SI-method for solving stiff nonlinear boundary value problems

# SI-method for solving stiff nonlinear boundary value problems

## Abstract.

In the present paper we thoroughly investigate theoretical properties of the SI-method, 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; SI-method; two point boundary value problem; stiff problems; singularly perturbed problems
###### 2010 Mathematics Subject Classification:
65L04, 65L05, 65L10, 65L20, 65L50, 65Y15

## 1. Introduction

The aim of the present paper is to provide a thorough theoretical justification of the SI-method introduced in [2] 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 [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 SI-method applied to BVP, which have been stated as propositions in [2] (without proof).

Continuing the research begun in [2], we focus on the boundary value problem

 (1) u′′(x)=N(u(x),x),
 (2) u(a)=0,u(b)=ub,a

where

 (3) N(u,x)≡N(u,x)u,N(u,x)∈C1(R×[a,b]),N′u(u,x)≥0,∀x∈[a,b],∀u∈R.

Condition (3) guarantees existence and uniqueness of the solution to BVP (1), (2) (see, [1, p. 331, Theorem 7.26]).

###### Lemma 1.

Let

 (4) N(u,x)≥0,∀u∈[0,+∞),x∈[a,b]

and condition (3) holds true. Then the solution to BVP (1), (2) is monotonically increasing on

###### Proof.

First of all, let us point out that

 (5) u′(a)≠0.

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

 (6) u(a)=u(b1)=0,x1∈(a,b1].

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) u′′(x)≥0,∀x∈[a,b].

On the other hand, in the light of (5), the positiveness of on immediately yields us

 (8) u′(0)>0.

Combining (7) and (8) we get the statement of the Lemma about monotonicity of on

Lemma 1 allows us to re-state the BVP (1), (2) in an equivalent form as a BVP with respect to inverse function

 (9) x′′(u)=−N(u,x(u))(x′(u))3,
 (10) x(0)=a,x(ub)=b,

## 2. SI-method for boundary value problems and its properties

In order to define the SI-approximation of the problem (1), (2) let us pick some and divide the intervals and into subintervals

 (11) δx={[xi−1,xi],i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯1, N1},
 x0=a,xN1=c,xi−1

and

 (12) δu={[ui−1,ui],i∈¯¯¯¯¯¯¯¯¯¯¯¯1,N2}
 u0=0,uN2=ub,ui−1

respectively.

We consider a pair of functions and satisfying the following conditions:

1. function is a solution to the equation

 (13) ~u′′(x)=α(Px(~u′(x)),Px(~u(x)),x)~u(x),x∈[a,c],~u(x)∈C1([a,c]),

where

 Px(f(x))=f(xi),x∈[xi,xi+1],∀i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N1−1,
 (14) α(Px(~u′(x)),Px(~u(x)),x)=αi(~u′(xi),~u(xi),x),x∈[xi,xi+1],∀i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N1−1,
 αi(u′,u,x)=(N′u(u,xi)u′+N′x(u,xi))(x−xi)+N(u,xi),∀i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N1−1

and satisfy the inequality

 (15) 0<~u(c)
2. function is a solution to the equation

 (16) ~x′′(u)=β(Pu(~x′(u)),Pu(~x(u)),u)(~x′(u))3,u∈[~u(c),b],~x(u)∈C1([~u(c),b]),

where

 Pu(f(u))=f(¯ui),u∈[¯ui,¯ui+1],
 (17) β(Pu(~x′(u)),Pu(~x(u)),u)=βi(~x′(¯ui),~x(¯ui),u),u∈[¯ui,¯ui+1],
 βi(x′,x,u)=−(N′u(¯ui,x)+N′x(¯ui,x)x′)(u−¯ui)−N(¯ui,x),
 (18) ¯ui={ui,ui>~u(c),~u(c)ui≤~u(c),∀i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N2−1.\lx@notefootnoteAsonecannotice,$¯ui=¯uj=~u(c),$$∀i,j∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N2−1|ui,uj≤~u(c)$
3. functions and satisfy the boundary conditions

 (19) ~u(0)=0,~x(ub)=b.

and the ”matching” conditions

 (20) ~x(~u(c))=c,~x′(~u(c))=1~u′(c).
###### Lemma 2.

Let a pair of functions, and satisfy the conditions 1, 2, 3. Then the inverse function exists on and belongs to

###### Proof.

Indeed, from (16), (17) it follows that

 (21) ~x′(u)=~u′(c) ⎷1−2(~u′(c))2(∑i|¯ui+1

The fact that is continuously differentiable on and condition (20) yield us

 ~u′(c)≠0,

which, in the light of formula (21), automatically guarantees that is monotone on This completes the proof. ∎

###### Definition 1.

Let functions satisfy conditions 1, 2, 3. Then function

 (22) u(x)={~u(x)x∈[a,c],~x−1(x)x∈[c,b]

is called an SI-approximation of the solution to BVP (1), (2).

###### Theorem 1.

Let conditions (3), (4) and

 (23) N′xu(u,x),N′uu(u,x)≥0,∀u∈[0,ub],∀x∈[a,b]

hold true. Then the SI-approximation (22) exists for arbitrary

 c∈(a,b).

In order to prove Theorem 1 we first need to prove a few auxiliary statements below.

###### Lemma 3.

Let denote the solution to equation (13), (14) subjected to initial conditions

 (24) ~uν(0)=0,~u′ν(0)=ν>0

and let conditions (3), (4) and (23) hold true. Then

 (25) ~uν(x)>~u¯ν(x),
 (26) ~u′ν(x)>~u′¯ν(x),

provided that

 (27) ν>¯ν.
###### Proof.

Let denote the solution of equation (13), (14) subjected to initial conditions

 (28) ~u(xi)=μ,~u′(xi)=ν,μ≥0,ν>0,∀i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N1−1

so that

 ~uν(x)≡~uν,0,0(x).

Let us fix some arbitrary

 j∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N1−1

and assume that

 (29) ~u′νj,μj,j(xj),~u′¯νj,¯μj,j(xj)>0,~u′νj,μj,j(xj)>~u′¯νj,¯μj,j(xj),
 (30) ~uνj,μj,j(xj),~u¯νj,¯μj,j(xj)≥0,~uνj,μj,j(xj)≥~u¯νj,¯μj,j(xj).

Under the conditions of Lemma and assumptions (29), (30) we are going to prove that

 (31) ~uνj,μj,j(x)>~u¯νj,¯μj,j(x),∀x∈(xj,xj+1],
 (32) ~u′νj,μj,j(x)>~u′¯νj,¯μj,j(x),∀x∈(xj,xj+1].

By definition, functions satisfy equations

 (33) ~u′′νj,μj,j(x)−αj(νj,μj,x)~uνj,μj,j(x)=0,∀x∈[xj,xj+1],
 (34) ~u′′¯νj,¯μj,j(x)−αj(¯νj,¯μj,x)~u¯νj,¯μj,j(x)=0,∀x∈[xj,xj+1]

respectively. It is easy to verify, that under conditions (3), (4), (23) the inequality

 (35) αj(νj,μj,x)≥αj(¯νj,¯μj,x)≥0,∀x∈(xj,xj+1]

holds true.

Subtracting (34) from (33) and using inequalities (35), we get the estimate

 (36) w′′(x)−αj(νj,μj,x)w(x)≥0,w(x)=~uνj,μj,j(x)−~u¯νj,¯μj,j(x),∀x∈[xj,xj+1].

From (29) and (30) it follows that

 w(xj)≥0,w′(xj)>0,

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

 w′(x)>0,∀x∈[xj,xj+1].

The latter automatically implies inequalities (31) (32).

By now we proved that if conditions (29), (30) hold true for some then they are also fulfilled for with

 νj+1=~u′νj,μj,j(xj+1),μj+1=~uνj,μj,j(xj+1)
 ¯νj+1=~u′¯νj,¯μj,j(xj+1),¯μj+1=~u¯νj,¯μj,j(xj+1).

Under the assumptions of the Lemma, inequality (27) implies conditions (29), (30) for with

 ν0=ν,μ0=0,
 ¯ν0=¯ν,¯μ0=0

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) ~uν∗(c)=ub.
###### Proof.

From conditions (3), (4) and (23) and the maximum principle it follows that

 ~uν(x)>ν(x−a),∀x∈[a,c],∀ν>0.

The latter yields us the inequality

 ~uν(c)>ub

provided that

 ν≥ubc−a,

which, in conjunction with the obvious equality

 ~u0(c)=0,

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.

Let the conditions of Lemma 3 hold true and let denote the solution to equation (16), (17) subjected to initial conditions

 (38) ~xν(~uν(c))=c,~x′ν(~uν(c))=1~u′ν(c),ν∈[0,ν∗]

where was introduced in Lemma 5. Then is a continuous function of and

 (39) limν↑ν∗ϕ(ν)=c.

Additionally to that, there exists such that

 (40) ϕ(ν∗)>b.
###### 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) ~xν(u)=~xν,i(u)=u∫¯ui~xν,i−1(¯ui)dη√1−2(~xν,i−1(¯ui))2η∫¯uiβi(~x′ν,i−1(¯ui),~xν,i−1(¯ui),ξ)dξ+~xν,i−1(¯ui),

where

 (42) ¯u0=~uν(c),~xν,−1(¯u0)=c,~x′ν,−1(¯u0)=1~u′ν(c).

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

 ~xν,i(¯ui+1) = ϕi(~x′ν,i−1(¯ui),~xν,i−1(¯ui),¯ui), (43) ~x′ν,i(¯ui+1) = ψi(~x′ν,i−1(¯ui),~xν,i−1(¯ui),¯ui), ¯ui = ¯ui(¯ui−1),

where

 (44) ϕi(x′,x,u)=¯ui+1∫ux′dη  ⎷1−2(x′)2η∫uβi(x′,x,ξ)dξ+x,u≤¯ui+1,
 (45) ψi(x′,x,u)=x′√1−2(x′)2¯ui+1∫uβi(x′,x,ξ)dξ,u≤¯ui+1,
 (46) ¯ui(u)={ui,u
 i∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯0,N2−1.

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

 ¯uN2−1≤~uν(c)<¯uN2=ub.

This allows us to reduce the limit in the left hand side of (39) to the following form

 limν↑ν∗~xν(ub)=c+limν↑ν∗ub∫~uν(c)dη√(~u′ν(c))2−2η∫~uν(c)βN2−1(1/~u′ν(c),c,~uν(c))dξ.

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).

 ~xν(u)−c=u∫~uν(c)⎡⎢ ⎢⎣(~u′ν(c))2−2η∫~uν(c)β(Pu(~x′η(ξ)),Pu(~xν(ξ)),ξ)dξ⎤⎥ ⎥⎦−12dη
 ≥u∫~uν(c)⎡⎢ ⎢⎣(~u′ν(c))2+2η∫~uν(c)(Tξ+M(ξ−~uν(c)))dξ⎤⎥ ⎥⎦−12dη
 =u∫~uν(c)[(~u′ν(c))2+T(η+~uν(c))(η−~uν(c))+M(η−~uν(c))2]−12dη
 ≥u∫~uν(c)⎡⎣(~u′ν(c))2+(T2√M(η+~uν(c))+√M(η−~uν(c)))2⎤⎦−12dη
 =1Qln(Qη+R~uν(c)+√(~u′ν(c))2+(Qη+R~uν(c))2)∣∣∣η=uη=~uν(c)
 =1Qln⎛⎜ ⎜ ⎜ ⎜⎝Qu+R~uν(c)+√(~u′ν(c))2+(Qu+R~uν(c))2T√M~uν(c)+√(~u′ν(c))2+(T√M~uν(c))2⎞⎟ ⎟ ⎟ ⎟⎠
 ≥1Qln⎛⎜ ⎜ ⎜ ⎜⎝T√MuT√M~uν(c)+√(~u′ν(c))2+(T√M~uν(c))2⎞⎟ ⎟ ⎟ ⎟⎠
 (47) ≥1Qln⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝T√Mu~u′ν(c)(T√M(c−a)+√1+(T√M(c−a))2)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

where

 T=max{N(u,x)|x∈[c,b],u∈[0,ub]},
 M=max{1,max{N′u(u,x)+|N′x|(u,x)(b−a)|x∈[c,b],u∈[0,ub]}}\lx@notefootnoteHereweusedanobviousinequality$¯ui~x′ν(¯ui)≤~xν(¯ui)−a≤b−a.$
 Q=T+2M2√M,R=T−2M2√M.

Taking into account that, according to Lemma 4, tends to as tends to from above, the estimate (47) yields us the limit equality

From the estimate (47) we get that

 ~uν(ub)>b

provided that

 (48) 0≤~u′ν(c)

According to Lemma 5, the latter inequality has nonempty set of solutions with respect to any of them can be taken for the mentioned in the Lemma. This concludes the proof. ∎

###### Proof of Theorem 1.

The statement of Theorem 1 immediately follows from Lemma 6 and the Bolzano’s intermediate value theorem being applied to function

 f(ν)=~xν(ub)−b,ν∈[0,ν∗],

where is defined in Lemma 5. ∎

###### Remark 1.

In scope of Lemma it was proved that if functions and satisfy conditions (13), (14), (15), (16), (17), (18), (19), (20), then is bounded from below (see inequality (48)) by a constant depending on the function and values

It is also not difficult to prove a similar estimate for from above as stated in the following lemma

###### Lemma 7.

Let functions and satisfy conditions (13), (14), (15), (16), (17), (18), (19), (20), then is bounded from above

 (49) ~u′(c)≤~u′∗\lx@stackreldef=ubb−c.

## 3. Error analysis of the SI-method applied to BVP

###### Lemma 8.

Let conditions (3), (4) hold true and function satisfies equation (1) and initial conditions

 ^u(a)=0,^u′(a)>0.

Then

 ^u′(x)>0∀x∈[a,a+δ].
###### Proof.

The statement can be proved using reasoning similar to that used to prove Lemma 1. ∎

###### Lemma 9.

Let conditions (3), (4) hold true and function satisfies equation (9) and initial conditions

 ˇx(ub)=b,ˇx′(ub)>0.

Then

 ˇx′(u)>0,∀u∈[ub−δ