Newton flows for elliptic functions II

Cyclic Identities Involving Jacobi Elliptic Functions. II

Abstract: Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at p equally shifted points on the real axis were recently found. These identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations. We derive four master identities, from which the identities discussed earlier are derivable ...

Trigonometric functions, elliptic functions, elliptic modular forms

A Newton-Like Method for Convex Functions

A Newton-like method for convex functions is derived. It is shown that this method can be better than the Newton method. Especially good results can be obtained if we combine these two methods. Illustrative numerical examples are given. Mathematics Subject Classification: 65H05

Altered Jacobian Newton Iterative Method for Nonlinear Elliptic Problems

We present an Altered Jacobian Newton Iterative Method for solving nonlinear elliptic problems. Effectiveness of the proposed method is demonstrated through numerical experiments. Comparison of our method with Newton Iterative Method is also presented. Convergence of the Newton Iterative Method is highly sensitive to the initialization or initial guess. Reported numerical work shows the robustn...

Heun functions versus elliptic functions

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integr...