SImethod for solving stiff nonlinear boundary value problems
Abstract.
In the present paper we thoroughly investigate theoretical properties of the SImethod, which was firstly introduced in [2] 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; SImethod; two point boundary value problem; stiff problems; singularly perturbed problems2010 Mathematics Subject Classification:
65L04, 65L05, 65L10, 65L20, 65L50, 65Y151. Introduction
The aim of the present paper is to provide a thorough theoretical justification of the SImethod introduced in [2] specifically for the case of boundary value problems. In what follows, we give a slightly different view on the SImethod as compared to that from [2] 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 [5] we prove error estimates for the SImethod applied to BVP, which have been stated as propositions in [2] (without proof).
Condition (3) guarantees existence and uniqueness of the solution to BVP (1), (2) (see, [1, p. 331, Theorem 7.26]).
Lemma 1.
Proof.
First of all, let us point out that
(5) 
Otherwise, according to the the PickardLindelof 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
(6) 
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
(7) 
On the other hand, in the light of (5), the positiveness of on immediately yields us
(8) 
Lemma 1 allows us to restate the BVP (1), (2) in an equivalent form as a BVP with respect to inverse function
(9) 
(10) 
2. SImethod for boundary value problems and its properties
In order to define the SIapproximation of the problem (1), (2) let us pick some and divide the intervals and into subintervals
(11) 
and
(12) 
respectively.
We consider a pair of functions and satisfying the following conditions:

function is a solution to the equation
(13) where
(14) and satisfy the inequality
(15) 
function is a solution to the equation
(16) where
(17) (18) 
functions and satisfy the boundary conditions
(19) and the ”matching” conditions
(20)
Lemma 2.
Proof.
Definition 1.
Theorem 1.
In order to prove Theorem 1 we first need to prove a few auxiliary statements below.
Lemma 3.
Proof.
Let us fix some arbitrary
and assume that
(29) 
(30) 
Under the conditions of Lemma and assumptions (29), (30) we are going to prove that
(31) 
(32) 
By definition, functions satisfy equations
(33) 
(34) 
respectively. It is easy to verify, that under conditions (3), (4), (23) the inequality
(35) 
holds true.
Subtracting (34) from (33) and using inequalities (35), we get the estimate
(36) 
From (29) and (30) it follows that
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
By now we proved that if conditions (29), (30) hold true for some then they are also fulfilled for with
Under the assumptions of the Lemma, inequality (27) implies conditions (29), (30) for with
and the rest obviously follows from what was proved above and the principle of mathematical induction. This completes the proof. ∎
Lemma 4.
Let the conditions of Lemma 3 hold true. Then the functions and as functions of parameter are continuous on
Proof.
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]). ∎
Lemma 5.
Let the conditions of Lemma 3 hold true. Then there exists a unique value such that
(37) 
Proof.
From conditions (3), (4) and (23) and the maximum principle it follows that
The latter yields us the inequality
provided that
which, in conjunction with the obvious equality
Lemma 4 and the Bolzano’s theorem, provides us the existence of mentioned in the Lemma. The uniqueness follows from the monotonicity properties of as a function of (Lemma 3). ∎
Lemma 6.
Proof.
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
(41) 
where
(42) 
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.
(45) 
(46) 
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).
(47) 
where
Proof of Theorem 1.
Remark 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 SImethod applied to BVP
Lemma 8.
Proof.
The statement can be proved using reasoning similar to that used to prove Lemma 1. ∎