- Research Article
- Open Access
- Published:
Analysis and Numerical Solutions of Positive and Dead Core Solutions of Singular Sturm-Liouville Problems
Advances in Difference Equations volume 2010, Article number: 969536 (2010)
Abstract
In this paper, we investigate the singular Sturm-Liouville problem ,
,
, where
is a nonnegative parameter,
,
, and
. We discuss the existence of multiple positive solutions and show that for certain values of
, there also exist solutions that vanish on a subinterval
, the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for
and for some model problems from the class of singular differential equations
discussed in Agarwal et al. (2007). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.
1. Introduction
In the theory of diffusion and reaction (see, e.g., [1]), the reaction-diffusion phenomena are described by the equation

where . Here
is the concentration of one of the reactants and
is the Thiele modulus. In case that
is radial symmetric with respect to
, the radial solutions of the above equation satisfying the boundary conditions

are solutions to a boundary value problem of the type

where denotes the radial coordinate. Baxley and Gersdorff [2] discussed problem (1.3), where
and
were continuous and
was allowed to be unbounded for
. They proved the existence of positive solutions and dead core solutions (vanishing on a subinterval
,
) of problem (1.3), and also covered the case of the function
approximated by some regular function
.
Problem (1.3) was a motivation for discussing positive, pseudo dead core, and dead core solutions to the singular boundary value problem with a -Laplacian,


see [3]. Here is a parameter, the function
is non-negative and satisfies the Carathéodory conditions on
,
for a.e.
, and
is positive and satisfies the Carathéodory conditions on
,
. Moreover, the function
is singular at
and
is singular at
.
Let us denote by the set of functions
which are absolutely continuous on
for arbitrary small
.
A function is called a positive solution of problem (1.4a)-(1.4b) if
on
,
,
satisfies (1.4b) and (1.4a) holds for a.e.
. We say that
satisfying (1.4b) is a dead core solution of problem (1.4a)-(1.4b) if there exists a point
such that
on
,
on
,
and (1.4a) holds for a.e.
. The interval
is called the dead core of
. If
,
on
,
,
satisfies (1.4b) and (1.4a) holds a.e. on
, then
is called a pseudo dead core solution of problem (1.4a)-(1.4b).
Since problem (1.4a)-(1.4b) is singular, the existence results in [3] are proved by a combination of the method of lower and upper functions with regularization and sequential techniques. Therefore, the notion of a sequential solution of problem (1.4a)-(1.4b) was introduced. In [3], conditions on the functions , and
were specified which guarantee that for each
, problem (1.4a)-(1.4b) has a sequential solution and that any sequential solution is either a positive solution, a pseudo dead core solution, or a dead core solution. Also, it was shown that all sequential solutions of (1.4a)-(1.4b) are positive solutions for sufficiently small positive values of
and dead core solutions for sufficiently large values of
.
The differential equation (1.5a) of the following boundary value problem satisfies all conditions specified in [3]:


Here, , and
. We note that in papers [2, 3] no information on the number of positive and dead core solutions of the underlying problem is given.
In this paper, we discuss the singular boundary value problem


where is a non-negative parameter, and the function
becomes unbounded at
. Problem (1.6a)-(1.6b) is the special case of problem (1.4a)-(1.4b).
A function is a positive solution of problem (1.6a)-(1.6b) if
satisfies the boundary conditions (1.6b),
on
and (1.6a) holds for
. A function
is called a dead core solution of problem (1.6a)-(1.6b) if there exists a point
such that
for
,
,
satisfies (1.6b) and (1.6a) holds for
. The interval
is called the dead core of
. If
, then
is called a pseudo dead core solution of problem (1.6a)-(1.6b).
The aim of this paper is twofold.
-
(1)
First of all, we analyze relations between the values of the parameter
and the number and types of solutions to problem (1.6a)-(1.6b), provided that
(1.7)or
(1.8) -
(2)
Moreover, we compute solutions
to the singular boundary value problem
(1.9a)

and the singular problem (1.5a), (1.9b). Note that (1.9a) is the special case of (1.6a) with satisfying (1.8).
In [4] similar questions in context of (1.6a) and the Dirichlet boundary conditions ,
have been discussed. For further results on existence of positive and dead core solutions to differential equations of the types
and
, we refer the reader to [5–9]. The Dirichlet conditions have been discussed in [5–7, 9], while [8] deals with the Robin conditions
,
,
,
.
We now recapitulate the main analytical results formulated in Theorems 2.10, 2.12, and 2.13. First, we introduce the auxiliary function

where satisfies (1.7). By Lemma 2.2, the equation
has a unique continuous solution
, and the function

is continuous on . Let
. Then the following statements hold.
-
(i)
Problem (1.6a)-(1.6b) has a positive solution if and only if
. In addition, for each
, problem (1.6a)-(1.6b) with
has a unique positive solution such that
,
.
-
(ii)
Problem (1.6a)-(1.6b) has a pseudo dead core solution if and only if
(1.12)This solution is unique.
-
(iii)
Problem (1.6a)-(1.6b) has a dead core solution if and only if
(1.13)
In addition, for all such , problem (1.6a)-(1.6b) has a unique dead core solution.
The final result concerning the multiplicity of positive solutions to problem (1.6a)-(1.6b) is given in Theorem 2.11. Let (1.8) hold and let . Then
and for each
, there exist multiple positive solutions of problem (1.6a)-(1.6b).
In Section 2 analytical results are presented. Here, we formulate the existence and uniqueness results for the solutions of the boundary value problem (1.6a)-(1.6b) and study the dependance of the solution on the parameter values . The numerical treatment of problems (1.9a)-(1.9b) and (1.5a)-(1.5b) based on the collocation method is discussed in Section 3, where for different values of
, we study positive, pseudo dead core, and dead core solutions of problem (1.9a)-(1.9b) and positive solutions of problem (1.5a)-(1.5b).
2. Analytical Results
2.1. Auxiliary Functions
Let assumption (1.7) hold, and let us introduce auxiliary functions , and
as

where ,


Here, the positive constants and
are identical with those used in boundary conditions (1.6b). Note that the function
is used in the analysis of positive and pseudo dead core solutions of problem (1.6a)-(1.6b), while the function
for its dead core solutions.
Properties of are described in the following lemma.
Lemma 2.1.
Let assumption (1.7) hold and let . Then
, and
is increasing on
.
Proof.
Let be arbitrary,
. Then
, and
is increasing on
by [4, Lemma
(where
is replaced by
)]. Since
is arbitrary, the result immediately follows.
In the following lemma, we introduce functions and
and discuss their properties.
Lemma 2.2.
Let assumption (1.7) hold. Then the following statements follow.
(i)The function is continuous on
, and
for
.
(ii)For each , there exists a unique
such that

and ,
for
,
.
(iii)The function

is continuous on .
Proof.
-
(i)
Let us define
on
by
(2.6)
Then . Let
and define
. Then, by (1.7),
. Hence

and consequently , which means that
is continuous at
. Let
. We now show that
is continuous at the point
. Let us choose an arbitrary
in the interval
. Then
for
and
, where

Since by [4, Lemma
(where 1 was replaced by
)], it follows that
is continuous at
. The continuity of
at
now follows from the fact that
is continuous at this point. Hence
is continuous on
, and from
we conclude
. Since

we have for
.
-
(ii)
Consider the equation
, that is,
(2.10)
The function is increasing on
,
, and, for
,
. Hence, for each
, there exists a unique
such that
and
. Clearly,
for
. In order to prove that
, suppose the contrary, that is, suppose that
is discontinuous at a point
,
. Then there exist sequences
in
such that
, and the sequences
are convergent,
,
,
. Let
in
and in
. This means
,
, and
by the definition of the function
, which contradicts
.
-
(iii)
By (ii),
(2.11)
and
for
. Hence, the function
is continuous on
and positive on
. From

we now deduce that . Since

where , and
on
,
, we conclude
. Hence
is continuous at
, and consequently
.
Let be the function from Lemma 2.2(ii) defined on the interval
. From now on,
denotes the value of
at
, that is,

In the following lemma, we prove a property of which is crucial for discussing multiple positive solutions of problem (1.6a)-(1.6b).
Lemma 2.3.
Let assumption (1.8) hold and let the function be given by (2.5). Then there exists
such that

Proof.
Note that d
. We deduce from [4, Lemma
(with 1 replaced by
)] that there exists an
such that

If for some
, then (2.16) yields

Consequently, inequality (2.15) holds for such an . If the statement of the lemma were false, then some
would exist such that
and

From the following equalities, compare (2.4),

and from , we conclude that

Finally, from

we have , which contradicts (2.18).
In order to discuss dead core solutions of problem (1.6a)-(1.6b) and their dead cores, we need to introduce two additional functions and
related to
and study their properties.
Lemma 2.4.
Assume that (1.7) holds and let be given by (2.3). Then for each
, there exists a unique
such that

The function is continuous and decreasing on
, and the function

is continuous and increasing on . Moreover,
.
Proof.
It follows from (1.7) that . Also,
is increasing w.r.t. both variables,
for any
, and
,
for any
. Hence, for each
, there exists a unique
such that
. In order to prove that
is decreasing on
, assume on the contrary that
for some
. Then
which contradicts
for
. Hence,
is decreasing on
. If
was discontinuous at a point
, then there would exist sequences
and
in
such that
and
,
are convergent,
, and
with
. Taking the limits
in
and
, we obtain
,
. Consequently,
by the definition of the function
, which is not possible.
By (2.22),

and therefore,

It follows from the properties of that the functions
,
are continuous, positive, and increasing on
. Hence (2.25) implies that
and
is increasing. Moreover,
since
d
is bounded on
.
Corollary 2.5.
Let assumption (1.7) hold. Then

and for each satisfying the inequality

there exists a unique such that

Proof.
The equalities for
and
imply that
. Since the function
defined by (2.23) is continuous and increasing on
, it follows that
for
; see (2.26). Let us choose an arbitrary
satisfying (2.27). Then
. Now, the properties of
guarantee that equation
has a unique solution
. This means that (2.28) holds for a unique
.
2.2. Dependence of Solutions on the Parameter
The following two lemmas characterize the dependence of positive and dead core solutions of problem (1.6a)-(1.6b) on the parameter .
Lemma 2.6.
Let assumption (1.7) hold and let be a positive solution of problem (1.6a)-(1.6b) for some
. Also, let
, and
. Then
,
,



where the function is given by (2.2).
Proof.
Since and
for
, we conclude that
on
and
,
. By integrating the equality
over
, we obtain

and consequently, since on
,

Finally, integrating

over yields (2.30). Now we set
in (2.30) and obtain (2.29). Equality (2.31) follows from
and from

Remark 2.7.
Let (1.7) hold and let be a pseudo dead core solution of problem (1.6a)-(1.6b). Then, by the definition of pseudo dead core solutions,
. We can proceed analogously to the proof of Lemma 2.6 in order to show that

where , and


From (2.38), we finally have . Consequently,
.
Remark 2.8.
If , then
,
is the unique solution of problem (1.6a)-(1.6b). This solution is positive.
Lemma 2.9.
Let assumption (1.7) hold and let be a dead core solution of problem (1.6a)-(1.6b) for some
. Moreover, let
. Then
and there exists a point
such that
for
,



where the function is given by (2.3). Furthermore,
is the unique dead core solution of problem (1.6a)-(1.6b) with
.
Proof.
Since is a dead core solution of problem (1.6a)-(1.6b) with
, there exists by definition, a point
such that
,
for
and
on
. Consequently,
on
, and
. We can now proceed analogously to the proof of Lemma 2.6 to show that

and (2.39) holds. Setting in (2.39), we obtain (2.40). Also, from (1.6b),
,

equality (2.41) follows.
It remains to verify that is the unique dead core solution of problem (1.6a)-(1.6b) with
. Let us suppose that
is another dead core solution of the above problem. Let
for
and
on
for some
. Then
for
, and consequently
on
and
. Hence, compare (2.40) and (2.41),


Since

by (2.41), and the function is increasing and continuous on
, we deduce from (2.45) and (2.46) that
. Then (2.40) and (2.44) yield
. Therefore,
d
for
. Finally, since
for
and since by Lemma 2.1 the function
is increasing on
,
follows. This completes the proof.
2.3. Main Results
Let the function be given by (2.5) and let us denote by
the range of the function
restricted to the interval
,

Since by Lemma 2.2(iii),
for
and
, we can have either (i)
for
, or (ii)
for some
. For (i), we have
, while in case of (ii),
with

holds. Clearly, .
Positive solutions of problem (1.6a)-(1.6b) are analyzed in the following theorem.
Theorem 2.10.
Let assumption (1.7) hold. Then problem (1.6a)-(1.6b) has a positive solution if and only if . Additionally, for each
, problem (1.6a)-(1.6b) with
has a unique positive solution
such that
and
.
Proof.
Let be a positive solution of problem (1.6a)-(1.6b) for
. By Lemma 2.6, (2.31) holds with
and
. Furthermore, by Lemmas 2.2(ii) and 2.6,
, which together with (2.29) implies that
. Consequently,
. For
, problem (1.6a)-(1.6b) has the unique positive solution
; see Remark 2.8. Since
,
. Consequently, if problem (1.6a)-(1.6b) has a positive solution, then
.
We now show that for each , problem (1.6a)-(1.6b) has a positive solution, and if
for some
, then problem (1.6a)-(1.6b) has a unique positive solution
such that
and
. Let us choose
. Then
for some
. If
, then
. Consequently,
and
is the unique solution of problem (1.6a)-(1.6b). Clearly,
and
since
. Let us suppose that
. If
is a positive solution of problem (1.6a)-(1.6b) and
, then, by Lemma 2.6; see (2.30), the equality
holds for
, where
is given by (2.1). Hence, in order to prove that for
problem (1.6a)-(1.6b) has a unique positive solution
such that
and
, we have to show that the equation

has a unique solution ; this solution is a positive solution of problem (1.6a)-(1.6b), and
,
. Since
,
is increasing by Lemma 2.1, and
, (2.49) has a unique solution
. It follows from
and
that
and
. In addition,

Hence, and
. In order to show that
is continuous at
, we set
. Then, compare (2.49),

and therefore,

Consequently, , and so
is continuous at
, or equivalently,
. Now (2.50) indicates that
and

Moreover, by the de L'Hospital rule,

As a result and
for
. Since
and, by (2.50),
, we have

by Lemma 2.2(ii). Thus, satisfies (1.6b), and therefore
is a unique positive solution of problem (1.6a)-(1.6b) such that
and
.
The following theorem deals with multiple positive solutions of problem (1.6a)-(1.6b).
Theorem 2.11.
Let assumption (1.8) hold. Then , with
given by (2.48), and for each
, there exist multiple positive solutions of problem (1.6a)-(1.6b).
Proof.
By Lemmas 2.2(iii) and 2.3, ,
, and
in a right neighbourhood of
. Hence,
. Let us choose
. Then there exist
such that
for
. Now Theorem 2.10 guarantees that problem (1.6a)-(1.6b) has positive solutions
and
such that
,
. Since
, we have
and therefore, for each
, problem (1.6a)-(1.6b) has multiple positive solutions.
Next, we present results for pseudo dead core solutions of problem (1.6a)-(1.6b). Note that here .
Theorem 2.12.
Let assumption (1.7) hold. Then problem (1.6a)-(1.6b) has a pseudo dead core solution if and only if

Moreover, for given by (2.56), problem (1.6a)-(1.6b) has a unique pseudo dead core solution such that
.
Proof.
Let us assume that is a pseudo dead core solution of problem (1.6a)-(1.6b) and let
. Then, by Remark 2.7, equalities (2.36), (2.38) hold, and
. Also, (2.37) implies that
is a solution of the equation

where and
are given by (2.1) and (2.56), respectively. The result follows by showing that equation (2.57) has a unique solution and that this solution is a pseudo dead core solution of problem (1.6a)-(1.6b). We verify these facts for solutions of (2.57) arguing as in the proof of Theorem 2.10, with
replaced by 0.
In the final theorem below, we deal with dead core solutions of problem (1.6a)-(1.6b).
Theorem 2.13.
Let assumption (1.7) hold and let be the function defined in Lemma 2.4. Then the following statements hold.
-
(i)
Problem (1.6a)-(1.6b) has a dead core solution if and only if
(2.58) -
(ii)
For each
satisfying (2.58), problem (1.6a)-(1.6b) has a unique dead core solution.
-
(iii)
If the subinterval
is the dead core of a dead core solution
of problem (1.6a)-(1.6b), then
and
(2.59)
Proof.
-
(i)
Let
be a dead core solution of problem (1.6a)-(1.6b) for some
and let
. Then there exists a point
such that
for
, and equalities (2.39), (2.40), and (2.41) are satisfied by Lemma 2.9. We deduce from (2.41) and from Lemma 2.4 that
. Therefore, compare (2.40),
(2.60)
Since

by Corollary 2.5, we have

Hence, if problem (1.6a)-(1.6b) has a dead core solution, then satisfies inequality (2.58).
We now prove that for each satisfying (2.58), problem (1.6a)-(1.6b) has a dead core solution. Let us choose
satisfying (2.58). Then, by Corollary 2.5, there exists a unique
such that

Let us now consider, compare (2.39),

where is given by (2.1). Since
and
is increasing on
by Lemma 2.1,
, and, by (2.63),
, there exists a unique solution
of (2.64) and
,
. In addition,

and consequently, and
. Since

by the Mean Value Theorem for integrals, where , we have

Therefore,

since . Hence,
is continuous at
, and
. Furthermore,

Let

Then ,
for
,
,
, and

Thus,

where is given by (2.3). Since
by Lemma 2.4,
satisfies the boundary conditions (1.6b). Consequently,
is a dead core solution of problem (1.6a)-(1.6b).
-
(ii)
Let us choose an arbitrary
satisfying (2.58). By (i), problem (1.6a)-(1.6b) has a dead core solution which is unique by Lemma 2.9.
-
(iii)
Let the subinterval
be the dead core of a dead core solution
of problem (1.6a)-(1.6b). Then, by Lemma 2.9, equalities (2.40) and (2.41) hold with
replaced by
and
. Since
by the definition of the function
, we have
. Equality (2.59) now follows from (2.40) with
and
replaced by
and
, respectively.
Example 2.14.
We now turn to the case study of the boundary value problem (1.9a)-(1.9b),

Note that (1.9a)-(1.9b) is a special case of (1.6a)-(1.6b) with satisfying (1.8). Since

we have

for , and
for
. By Lemma 2.2, the equation
has a unique solution
for
,
,
for
, and
. Let

Then ,
, and
. In order to show that
is increasing on
it is sufficient to verify that
is injective. Let us assume that this is not the case, then there exist
,
, such that
. From
,
, or equivalently, from

it follows that , and
which is a contradiction. Hence,
is increasing on
and therefore, there exists the inverse function
mapping
onto
. Since

and for
, we have

Consequently,

In order to discuss the range of the function
and the value of
for
, we first consider properties of the function

defined on . Let

Then

where . The function
vanishes only at point

in the interval , and
, because
and
. Since
,
on
,
on
and

for , we have
on
and
on
. Let us define
. Then
, and it follows from
that
on
and
on
. Consequently,
is increasing on
and decreasing on
. It follows from the equality
for
and from the properties of the functions
and
that
is increasing on
and decreasing on
. Hence,
, where
. Also,

Using properties of the function and the results of Theorems 2.10–2.13, we can now characterize the structure of the solution
.
-
(i)
For each
, there exists only a unique dead core solution of problem (1.9a)-(1.9b).
-
(ii)
For
, there exist a unique dead core solution and a unique positive solution of problem (1.9a)-(1.9b).
-
(iii)
For each
, there exist a unique dead core solution and exactly two positive solutions of problem (1.9a)-(1.9b).
-
(iv)
For
, there exist the unique pseudo dead core solution
and a unique positive solution of problem (1.9a)-(1.9b).
-
(v)
For each
, there exist only a unique positive solution of problem (1.9a)-(1.9b).
Using Theorem 2.10, Lemma 2.6, and the properties of the function , we can specify further properties of positive solutions of problem (1.9a)-(1.9b).
-
(i)
If
is the (unique) positive solution of problem (1.9a)-(1.9b) with
, then
, where
is the root of the equation
.
-
(ii)
If
are the (unique) positive solutions of problem (1.9a)-(1.9b) with
, then
,
, where
, are the roots of the equation
.
We are also able to give some more information on the dead core solutions of problem (1.9a)-(1.9b). Since

the function ,
, is the solution of the equation
. Let us choose an arbitrary
. By Corollary 2.5, the equation; see (2.28),

has a unique solution . Consequently,

One can easily show that the function

is the unique dead core solution of problem (1.9a)-(1.9b). Additionally, it follows from Theorem 2.13(iii) that since
.
3. Numerical Treatment
We now aim at the numerical approximation to the solution of the following two-point boundary value problem:

For the numerical solution of (3.1), we are using the collocation method implemented in our Matlab code bvpsuite. It is a new version of the general purpose Matlab code sbvp, compare [10–12]. This code has already been used to treat a variety of problems relevant in application; see, for example, [13–17]. Collocation is a widely used and well-studied standard solution method for two-point boundary value problems, compare [18] and the references therein. It can also be successfully applied to boundary value problems with singularities.
In the scope of the code are systems of ordinary differential equations of arbitrary order. For simplicity of notation we present a problem of maximal order four which can be given in a fully implicit form,


In order to compute the numerical approximation, we first introduce a mesh

The approximation for is a collocation function

where we require in case that the order of the underlying differential equation is
. Here,
are polynomials of maximal degree
which satisfy the system (3.2a) at
inner collocation points

and the associated boundary conditions (3.2b).
Classical theory, compare [18], predicts that the convergence order for the global error of the method is at least , where
is the maximal stepsize,
To increase efficiency, an adaptive mesh selection strategy based on an a posteriori estimate for the global error of the collocation solution is utilized. A more detailed description of the numerical approach can be found in [4].
The code bvpsuite also allows to follow a path in the parameter-solution space. This means that in the following problem setting, parameter is unknown:


where is given. The path following strategy can also cope with turning points in the path. The theoretical justification for the path following strategy implemented in bvpsuite has been given in [19].
We first study the boundary problem (1.9a)-(1.9b). Positive solutions of problem (1.5a)-(1.5b) will be discussed in Section 3.4.
The above analytical discussion indicates that depending on the values of ,
,
, the problem has one or more positive solutions, a pseudo dead core solution or a dead core solution. All numerical approximations have been calculated on a fixed mesh with
subintervals and collocation degree
. Figure 1 shows
for our choice of parameters used in the following sections. Here,
is given by (2.81).
3.1. Positive Solutions
For ),
, there exist a unique positive solution. This solution was found numerically by using the original problem formulation (1.9a)-(1.9b). For
we obtain
. In Figure 2 we display the numerical solution, the error estimate and the residual for
. The residual
is calculated by substituting the numerical solution
into the differential equation,

Due to the very small size of the error estimate and residual, it is obvious that the numerical approximation is very accurate. According to the analytical results, a solution to the problem satisfies where
is a root of
. Here, we have
and
which again shows the high quality of the numerical solution. In Figure 3 we depict the results for the parameter
,
and
. For this choice of parameters
and
.
For there exists a unique positive solution. To compute its numerical approximation, we rewrite the problem (1.9a)-(1.9b) and consider

The numerical results related to parameter sets ,
, and
,
are shown in Figure 4 and Figure 5, respectively.
Again, the error estimate and the residual are both very small and , so
. Moreover, for the second set of parameters,
and
.
For with
there exist two positive solutions. These two different solutions for a fixed value of
can be characterized via the roots
of
for
. The choice of parameters remains the same. For
,
and
the solution corresponding to
is shown in Figure 6. The solution corresponding to
is depicted in Figure 7. Note that for these values of
and
we have
.
The first of those two solutions was found using the reformulated problem (3.8) with as the right-hand side. For the second solution it was necessary to rewrite the problem again and use

with as a free unknown parameter and
as a necessary additional boundary condition. Here,
and
. For comparison,
and
. In Figures 8 and 9, two different positive solutions for the second parameter set,
,
, and
, are shown. Note that
,
and
. For this example
and
. Here,
and
. Finally, for
, there exists a unique positive solution. In Figures 10 and 11 we display the numerical results for
,
and for
,
, respectively. In this example,
and
. Using this latter set of parameters, we obtain
and
. All positive solutions could be easily found and they all show a very satisfactory level of accuracy.
3.2. Pseudo Dead Core Solutions
In order to calculate the pseudo dead core solutions, we solved the following problem:

where the differential equation has been premultiplied by the factor . Otherwise, the problem formulated as (3.1) or (3.8), would have not been well defined at all points
such that
. In Figures 12 and 13, we report on the pseudo dead core solutions for

In this case, the analytical unique pseudo dead core solution is known,

Therefore, the exact global error is accessible. In Table 1, we show the values for the global error, where
is the numerical solution at
.
3.3. Dead Core Solutions
We now deal with the dead core solutions of the problem. Note that they only occur for

Moreover, the relation between and
, where
is such that the solution vanishes on
, is given by

Also, the dead core solution is known,

For the experiments, we used and
, in order to solve the problem,

Clearly, if we approached the problem (3.16) directly, we had to use the knowledge of which is not available in general. Therefore, it is especially important to note that we were able to find the dead core solution without explicit knowledge of
by treating the problem (3.10), formulated on the whole interval
,

instead of solving (3.16). In Figures 14 and 15, we report on the numerical test runs for ,
, and two values of
,
and
, respectively. In Figures 16 and 17, analogous results for
,
, and
,
, respectively, can be found.
Table 2 contains the information on the exact global error of the numerical dead core solution. We report on its maximal value for a wide range of parameters. Obviously, dead core solutions can be found without exact use of the known solution structure, but the initial profile must be chosen carefully to guarantee the Newton iteration to convergence.
3.4. Positive Solutions of Problem (1.5a)-(1.5b)
In this section, we deal with problem (1.5a)-(1.5b). Since this problem is very involved, we decided to simulate it numerically first in order to provide some preliminary information about its solution. The numerical treatment of (1.5a)-(1.5b) turned out to be not at all straightforward, but nevertheless, for a certain choice of parameters, ,
,
and
,
, we were able to solve the problem and provide the error estimate and the residual for its approximative solution. We have applied the path following strategy implemented in bvpsuite to the boundary value problem

In Figures 19 to 28, we present numerical results for problem (3.18). The values of for which we were able to calculate the associated numerical solutions, are shown in Figure 18. According to Figure 18, we have found a turning point at
In a certain region below this value, there exist for any
two different positive solutions.
In order to start the path following procedure we set and used
as an initial profile. For each further step, we used the solution from the previous step as an initial profile. The solution corresponding to the values of
shown in Figures 19 and 20 is unique. For
we have found two different positive solutions, compare Figures 21 and 22. Also, for
, two different positive solutions exist; see Figures 23 and 24. Interestingly, solutions found in the vicinity of the turning point change rather fast, although the values of
do not; see Figures 25 to 26. Finally, in the last step of the procedure, we obtained a solution which nearly reaches a pseudo dead core solution with
.
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Acknowledgments
This work was supported by the Austrian Science Fund Project P17253 and supported by Grant no. A100190703 of the Grant Agency of the Academy of Science of the Czech Republic and by the Council of Czech Government MSM 6198959214.
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Pulverer, G., Staněk, S. & Weinmüller, E.B. Analysis and Numerical Solutions of Positive and Dead Core Solutions of Singular Sturm-Liouville Problems. Adv Differ Equ 2010, 969536 (2010). https://doi.org/10.1155/2010/969536
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DOI: https://doi.org/10.1155/2010/969536
Keywords
- Collocation Method
- Global Error
- Singular Boundary
- Initial Profile
- Unique Positive Solution