A symbolic algorithm for solving linear two-point boundary value problems by modified Picard technique

In this paper, we consider linear two-point boundary value problems with constant coefficients. We introduce a relatively new algorithm as well as a symbolic implementation of the algorithm. We generate a rapidly convergent series solution by using a Gauss Seidel version of the Picard method for solving initial value problems introduced by the authors, [I.K. Youssef, H.A. El-Arabawy, Picard iteration algorithm combined with Gauss Seidel technique for initial value problems, Applied Mathematics Computation 190 (2007) 345–355]. We replace the boundary condition by an initial one with an unknown value “γγ value”. Unlike the well known shooting technique we determine this γγ value through an algebraic process and simultaneously we find the new modified Picard solution. Examples and different comparison techniques with other methods have illustrated the efficiency as well as the accuracy of the proposed technique. We have implemented our treatment as a simple Mathematica function using a computer algebra software “Mathematica 6.0”. Other extensions and applications for further work are mentioned.