Algorithm 927

Abstract

In this article we describe the code bvptwp.m, a MATLAB code for the solution of two point boundary value problems. This code is based on the well-known Fortran codes, twpbvp.f, twpbvpl.f and acdc.f, that employ a mesh selection strategy based on the estimation of the local error, and on revisions of these codes, called twpbvpc.f, twpbvplc.f and acdcc.f, that employ a mesh selection strategy based on the estimation of the local error and the estimation of two parameters which characterize the conditioning of the problem. The codes twpbvp.f/tpbvpc.f use a deferred correction scheme based on Mono-Implicit Runge-Kutta methods (MIRK); the other codes use a deferred correction scheme based on Lobatto formulas. The acdc.f/acdcc.f codes implement an automatic continuation strategy. The performance and features of the new solver are checked by performing some numerical tests to show that the new code is robust and able to solve very difficult singularly perturbed problems. The results obtained show that bvptwp.m is often able to solve problems requiring stringent accuracies and problems with very sharp changes in the solution. This code, coupled with the existing boundary value codes such as bvp4c.m, makes the MATLAB BVP section an extremely powerful one for a very wide range of problems.

Evaluation of the general three-loop vacuum Feynman integral

“…The Runge-Kutta running is performed with the 6-stage, 5th-order Cash-Karp algorithm [44] with automatic step-size adjustment. However, in some cases, the endpoint t = 1 is also a pole of one or more of the coefficients c ij and c i , even though all of the I, F , G, and H integrals are welldefined there.…”

Symplectic integration of boundary value problems

“…The AUTO software [8] is based on Gauss collocation, which is symplectic when the equations are presented in canonical variables. The two-point boundary-value codes MIRKDC [9], TWPBVP [7] and TWPBVPL [3,6] are based on non-symplectic Runge-Kutta methods. MATLABs bvp4c uses 3-stage Lobatto IIIA [15], which is not symplectic.…”

BVPs Codes for Solving Optimal Control Problems

“…The Matlab environment allows the use of two functions, named bvp4c [16] and bvp5c [17], for solving BVPs. Other interesting codes that are usable in Matlab are bvptwp [18], TOM [19], HOFiD_bvp [20] and bvpSuite2.0 [21], based on the code sbvp [22] for the solution of singular problems. The code bvpSuite2.0 could be used also for singular BVPs and differential algebraic problems of index 1.…”