Abstract
We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.
Highlights
We propose a new computer assisted method for rigorously proving invertibility and bounding the norm of the inverse of operators of the form2 tF [v](t) = v(t) − v(0) − A(s)v(s)ds, B0v(0) + B1v(1), A(t), Bi ∈ Rn×n. (1)Operators of this form correspond to linear two point boundary value problems (BVPs) and arise naturally as the Frechet derivative of operators defining nonlinear BVPs
Generalizations to elliptic and parabolic PDEs are discussed in, for example, [26, 25, 28, 31]. As it pertains to two point boundary value problems, computer assisted proofs are typically based on a fixed point theorem [40, 30, 27]
The use of the BVP fundamental solution makes the method presented here applicable in a range of situations for which initial value problem (IVP) based methods are ill suited, namely when the IVP fundamental solution or its inverse has a ‘large’ norm but the corresponding BVP fundamental solution and Green’s function has a ‘moderately sized’ norm
Summary
Operators of this form correspond to linear two point boundary value problems (BVPs) and arise naturally as the Frechet derivative of operators defining nonlinear BVPs. The use of the BVP fundamental solution makes the method presented here applicable in a range of situations for which IVP based methods are ill suited, namely when the IVP fundamental solution or its inverse has a ‘large’ norm but the corresponding BVP fundamental solution and Green’s function has a ‘moderately sized’ norm.
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