(p)=  f(p) 1 –  K(p) ,(16.2.3.3) which can be written equivalently in the form y(p)=  f(p)+  R(p)  f(p),  R(p)=  K(p) 1 –  K(p) .(16.2.3.4) On applying the Laplace inversion formula to (16.2.3.4), we obtain the solution of equation (16.2.3.1) in the form y(x)=f(x)+  x 0 R(x – t)f(t) dt, R(x)= 1 2πi  c+i∞ c–i∞  R(p)e px dp. (16.2.3.5) To calculate the corresponding integrals, tables of direct and inverse Laplace transforms can be applied(see Sections T3.1 and T3.2), and, in many cases, to find the inverse transform, methods of the theory of functions of a complex variable are applied, including formulas for the calculation of residues and the Jordan lemma (see Subsection 11.1.2). Remark. If the lower limit of the integral in the Volterra equation with kernel depending on the difference of the arguments is equal to a, then this equation can be reduced to equation (16.2.3.1) by the change of variables x = ¯x – a, t = ¯ t – a. Figure 16.1 depicts the principal scheme of solving Volterra integral equations of the second kind with difference kernel by means of the Laplace integral transform. Example. Consider the equation y(x)+A  x 0 sin  λ(x – t)  y(t) dt = f(x), (16.2.3.6) which is a special case of equation (16.2.3.1) for K(x)=–A sin(λx). We first apply the table of Laplace transforms (see Subsection T3.1.6) and obtain the transform of the kernel of the integral equation in the form  K(p)=– Aλ p 2 + λ 2 . 814 INTEGRAL EQUATIONS Solution of the equation for the transform Figure 16.1. Scheme of solving Volterra integral equations of the second kind with difference kernel by means of the Laplace integral transform; R(x) is the inverse transform of the function  R(p)=  K(p)/[1 –  K(p)]. Next, by formula (16.2.3.4) we find the transform of the resolvent:  R(p)=– Aλ p 2 + λ(A + λ) . Furthermore, applying the table of inverse Laplace transforms (see Subsection T3.2.2) we obtain the resolvent R(x)= ⎧ ⎪ ⎨ ⎪ ⎩ – Aλ k sin(kx)forλ(A + λ)>0, – Aλ k sinh(kx)forλ(A + λ)<0, where k = |λ(A + λ)| 1/2 . Moreover, in the special case λ =–A,wehaveR(x)=A 2 x. On substituting the expressions for the resolvent into formula (16.2.3.5), we find the solution of the integral equation (16.2.3.6). In particular, for λ(A + λ)>0, this solution has the form y(x)=f(x)– Aλ k  x 0 sin  k(x – t)  f(t) dt, k =  λ(A + λ). (16.2.3.7) Remark. The Laplace transform can be applied to solve systems of Volterra integral equations of the form y m (x)– n  k=1  x 0 K mk (x – t)y k (t) dt = f m (x), m = 1, , n. 16.2.3-2. Method based on the solution of an auxiliary equation. Consider the integral equation Ay(x)+B  x a K(x – t)y(t) dt = f(x). (16.2.3.8) . integrals, tables of direct and inverse Laplace transforms can be applied(see Sections T3.1 and T3.2), and, in many cases, to find the inverse transform, methods of the theory of functions of a complex. case of equation (16.2.3.1) for K(x)=–A sin(λx). We first apply the table of Laplace transforms (see Subsection T3.1.6) and obtain the transform of the kernel of the integral equation in the form  K(p)=– Aλ p 2 +. truncation error) of formula (16.1.5.1). As a rule, A i ≥ 0 and n  i=1 A i = b – a. There are quite a few quadrature formulas of the form (16.1.5.1). The following formulas are the simplest and most