On Taylor-series expansion methods for the second kind integral equations
چکیده
P. Huabsomboon et al. / Journal of Computational and Applied Mathematics 234 (2010) 1466–1472 1467 The first step in the Taylor expansion method presented in [1] is to write x(t) ≈ x(s)+ x(s)(t − s)+ · · · + 1 n! x(n)(s)(t − s)n. (1.2) Substituting (1.2) for x(t) in the integral in (1.1), we obtain [ 1− ∫ 1 0 k(s, t)dt ] x(s)− [∫ 1 0 k(s, t)(t − s)dt ] x(s)− · · · − [ 1 n! ∫ 1 0 k(s, t)(t − s)ndt ] x(n)(s) ≈ y(s), 0 0, then the current method converges as n → ∞. Throughout our numerical experiments, we found the method to be stable relative to the choice of n and to provide accurate approximate solutions. As an additional example, we solve Example 2 of [1] and compare the results between the two methods. Example 2.2. Consider x(s)− 2 π ∫ 1 0 k(s, t)x(t)dt = y(s), 0 ≤ s ≤ 1, where k(s, t) = [4 + (s − t)2]−1 and y(s) is chosen so that x(s) = 1 + s2 + s5 is the solution. Note that when y(s) = 1, this represents the well known Love’s equation which arises in electrostatics. Numerical results with n = 5 gave excellent approximations for x as well as for its derivatives. The numerical approximation for x(s) and its first derivative with n = 5 are shown in Tables 1 and 2, respectively. For each value of s, we recorded computational time spent to obtain the result. The average elapsed time using our technique is 0.05678 s while the average elapsed time using Ren’s technique is 0.03253 s. 1470 P. Huabsomboon et al. / Journal of Computational and Applied Mathematics 234 (2010) 1466–1472 Table 2 Numerical approximation for x(s)with n = 5 for Example 2.2. s x(s) Absolute error Exact Our approx. Ren’s approx. [1] Our approx. Ren’s approx. [1] 0.0 0.00000 0.00000 −0.02277 1.7018e−008 2.2772e−002 0.1 0.20050 0.20050 0.17962 2.2075e−008 2.0875e−002 0.2 0.40800 0.40800 0.38974 2.0652e−008 1.8261e−002 0.3 0.64050 0.64050 0.62517 5.9138e−008 1.5327e−002 0.4 0.92800 0.92800 0.91545 5.9923e−008 1.2548e−002 0.5 1.31250 1.31250 1.30197 1.4485e−007 1.0534e−002 0.6 1.84800 1.84800 1.83788 2.1443e−007 1.0118e−002 0.7 2.60050 2.60050 2.58801 1.8399e−007 1.2487e−002 0.8 3.64800 3.64800 3.62870 2.4116e−007 1.9304e−002 0.9 5.08050 5.08050 5.04769 1.7642e−007 3.2807e−002 1.0 7.00000 7.00000 6.94413 6.4720e−008 5.5867e−002 3. System of Fredholm integral equations In this section, we apply the modified Taylor-series method to a system of Fredholm integral equations of the second kind. In a recent paper [7], the technique in [1] was extended to a system of Fredholm equations. Following the paper [1], authors of the paper [7] also replace, in each of the derivative equations, x j (t) by x (r) j (s). Again the success of this technique depends heavily upon rapidly decaying kernels kij(|s − t|) for each i, j as |s − t| increases. We note that Examples 1 and 2 of [7] do not satisfy this critical condition of decaying convolution kernel. The kernels in Example 1 are of convolution type but do not decay to zero as |s− t| increases, whereas the kernels in Example 2 are not of convolution type. This is more likely the explanation of why the accuracies of approximations reported in [7] are not high despite high order derivatives taken in their computations. We will demonstrate in Examples 3.1 and 3.2 below that the exact solutions of Example 1 in [7] can be found by the current method and far more accurate solutions can be obtained for Example 2 in [7]. Consider a system of Fredholm equations of the second kind, X(s)− ∫ 1 0 K(s, t)X(t)dt = Y(s), 0 ≤ s ≤ 1, (3.1) where X(s) = [x1(s), x2(s), . . . , xm(s)] , Y(s) = [y1(s), y2(s), . . . , ym(s)] , K(s, t) = [kij(s, t)], i, j = 1, 2, . . . ,m. The ith equation of (3.1) is given by xi(s)− ∫ 1 0 m ∑ j=1 kij(s, t)xj(t)dt = yi(s), i = 1, 2, . . . ,m. (3.2) Since xj(t) ≈ xj(s)+ x′j(s)(t − s)+ · · · + 1 n! x j (s)(t − s) n (3.3) substituting (3.3) into (3.2), we obtain xi(s)− n ∑