A general framework for solving differential equations

Abstract

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along a suitable orthonormal basis. Interestingly, this approach can be extended to cope with more general differential problems. In this paper we sketch this fact, by considering some relevant examples.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

“…with prescribed a j , b j , d j , j ≥ 0. In fact, by setting φ(c) = c, and using the scalar product (9), to enforce (7), for P 0 , . .…”

“…In this paper, we consider a major improvement of the recent solution approach described in [2,9], based on previous work on Hamiltonian Boundary value Methods (HBVMs) [7,8,11,14] (also used as spectral methods in time [3,5,12,13]), for solving fractional initial value problems in the form:…”

Recent Advances in the Numerical Solution of the Nonlinear Schrödinger Equation

“…We refer to [68,71] (and, in particular, to the monograph [67] 1 ) for the derivation and analysis of such methods, which have been devised within the framework of the so called line integral methods (see, e.g., the review paper [72]). We mention that generalizations and extensions of such approach have been also considered in [73][74][75][76][77][78][79][80][81][82][83][84][85]. HBVMs have been also considered for the efficient numerical solution of a number of Hamiltonian PDEs (see, e.g., [86][87][88][89][90]), among which the NLSE [91] and Manakov systems [92].…”