Analysis and Numerical Solutions of Positive and Dead Core Solutions of Singular SturmLiouville Problems
 Gernot Pulverer^{1},
 Svatoslav Staněk^{2}Email author and
 Ewa B. Weinmüller^{1}
DOI: 10.1155/2010/969536
© Gernot Pulverer et al. 2010
Received: 20 December 2009
Accepted: 28 April 2010
Published: 6 June 2010
Abstract
In this paper, we investigate the singular SturmLiouville 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 socalled 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
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 .
see [3]. Here is a parameter, the function is nonnegative 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 .
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.
where is a nonnegative 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).
 (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 , , , .
 (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
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 .
and , for , .
is continuous on .
 (i)Let us define on by(2.6)
 (ii)Consider the equation , that is,(2.10)
 (iii)By (ii),(2.11)
where , and on , , we conclude . Hence is continuous at , and consequently .
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.
Proof.
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.
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.
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.
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.
where the function is given by (2.2).
Proof.
Remark 2.7.
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.
where the function is given by (2.3). Furthermore, is the unique dead core solution of problem (1.6a)(1.6b) with .
Proof.
equality (2.41) follows.
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
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 .
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.
Moreover, for given by (2.56), problem (1.6a)(1.6b) has a unique pseudo dead core solution such that .
Proof.
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.
 (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)
 (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)
Hence, if problem (1.6a)(1.6b) has a dead core solution, then satisfies inequality (2.58).
 (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.
 (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).
 (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 .
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
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 wellstudied standard solution method for twopoint boundary value problems, compare [18] and the references therein. It can also be successfully applied to boundary value problems with singularities.
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].
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.
3.1. Positive Solutions
Again, the error estimate and the residual are both very small and , so . Moreover, for the second set of parameters, and .
3.2. Pseudo Dead Core Solutions
Problem (3.10): Exact global error of the pseudo dead core solution.




1  1 

5  0.5 

3.3. Dead Core Solutions
Maximum of the exact global error of the numerical dead core solution.





1  1  0.2 

1  1  0.5 

1  1  0.8 

0.5  1.5  0.2 

0.5  1.5  0.5 

0.5  1.5  0.8 

0.5  0.8  0.3 

0.5  0.8  0.5 

0.5  0.8  0.8 

0.3  5  0.2 

0.3  5  0.5 

0.3  5  0.8 

5  0.5  0.2 

5  0.5  0.5 

5  0.5  0.8 

3.4. Positive Solutions of Problem (1.5a)(1.5b)
Declarations
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.
Authors’ Affiliations
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