An inverse convergence approach for arguments of Jacobian elliptic functions

Computing the value of the Jacobian elliptic functions, given the argument u and the parameter m, is a problem, whose solution can be found either tab ulated in tables of elliptic functions [1] or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the:Jacobian elliptic function and the parameter tn, is a problem whose sol ution is found only in tables of elliptic functions. Standard polynomial in verse interpolation procedures fail, due to ill conditioning of the system of the unknowns. In this paper, we describe a numerical procedure based on the convergence of the unknowns of the solution, by the use of arithmetical method, as an alternative way of solving the problem. The method gives ver y good results with no significant error, in the computation of the argumen t of the Jacobian elliptic function given the Jacobian elliptic function an d the parameter. This new procedure is important in problems involving cavi ties or inclusions of ellipsoidal shape encountered in the mechanical desig n of bearings, filters, and composite materials. They are also important in the modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone mineral density in humans with bad consequenc es. Also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic held for large area Josephson junct ions with overlap boundary conditions. (C) 1999 Elsevier Science Ltd. All r ights reserved.