Solving Singular Two-Point Boundary Value Problems Using Continuous Genetic Algorithm

and Applied Analysis 3 and engineering community. CGA is well suited for a broad range of problems encountered in science and engineering 14–22 . CGA was developed by the second author 14 as an efficient method for the solution of optimization problems in which the parameters to be optimized are correlated with each other or the smoothness of the solution curve must be achieved. It has been successfully applied in the motion planning of robot manipulators, which is a highly nonlinear, coupled problem 15, 16 , in the numerical solution of regular two-point BVPs 17 , in the solution of optimal control problems 18 , in the solution of collision-free path planning problem for robot manipulators 19 , in the numerical solution of Laplace equation 20 , and in the numerical solution of nonlinear regular system of second-order BVPs 21 . Their novel development has opened the doors for wide applications of the algorithm in the fields of mathematics and engineering. It has been also applied in the solution of fuzzy differential equations 22 . The reader is asked to refer to 14–22 in order to know more details about CGA, including their justification for use, conditions on smoothness of the functions used in the algorithm, several advantages of CGA over conventional GA discrete version when it is applied to problems with coupled parameters and/or smooth solution curves, and so forth. The work presented in this paper is motivated by the needs for a new numerical technique for the solution of singular two-point BVPs with the following characteristics. 1 It does not require anymodificationwhile switching from the linear to the nonlinear case; as a result, it is of versatile nature. 2 This approach does not resort to more advanced mathematical tools; that is, the algorithm should be simple to understand, implement, and should be thus easily accepted in the mathematical and engineering application’s fields. 3 The algorithm is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical and engineering problems. However, being a variant of the finite difference scheme with truncation error of the order O h10 , the method provides solutions with moderate accuracy. The organization of the remainder of this paper is as follows: in the next section, we formulate the singular two-point BVPs. Section 3 covers the description of CGA in detail. Numerical results and discussion are given in Section 4. Finally, concluding remarks are presented in Section 5. 2. Formulation of the Singular Two-Point BVPs In this section, 1.1 and 1.2 are first formulated as an optimization problem based on the minimization of the cumulative residual of all unknown interior nodes. After that, a fitness function is introduced in order to convert the minimization problem into a maximization problem. To approximate the solution of 1.1 and 1.2 , we make the stipulation that the mesh points are equally distributed through the interval I. This condition is ensured by setting xi a ih, i 0, 1, . . . ,N, where h b − a /N. Thus, at the interior mesh points, xi, i 1, 2, . . . ,N − 1, the equation to be approximated is given as F ( xi, y xi , y′ xi , y′′ xi ) : y′′ xi − f ( xi, y xi , y′ xi ) 0, x1 ≤ xi ≤ xN−1, 2.1 4 Abstract and Applied Analysis subject to the boundary conditions