Analysis and Numerical Solutions of Positive and Dead Core Solutions of Singular Sturm-Liouville Problems

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

(1.1)

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

(1.2)

are solutions to a boundary value problem of the type

(1.3)

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,

(1.4a)

(1.4b)

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]:

(1.5a)

(1.5b)

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

(1.6a)

(1.6b)

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)

(1.9b)

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

(1.10)

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

(1.11)

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

(2.1)

where ,

(2.2)

(2.3)

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

(2.4)

and , for , .

(iii)The function

(2.5)

is continuous on .

Proof.

• (i) Let us define on by

  

  (2.6)

Then . Let and define . Then, by (1.7), . Hence

(2.7)

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

(2.8)

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

(2.9)

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

(2.12)

we now deduce that . Since

(2.13)

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,

(2.14)

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

(2.15)

Proof.

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

(2.16)

If for some , then (2.16) yields

(2.17)

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

(2.18)

From the following equalities, compare (2.4),

(2.19)

and from , we conclude that

(2.20)

Finally, from

(2.21)

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

(2.22)

The function is continuous and decreasing on , and the function

(2.23)

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),

(2.24)

and therefore,

(2.25)

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

(2.26)

and for each satisfying the inequality

(2.27)

there exists a unique such that

(2.28)

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 , ,

(2.29)

(2.30)

(2.31)

where the function is given by (2.2).

Proof.

Since and for , we conclude that on and , . By integrating the equality over , we obtain

(2.32)

and consequently, since on ,

(2.33)

Finally, integrating

(2.34)

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

(2.35)

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

(2.36)

where , and

(2.37)

(2.38)

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 ,

(2.39)

(2.40)

(2.41)

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

(2.42)

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

(2.43)

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),

(2.44)

(2.45)

Since

(2.46)

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 ,

(2.47)

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

(2.48)

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

(2.49)

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,

(2.50)

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

(2.51)

and therefore,

(2.52)

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

(2.53)

Moreover, by the de L'Hospital rule,

(2.54)

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

(2.55)

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

(2.56)

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

(2.57)

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

(2.61)

by Corollary 2.5, we have

(2.62)

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

(2.63)

Let us now consider, compare (2.39),

(2.64)

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,

(2.65)

and consequently, and . Since

(2.66)

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

(2.67)

Therefore,

(2.68)

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

(2.69)

Let

(2.70)

Then , for , , , and

(2.71)

Thus,

(2.72)

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),

(2.73)

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

(2.74)

we have

(2.75)

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

(2.76)

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

(2.77)

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

(2.78)

and for , we have

(2.79)

Consequently,

(2.80)

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

(2.81)

defined on . Let

(2.82)

Then

(2.83)

where . The function vanishes only at point

(2.84)

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

(2.85)

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,

(2.86)

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

(2.87)

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

(2.88)

has a unique solution . Consequently,

(2.89)

One can easily show that the function

(2.90)

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:

(3.1)

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,

(3.2a)

(3.2b)

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

(3.3)

The approximation for is a collocation function

(3.4)

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

(3.5)

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:

(3.6a)

(3.6b)

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).

Figure 1

for (a) and for , (b).

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,

(3.7)

Figure 2

Problem (1. 9a)-(1.9b): The numerical solution, the error estimate, and the residual for , and .

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 .

Figure 3

Problem (1. 9a)-(1.9b): The numerical solution, the error estimate, and the residual for , and .

For there exists a unique positive solution. To compute its numerical approximation, we rewrite the problem (1.9a)-(1.9b) and consider

(3.8)

The numerical results related to parameter sets , , and , are shown in Figure 4 and Figure 5, respectively.

Figure 4

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and .

Figure 5

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and .

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 .

Figure 6

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and . The associated root is .

Figure 7

Problem (3. 9): The numerical solution, the error estimate, and the residual for , and . The associated root is .

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

(3.9)

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.

Figure 8

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and . The associated root is .

Figure 9

Problem (3. 9): The numerical solution, the error estimate, and the residual for , and . The associated root is .

Figure 10

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and .

Figure 11

Problem (3. 8): The numerical solution, the error estimate, and the residual for , and .

3.2. Pseudo Dead Core Solutions

In order to calculate the pseudo dead core solutions, we solved the following problem:

(3.10)

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

(3.11)

Figure 12

Problem (3. 10): The numerical solution, the error estimate, and the residual for , and .

Figure 13

Problem (3. 10): The numerical solution, the error estimate, and the residual for , and .

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

(3.12)

Therefore, the exact global error is accessible. In Table 1, we show the values for the global error, where is the numerical solution at .

Table 1 Problem (3.10): Exact global error of the pseudo dead core solution.

3.3. Dead Core Solutions

We now deal with the dead core solutions of the problem. Note that they only occur for

(3.13)

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

(3.14)

Also, the dead core solution is known,

(3.15)

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

(3.16)

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 ,

(3.17)

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.

Figure 14

Problem (3. 10): The initial profile, the numerical solution, the error estimate, and the residual for , and .

Figure 15

Problem (3. 10): The initial profile, the numerical solution, the error estimate, and the residual for , and .

Figure 16

Problem (3. 10): The initial profile, the numerical solution, the error estimate, and the residual for , and .

Figure 17

Problem (3. 10): The initial profile, the numerical solution, the error estimate, and the residual for , and .

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.

Table 2 Maximum of the exact global error of the numerical dead core solution.

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

(3.18)

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.

Figure 18

Graph of the path obtained in 76 steps of the path following procedure, where . The turning point has been found at .

Figure 19

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

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 .

Figure 20

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 21

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 22

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 23

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 24

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 25

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 26

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 27

Problem (3. 18): The numerical solution, the error estimate, and the residual for .

Figure 28

Problem (3. 18): The numerical solution, the error estimate, and the residual for .