Abstract
We study the Dirichlet boundary value problem generally and develop a schema for determining the relationship between the values of its parameters and the number of positive solutions. Then, we focus our attention on the special cases when and , respectively. We prove first that all positive solutions of the first problem are less than or equal to , obtain more specific lower and upper bounds for these solutions, and compute a curve in the -plane with accuracy up to , below which the first problem has a unique positive solution and above which it has exactly three positive solutions. For the second problem, we determine its number of positive solutions and find a formula for the value of that separates the regions of , in which the problem has different numbers of solutions. We also computed the graphs for some special cases of the second problem, and the results are consistent with the existing results. Our code in Mathematica is available upon request.
1. Introduction
As is well known, Dirichlet boundary value problems have a variety of applications in physics, chemistry, and mathematics [1–7]. In recent years, there has been considerable work on the study of the number of positive solutions and bifurcation diagrams of various special cases of the following Dirichlet boundary value problem (DBVP) [1–21].where, is a positive parameter. In this paper, we study DBVP (1) generally first and then develop a schema that can be used to determine the exact number of positive solutions of DBVP (1). Then, we focus our attention on two specific problems of DBVP (1) for and . The first problem was derived from a chemical reaction by Aris R. [7], and the second one is a generalization of the problems studied in [2, 4]. We apply our general schema to the first problem, establish an algorithm for efficient computation, and improve the results obtained by K. J. Brown, M.M.A. Ibrahim, and R. Shivaji [4] significantly. More specifically, K. J. Brown, M.M.A. Ibrahim, and R. Shivaji show that the first problem has a unique solution when or and , we use a new idea that converts the corresponding boundary value problem into its equivalent integral equation and establishes some precise upper and lower bounds for its solutions, and then use our algorithm to compute a curve in the -plane with accuracy up to , below which the first problem has a unique positive solution and above, which it has exactly three positive solutions. To our knowledge, this curve is completely new. The second problem is a generalization of the problems studied in [2, 4]. While Korman et al. [4] studied a special case of our second problem for to be a third degree polynomial, and Kirichuka and Sadyrbaev studied the special case for to be a seventh degree polynomial, we apply our schema to polynomials of any degree and give a formula of the parameter that separates the regions of the values of , in which the problem has different numbers of solutions. Our results recover the ones obtained in [2, 4].
2. General Theorem
It has been proved [8] that all the positive solutions of DBVP (1) are even functions with for and hence, any positive solution is uniquely identified by the value . Now, we consider the initial value problem
Integrating (2) once, we get the following equation:
Because and , it is necessary to have
Since over (0,1), one may get
With one more integration, we get part of an implicit solution of problem (2)
Because is an even function, we may set
As a solution of DBVP (2), (6) satisfies
It is also easy to see that, for any given , if is determined by (8), the function determined by (7) is a positive solution of DBVP (1) with its maximum at .
Theorem 1. The number of positive solutions of DBVP (1) is equal to the number of roots of the equationwhereFurthermore, the corresponding positive solutions of DBVP (1) are given by (7).
Remark 1. If we get the roots of (9), we get implicit solutions by (7) for Since (10) is an improper double integral, direct computation is very inefficient. We need to develop an algorithm to improve its computation efficiency and increase its accuracy.
In the following sections, we focus our attention on the specific problems when and , respectively.
3. Bounds and the Bifurcation Diagram of the Solutions of DBVP (1) for
In this section, we consider the special case of DBVP (1):where and are positive constants, which is a problem arising in chemical reaction theory [7].
First, we establish some bounds for all the positive solutions to this problem by proving the following theorem.
Theorem 2. If is a positive solution of DBVP (13), then has only one extreme point that has a maximum value at and satisfies(1)(2); and(3)for
Proof. First, we prove that for all by dividing it into two cases.(a) has a local maximum . Assume that is the first value of such that and . Then, is decreasing for all and for all . Hence, is decreasing over . This is a contradiction to the assumption and therefore, has as its only extremum and by its even property.(b) does not have a local maximum before it reaches at . Now, one has . If for some small interval , then will never be able to get back to as we discussed in case 1. If for some small interval after , then will never be able to get back to and cannot satisfy the boundary condition since in this interval and cannot have a maximum.To prove the rest of our result in this theorem, we need to convert DBVP (13) into its equivalent integral equation.where, Green’s function is calculated as follows:It is clear that for all and , and . Since is an increasing function with respect to its variable , one hasIn particular, we haveBy using (14) again, we getSetting in (18), one can getCombining (17) and (18), we haveThe theorem is proved.
In [4], Brown et al. proved that DBVP (13) has a unique solution if or and . To explain our approach below, we first recall the idea of their proof. Let and , equation (9) can be written as follows:Now, we denote the left side of (21) by and take its derivative with respect to .where, . For the solution of DBVP (13) to be unique, we need to be monotone. We take the derivative of with respect to .Therefore, and then when or and for . In turn, we have when or and for all . Therefore, is monotone and DBVP (13) has a unique solution when or and . Hence, the condition and or is necessary for DBVP (13) to have multiple (two or three) solutions. Since the integral can still be positive when is less than zero in a small interval, there should be some such that DBVP (13) has a unique solution when .
Let’s first work on this value of heuristically first. When (this is actually a reasonable assumption when we realize that is forced to go to infinity by the condition when is getting close to 4), DBVP (13) is approaching the DBVPwhich brings us back to the well-known one-dimensional perturbed Gelfand two-point DBVPIn [19], we get that DBVP (25) has exactly three solutions when and < Using the transformation , one may convert problem (25) toComparing (24) with (26), we can conclude that DBVP (23) or (13) has exactly three positive solutions when . In fact, this heuristic result is quite close to our result obtained in the following numerical computation:
Through the above discussion, we see that there is a region contained in and in the -plane such that DBVP (13) has exactly three solutions. Now we use the result in Section 2 to develop an efficient algorithm for computing this region.
Let’s introduce a curve defined bySince it is very inefficient to compute improper double integrals, we apply integration by parts on the integral in (11), expand into its power seriesand plug back to equation (10) to getwhereIt follows (28) and (29) that and is defined in . Hence, it is clear that DBVP (13) has positive solutions. By equation (23), we can find that the left side of equation (9) for the problem in this section may have zero or one local maximum, which implies that equation (9) may have one or two roots or three roots for given parameters , and . This recovers the result for this problem given by Brown et al. [9].
Using numerical methods, we look for and so that the bifurcation diagram is monotone when and it is S-shaped when . The following results are obtained by using our code in Mathematica.
Curve fitting of the data in Table 1 tells us the following formulas for functions and and the figures:Figure 1 shows that the fitting between (32) and the data in Table 1 work very well. Functions in (32) also show that and almost coincide. Using the fitting functions and determined by (32), we get the following values Figure 2
For the data in Table 2, we draw to verify the correctness of the above conclusion. As one can see from the second figure, when is on , is monotonically increasing overall, so (29) has only one root. One can also see from Figure 3, when is on and has a maximum and a minimum overall, so (29) has three roots.
To refine the curves, we use our code in Mathematica to get the following data. Curve fitting of the data in Table 3 provides us the following functions and :Since the functions and in (33) are so close to each other, we can use either or for if we limit our accuracy up to . Now, let’s take , we get the following values for some :
If we add to each piece of the data in Table 4, the corresponding graph of must have a minimax value. Their images are shown in Figure 4 below.
Because both the extremum interval and the difference between the extrema are very small, the curve drawing is very time-consuming and error-prone. To avoid these problems, we take 41 points on the curve. All the curves in Figure 4 have clear maximum and minimum, so DBVP (13) has exactly three solutions in this case. The curve separates the first quarter of the -plane such that DBVP (13) has exactly one solution when is below the curve and has exactly three solutions when is above the curve .
4. Number of Solutions of DBVP (1) for
In this section, we consider the special case of DBVP (1): where, are constants satisfying for and for . Let , thenwhere
By (34), we see thatwhere, and
When or for odd m, by (37) and (38), it is clear that the solution of the initial value problem (2) cannot satisfy DBVP (1). However, under some conditions, there are still multiple solutions for DBVP (1).
Lemma 1. If letThen, the functionSatisfies
Proof. By the mean value theorem of integrals,where, for . Thus,One can do it similarly to get
Theorem 3. Under the assumption of Lemma 1, if , DBVP (34) has at least two positive solutions for all and has minimum values , such thatwhere, is the minimum point, . Further, the following result holds:(1)DBVP (34) has m positive solutions for and , where are the roots of the equation(2)There are 3 to positive solutions of DP (1) for
Proof. By Lemma 1, (45) holds. By (36), we haveBy (43) and (47), we haveBy (44) and (49) and the continuity of over and , the solution of DBVP (34) exists on each of these two intervals, i.e.,for and
Without loss of generality, suppose . If , in addition to the two solutions (50), the functionis also a solution of DBVP (34). If , in addition to the two solutions (50), the functionsare also two solution of DBVP (1), where and are two roots of (47) in . Working up similarly for other pairs of , we complete the prove of the theorem.
Theorem 4. Under the assumption of Lemma 1, if , the functionhas minimum values , such thatwhere, is the minimum point with for . So we have(1)If , let be such that . Then, a unique solution of DBVP (34) is(2)If , let be such that . There are m positive solutions of DBVP (34), one of the solutions is and the other solutions are where, are the roots of (47) and , (3)There are 2 to positive solutions of DBVP (13) for .
Proof. When notice that is undefined for . The rest of the proof is similar to the proof of Theorem 3, so it is omitted.
Example 1. If , then DBVP (34) has at least two positive solutions by Theorem 3 for all . For (48) becomesLet , the two roots are and for (58). The graph of and the two corresponding positive solutions of DBVP (34) are shown in Figure 5.
Example 2. If , then , in (4.9) and DBVP (13) has three positive solutions by Theorem 4 for . For , the minimum value of is , where . There are two positive solutions to DBVP (1) for and the three positive solutions to DBVP (1) for respectively. They are shown in Figure 6.
Remark 2. When , the condition in Example 2 is consistent with [2]. However, the does not only exist but also is clearly identified by .
Example 3. If , then DBVP (1) has at least four positive solutions by Theorem 3 for all . However, unlike Example 2, it is difficult to determine a simplified form for the condition of .
5. Conclusion
In this paper, we studied the multiplicity and bifurcation diagram of the Dirichlet boundary value problem (1). Our major results are on the problem for and , respectively. We first proved that all positive solutions of the first problem are less than or equal to σ, obtained more specific lower and upper bounds for these solutions, and computed a curve with accuracy up to , below which the first problem has a unique positive solution and above which it has exactly three positive solutions. This curve is completely new to our knowledge. For the second problem, we determine its exact number of positive solutions and find a formula used to compute the values of that separates the regions of for different numbers of solutions.
Data Availability
The programming code used to generate the data in this article is available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Huizeng Qin was supported by the National Natural Science Foundation of China [grant numbers: 61379009, 61771010]. Youmin Lu was supported by Bloomsburg University.