Enclosing all solutions of two-point boundary value problems for ODEs

Abstract

The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering and science, including the modeling of chemical reactions, heat transfer, and diffusion, and the solution of optimal control problems. A TPBVP may have no solution, a single solution, or multiple solutions. A new strategy is presented for reliably locating all solutions of a TPBVP.The method determines narrow enclosures of all solutions that occur within a specified search interval. Key features of the method are the use of a new solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with intervalvalued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method.

Rigorous Global Optimization for Dynamic Systems Subject to Inequality Path Constraints

“…We then fixed the Taylor model order at q = 3 and considered different ITS truncation orders κ, with results shown in Table 10, again for p ≤ 4. The ranges of the κ and q values used covers values that we have found useful in various other applications 33,34,55,56,63 involving VSPODE. For each value of p, the ǫ-global optimum found is the same, within the ǫ tolerance, for all the combinations of κ and q values.…”

“…It has been shown previously how efficient constraint propagation schemes, based on hull consistency and using Taylor models, can be developed for inequality constraints, 32 bound constraints, 55 and equality constraints. 56 In the method described below, we use the procedure given by Lin and Stadtherr 32 for constraint propagation with Taylor models on an inequality constraint c(x) ≤ 0, with x ∈ X. Using this constraint propagation procedure (CPP) with a Taylor model T c of the constraint, a region X nf ⊆ X that is guaranteed not to satisfy the constraint may be identified and removed from X to obtain an updated X;…”

Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

“…One can generate a set of differential equations known as the reaction-diffusion problem. Owing to the strong nonlinearity of the reaction rate, mainly from the effect of temperature, reaction-diffusion equations are paid more attention in analyzing and designing chemical and catalytic reactors [1]. The same phenomena exist in electrochemical processes, with the add complexity of a varying potential field, and considerable research has been reviewed for electrochemical reactions occurring in the porous electrode [2].…”

B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

“…Many dierent techniques, such as decomposition method [3], homotopy perturbation technique [4], Laplace transform decomposition method [5], differential transform method [6], variational iteration method [7], initial value method [1], Adomian decomposition method [8], validating solver for parametric ordinary dierential equations (ODEs) (VSPODE) [9] have been used to solve the Troesch problem. They have also shown that the numerical results do not converge to sufcient accuracy for λ > 1 [3,4].…”