Abstract
In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of was achieved with smooth functions.
1. Introduction
In physics, chemistry, biology, sociology, and many other disciplines, there are tremendous problems that can be described by differential equations (DEs), but it is difficult to get their explicit expressions. So, people began to seek the numerical solutions of these problems, which can also be applied to scientific research and engineering practice if their accuracy satisfies the needs. Especially the advent of computers makes it possible to quickly carry out a large number of calculations, which also makes the numerical solution method of DEs become one of the most important branches of computational mathematics. Due to its high smoothness, low power, and easy calculation, the spline function has been widely used in computer graphics, data interpolation and fitting, shape control, and numerical solutions of DEs. There are two main schemes in numerical solutions of DEs using spline functions: the spline finite element method and the spline difference method. The first has a wide range of application and can be applied to many types of equations, but it requires a large amount of calculation. While the second is simple process with a small amount of calculation and high accuracy, but it can only be applied to specific types of equations.
In this paper, we mainly focus a class of second-order ordinary differential equations (ODEs):which meets one of the following boundary conditions.(1)First boundary condition:(2)Second boundary condition:(3)Third boundary condition:(4)Fourth boundary condition:where and are constants and , , and are continuous functions in the interval of .
Many scholars have been studying such two-point boundary value problems. Albasiny and Hoskins [1] and Raghavarao et al. [2] used cubic polynomial splines to solve such problems. Blue [3] used quintic polysplines to solve such problems. Caglar et al. [4] used cubic B-splines in their solving scheme. Chawla and Shiva Kumar[5] extended the problem to semi-infinite regions.
In recent years, with the deepening of research, people have begun to use nonpolynomial splines to solve such problems. Zahra [6], Rao and Kumar [7], Tirmizi et al. [8], Ramadan et al. [9], Surla and Stojanović [10], Jha [11, 12], and Kadalbajoo and Patidar [13] have carried out a lot of research in this area and achieved very high computational accuracy.
However, there are still many theoretical problems to be broken in the study of nonpolynomial splines. Due to the diversity of nonpolynomial splines, it is crucial to choose the basis and parameters in solving the problem. However, there is still no reference in this regard. In this paper, a selected set of spline basis functions was used to deduce the relationship between the derivative and the function value and then to obtain the second-order difference scheme for solving second-order ODEs, which provides a method for solving such problems.
2. Exponential Spline Function Space
Exponential spline refers to a type of spline in which the nonpolynomial factors of spline basis functions contain only exponential functions. The exponential spline in this sense is not very specific; it can contain many forms of exponential spline, which can produce substantially different splines, and is inconvenient to study. Therefore, the exponential spline refers to that with a specific form in the rest of this work.
Next, we define an exponential spline function space. Letwhere , , , and are coefficients and and are parameters with .
Definition 1. The following function spaceis called the cubic -order exponential spline function space.
Obviously, the function in must meetThe dimension of is . Then, we find a set of basic functions with local supports for .
Assume , for given , ; letWe can obtainwhere is the matrix obtained by replacing the column of with , , and .
For even splitting, i.e., , and , the results obtained arewhereDefine a functionwhere the coefficients of are given by the solution of function (9). Besides, for , an interval expansion will be conducted, i.e., will be extended to beThus, in (11) can take the values of . Of course, the function domain is still .
For the space basis of , we have the following proposition.
Proposition 1. When , the function set of is a set of basis of .
For special cases, we can prove the following.
Theorem 1. When and , the function set of is a set of basis of .
Proof. Obviously, , and the number of functions equals the dimension of . Therefore, we need to show that is linearly independent.
For , let , if matrix is invertible, then is linearly independent. It is easy to know that is a tridiagonal matrix, andwhere . Because is a strictly tridiagonal matrix. Thus, is invertible. This proves that is linearly independent.
When , has the following properties.
Proposition 2. For any ,where is not related to , and
The basis function has a local supporting set of . Figure 1 shows all functions in the function set of at the same coordinate.
3. Properties of Exponential Spline Functions
The relationship between the function value and the first derivative and that between the function value and the second derivative are commonly used in numerically solving DEs. Next, we will derive some properties of the functions in , and these relationships will be used in numerically calculating DEs.
Let ; then,
Denotesince , the factors of , and can be written aswhere (the same applies hereinafter). Using the continuous condition of the first derivativewe obtainwhere ,
If and , then the corresponding coefficients become
Let ; then, (24) is equivalent to
In a similar way, we can obtain the relationship between the function value and the first derivative:where
If and , then
Let ; then, (28) is equivalent to
This is consistent with the cubic 2nd-order polynomial spline function relationship.
Besides, usingwe can obtain
Then, using continuous condition of , we obtain
Solving (39) yields
Meanwhile, by eliminating with (39) and (40) and rearranging, we obtain
We can also use to obtainwhere . Fast Hermite interpolation can achieved by using this set of relations.
4. Exponential Spline Difference Method
The spline difference method uses the relationship between the spline function and its derivative to construct the differential expression to numerically solve DEs, by which the numerical solution at nodes can be obtained, and that within the subintervals can also be calculated by using the spline function expressions. It is the advantage of this method compared with the general difference schemes. In fact, it can be said that the approximate analytical solution using the splines is obtained.
4.1. Differential Expression
The following presents a spline difference method for solving (1) which satisfies one of the boundary conditions (2)–(5) for the boundary value problems of ODEs. Due to the limitation of this method, we only consider the case of in this section. For convenience of description, we first consider the boundary condition (2). From (1), we can obtain
By discretization the above equation, we obtainwhere .
Substitute with and with in (24), and we obtainwhere . is the local truncation error at . By substituting (46) into (47) and rearranging, we obtainwhere . Thus, we get equations about .
The following equation can also be derived from the boundary conditions (2):
So, there will be equations in (48) and (49) in total, and it matrix form can be written aswhere , with
If is nonsingular, then equation (50) has a unique solution. Solving (50) yields the approximate values of in at the splitting points . Since is a tridiagonal matrix, the catch-up method can be used to reduce the calculation in practice.
To obtain the spline function expression, we can calculate from (2) after finding and substitute them into (22) to obtain so that the approximate solution of the spline over the entire interval will be found.
The case of other boundary conditions will be discussed below. For the second boundary conditions, from (41), (42), and (3), we obtainwhere
Solving this type of boundary problem only needs to modify the first line and the last row of , , and in (47), namely,
And the third and the fourth boundary conditions should also be modified accordingly.
4.2. Error Estimation
Suppose is sufficiently smooth in , by Taylor expanding in difference equation (47) at , we obtainwhere
Let and , we find and . Whenwe obtain ,
for an even splitting, i.e., , and . This indicates that the even splitting has higher accuracy if is sufficiently smooth.
4.3. Convergence Analysis
We mainly discuss the convergence of equation (47) and the differential expression of (49) in the sense of .
Lemma 1 (see [14]). If the -order matrix satisfies one of the following two conditions:(1) is a strictly diagonally dominant matrix(2) is a second half strong diagonally dominant matrixThen, is nonsingular and , where and is the identity matrix.
By (14),
If is reversible, combining (50), we obtainwhere , and , , and are given by (51)–(53). The reversibility of can be proved by 1. From the boundary condition (2), we can get . Thus, discussing the convergence of is consistent with that of . So, in the following convergence discussion, the first and the last row and the first and the last column will be removed from the original matrices of , , and to obtain the -order ones, keeping the subscript value unchanged.
Ifthen
To calculate , the following lemma is needed:
Lemma 2 (see [15]). Let the square matrix be an -order tridiagonal one, with the following expression:and , ; then, the expression of can be written aswhere denotes the matrix consisting of the elements of rows crossed with the columns in . Specifically, for .
Using (2), in (65) can be written as (with )then . Thus,
According to (27), we find . Thus,
So, if , then when .
So far, we have proved that
Theorem 2. For given , and , , , if , thenthat is, the differential expression of (47) is convergent if .
5. Example
Example 1. Solve the following singular boundary value problem (see [7]):where and , and its analytical solution isTable 1 lists the results of the exponential spline difference method for and that of the method proposed in [7].
Figure 2 shows the results for , and and 10, respectively. To show the difference between the exact solution and the numerical one, we deliberately use fewer split points and simply connect two adjacent solutions with straight lines. A smoother exponential spline function which is closer to the exact solution can be constructed using these obtained numerical solutions.
Example 2. Consider the convection-dominated equation (see [11]):The analytical solution is given byThe computational results are shown in Table 2 for and various values of (Figure 3).
Example 3. Consider the convection-dominated equation (see [16–19]) Figure 4:The computational results are shown in Table 3 for and various values of .
6. Conclusions
Exponential spline, which is a generalization of polynomial spline, is an ideal function approximation tool due to its excellent curve fitting ability. High accuracy can be achieved in solving second-order ODEs using the exponential spline scheme. The spline difference method is an ideal scheme because it can give not only the numerical results but also the spline function expressions by reusing these numerical results at the same time, whereas it is only suitable for solving certain types of equations and does not have generality. And the selection of the appropriate parameters is also needed for this method, but there is no better guideline for the selecting.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Education Department of Guangxi Province in 2015 (KY2015YB081).