A Study and development of numerical techniques for the solution of integral equations

Abstract

We have shown that given any function f(x), there exists a unique f?(x), as defined by the relation (2.1), which exactly coincides with f(x) at the points jh. By using the package MATHEMATICA, we can choose a k, such that kh is not a zero of the function. Also, the interpolation function has been expressed as a polynomial of degree n and two correction terms. In the particular choice of U?(kx) = cos(kx) and U?(kx) = sin(kx), the results derived in this chapter tend to the mixed trigonometric interpolation formula. Thus, we have generalized the concept of mixed interpolation (see Chakrabarti and Hamsapriye [18]), by generalizing the operator of Meyer et al. [80]. We have supported the work through several numerical examples, with different choices of U?(kx) and U?(kx). The derivation given in section 3.2 and the error analysis given in section 3.3 hold for any arbitrary U?(kx) and U?(kx), which are the two linearly independent solutions of a second-order linear ODE. In Chapter 4, we have derived the Newton-Cotes quadrature formulae, based on the newly derived generalized mixed interpolation formula. Our main aim in this chapter is to derive the various quadrature rules, which integrate exactly a linear combination of a polynomial up to a certain degree and two other functions U?(kx) and U?(kx). We have established the (n+1)-point generalized Newton-Cotes quadrature formulae, which we have called GMNCF, of the closed type. We remark that the open-type formulae can also be derived on the same lines, as done for the closed type. These formulae (both open and closed) are obtained by replacing the integrand by the mixed interpolation function of the form f?(x) = aU?(kx) + bU?(kx) + ? c?x?, based on the equally spaced grid points x? = jh. It is to be noted that the quadrature formulae derived in section 4.2 and the error analysis given in section 4.3 are independent of the choice of the functions U?(kx) and U?(kx). We have worked out three examples, with two sets of functions, which are the two linearly independent solutions of linear second-order ODEs y''(x) + k²[(kx + 1)²]y(x) = 0 and y''(x) - 2ky'(x) + 2k²y(x) = 0, and which also satisfy the requirement that lim h ? 0 = Ch², for some non-zero constant C. The tables show the validity of the theory of generalized modified quadrature formulae. In the next chapter, i.e., Chapter 5, we have discussed the derivation of various Gregory quadrature rules, based on the generalized mixed interpolation formula of Chapter 3. We have derived the generalized Gregory rules, which are based on the generalized mixed interpolation theory. The interpolation function is of the form U?(kx) + U?(kx) + ? c?x?, which clearly is a combination of a polynomial of a certain degree and two other linearly independent functions U?(kx) and U?(kx). We have made the choice of these functions based on the oscillation theory of ODEs. The generalized Gregory rules are derived by considering the well-known Euler-Maclaurin formula, in which the derivatives have been replaced by the corresponding finite difference formulae. We have derived the generalized Gregory rules associated with both the composite trapezium rule, as well as the composite Simpson’s rule. We have given the error analysis in brief, for both the classes of Gregory quadrature rules. We have also discussed how to choose the appropriate k? and k?’s, which help in controlling the error. We have worked out a few numerical examples, which show the efficiency of the generalized Gregory rules over the other known rules. In these examples, we have worked with two different pairs of the functions U?(kx) and U?(kx) and in both the cases we have obtained better results. In the particular case when U?(kx) = cos(kx) and U?(kx) = sin(kx), we retrieve the results of Bodier et al. [10,11]. Also, in the limiting case as k ? 0, it is verified that the generalized Gregory rules reduce to the corresponding classical Gregory rules. In Chapter 6, we have discussed the utility of these Gregory rules in solving integral equations of the second kind, Fredholm type, following an iterative procedure. We have presented here a numerical method for solving Fredholm integral equations of the second kind. This method is based on the generalized Gregory quadrature rules, which in turn are generalizations of the modified Gregory quadrature rules based on the mixed-trigonometric interpolation formula. The iterative methods are explained and the iterative method II is a modified version of the Fox and Goodwin iterative approach. The condition for the convergence of the iterative scheme is also discussed. The truncation error involved in the approximate solution of the integral equation is also explained. A method of choosing values for the free parameters k? and k? (originally present in the generalized mixed interpolation formula) has also been explained. Several numerical examples are studied, which exhibit both the advantages as well as the disadvantages of the numerical method. Though the method is a little more time-consuming, at times it may give better results. In Chapter 7, we have taken up the study of a singular integro-differential equation, which is of practical interest. We have discussed four different methods for handling the equation.