SI-method for solving stiff nonlinear boundary value problems
In the present paper we thoroughly investigate theoretical properties of the SI-method, which was firstly introduced in  and proved to be remarkably stable when applied to a certain class of stiff boundary value problems. In particular, we provide sufficient conditions for the method to be applicable to the given two point boundary value problem for the second order differential equation as well as the corresponding error estimates. The implementation details of the method are addressed.
Key words and phrases:Ordinary differential equation; SI-method; two point boundary value problem; stiff problems; singularly perturbed problems
2010 Mathematics Subject Classification:65L04, 65L05, 65L10, 65L20, 65L50, 65Y15
The aim of the present paper is to provide a thorough theoretical justification of the SI-method introduced in  specifically for the case of boundary value problems. In what follows, we give a slightly different view on the SI-method as compared to that from  and provide sufficient conditions that guarantee the method’s applicability (existence of the method’s approximation) to the given two point boundary value problem for the second order ordinary differential equation (ODE). Using results from  we prove error estimates for the SI-method applied to BVP, which have been stated as propositions in  (without proof).
Continuing the research begun in , we focus on the boundary value problem
First of all, let us point out that
Otherwise, according to the the Pickard-Lindelof Theorem (see, for example, [1, p.350]), whose conditions are fulfilled, must totally coincide with on which contradicts the condition
Second, let us prove that Assume that the latter is not true and there exists at least one point such that This immediately implies the existence of a point such that
Obviously, function also satisfies equation (1) which, in conjunction with condition (3), allows us to apply the result of Theorem 21 from [3, p. 48] (the maximum principle) and prove that neither nor can achieve positive maximum on and, hence, The latter means that which contradicts to (5)!?
The fact that is positive on together with condition (4) means that
On the other hand, in the light of (5), the positiveness of on immediately yields us
2. SI-method for boundary value problems and its properties
We consider a pair of functions and satisfying the following conditions:
function is a solution to the equation
and satisfy the inequality
function is a solution to the equation
functions and satisfy the boundary conditions
and the ”matching” conditions
In order to prove Theorem 1 we first need to prove a few auxiliary statements below.
Let us fix some arbitrary
and assume that
By definition, functions satisfy equations
which, in conjunction with the maximum principle (which is applicable to as well, see, for example, [3, Theorem 3, 4, p. 6–7])), yields us a fact that
and the rest obviously follows from what was proved above and the principle of mathematical induction. This completes the proof. ∎
Let the conditions of Lemma 3 hold true. Then the functions and as functions of parameter are continuous on
The statement of the Lemma almost immediately follows from the corresponding theorem about continuity of solutions of IVPs with respect to initial conditions and parameters (see, for example, [1, Theorem 8.40, p 372]). ∎
Let the conditions of Lemma 3 hold true. Then there exists a unique value such that
The latter yields us the inequality
which, in conjunction with the obvious equality
We start by proving that the function is continuous on
It is easy to see that on each interval function can be expressed in a recursive way
According to the definition of given in (18), some intervals have zero measure, containing a single point This, however, does not affect correctness of the reasoning below.
From (41) it follows that
It is easy to see that the functions (44), (45) and (46) are all the continuous functions of their arguments, which, in conjunction with the recursive formulas (2) and initial conditions (42), implies that is continuously dependent on On the other hand, according to Lemma 4 the latter two quantities are also continuous functions of the parameter which completes the first part of the proof.
To prove equality (39) we can, without loss of generality, to assume that is so close to that
This allows us to reduce the limit in the left hand side of (39) to the following form
Apparently, the limit of the last term in the right hand side of the later equality is equal to (since tends to as tends to ), which proofs the target equality (39).
Proof of Theorem 1.
It is also not difficult to prove a similar estimate for from above as stated in the following lemma
3. Error analysis of the SI-method applied to BVP
The statement can be proved using reasoning similar to that used to prove Lemma 1. ∎