The purpose of this paper is to present a method for approximate solution of initial value problems of ordinary differential equation by the double exponential transformation. The original problem is transformed into a Volterra integral equation and it is solved via the indefinite integration formula derived by Muhammad and Mori. A remarkable advantage of the double exponential transformation technique for solving initial value problems in this method is that it is easily implemented and gives a result with high accuracy also for problems with end point singularities and for stiff problems. The high accuracy of the method proposed in this paper is confirmed by numerical examples and an exponential convergence rate exp(−cN/ log N ) is attained in almost all cases. §1. IntroductionSuppose that a first order initial value problem of the formis given. Our purpose is to solve this problem using the indefinite integration formula derived by Muhammad and Mori [2] based on the double exponential
A formula for numerical evaluation of two dimensional iterated integrals of the formis derived by means of the sinc approximation based on the double exponential transformation. The integrand f (x, y) is assumed to be analytic as a function of x on a < x < b and also of y on a' < y < b', and q(x) is assumed to be an analytic function of x on a < x < b. The order of the error of the formula derived is 0 (exp(-ßN/ log(yN) )) as a function of N = ntotai/2 where as total is the total number of function evaluations. Numerical examples also proves high efficiency of the formula. When the integrand is of a product type, i.e. f (x, y) = X(x)Y(y), we can obtain an approximate value of I by evaluating only 2 x (2N+1) values (X(x^), -N
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