An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

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. (e results show that the algorithm is high accurate and conditionally convergent, and an accuracy of (1/240)h6 was achieved with smooth functions.


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. ere are two main schemes in numerical solutions of DEs using spline functions: the spline finite element method and the spline difference method. e 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.
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.

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. e 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. erefore, the exponential spline refers to that with a specific form in the rest of this work.
Next, we define an exponential spline function space. Let where a i , b i , c i , and d i are coefficients and τ 1 and τ 2 are parameters with τ 1 ≠ τ 2 .
Definition 1. e following function space is called the cubic r-order exponential spline function space. Obviously, the function s(x) in E 2 3 (Δ n ) must meet en, we find a set of basic functions with local supports for E 2 We can obtain where 2 Discrete Dynamics in Nature and Society A k (k � 1, 2, . . . , 16) is the matrix obtained by replacing the k column of A with [0, . . . , 0, 1] T , θ i � τ 1 h i , and η i � τ 2 h i .

Of course, the function domain is still
For the space basis of E 2 3 (Δ n ), we have the following proposition.
For special cases, we can prove the following.
j�− 1 has the following properties.
where C is not related to x, and

Properties of Exponential Spline Functions
e 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 E 2 3 (Δ n ), and these relationships will be used in numerically calculating DEs.
, the factors of a i , b i , c i , and d i can be written as where . . , n (the same applies hereinafter). Using the continuous condition of the first derivative we obtain where i � 1, 2, . . . , n − 1, If h i � h > 0, i � 1, 2, . . . , n and τ 2 � − τ 1 � τ, then the corresponding coefficients become 6 Discrete Dynamics in Nature and Society Let τ ⟶ 0; then, (24) is equivalent to In a similar way, we can obtain the relationship between the function value and the first derivative: Discrete Dynamics in Nature and Society If τ 2 � − τ 1 � τ and h i � h, i � 1, 2, . . . , n, then Let τ ⟶ 0; then, (28) is equivalent to is is consistent with the cubic 2nd-order polynomial spline function relationship.
Besides, using we can obtain en, using continuous condition of Discrete Dynamics in Nature and Society Solving (39) yields Meanwhile, by eliminating D i+1 with (39) and (40) and rearranging, we obtain We can also use Discrete Dynamics in Nature and Society where Fast Hermite interpolation can achieved by using this set of relations.

Exponential Spline Difference Method
e 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.

Differential Expression.
e 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 q(x) � 0 in this section. For convenience of description, we first consider the boundary condition (2). From (1), we can obtain (45) By discretization the above equation, we obtain where g i � g(x i ) and p i � p(x i ). Substitute S with u and M with u ″ in (24), and we obtain . t i is the local truncation error at x i . By substituting (46) into (47) and rearranging, we obtain where i � 1, 2, . . . , n − 1. us, we get n − 1 equations about u 0 , u 1 , . . . , u n . e following equation can also be derived from the boundary conditions (2): So, there will be n + 1 equations in (48) and (49) in total, and it matrix form can be written as where A � B − W, with 10 Discrete Dynamics in Nature and Society T .

(53)
If A is nonsingular, then equation (50) has a unique solution. Solving (50) yields the approximate values of u i in u(x) at the splitting points x i , i � 0, 1, . . . , n. Since A 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 u i ″ from (2) after finding u i and substitute them into (22) to obtain a i , b i , c i , and d i so that the approximate solution of the spline over the entire interval will be found. e case of other boundary conditions will be discussed below. For the second boundary conditions, from (41), (42), and (3), we obtain where Solving this type of boundary problem only needs to modify the first line and the last row of B, W, and F in (47), namely, W n � 0 · · · 0 α n1 p n− 1 α n2 p n , And the third and the fourth boundary conditions should also be modified accordingly.
is indicates that the even splitting has higher accuracy if u(x) is sufficiently smooth.

Convergence Analysis.
We mainly discuss the convergence of equation (47) and the differential expression of (49) in the sense of ‖E‖ ∞ .
Lemma 1 (see [14]). If the n-order matrix B satisfies one of the following two conditions: (1) B is a strictly diagonally dominant matrix (2) B is a second half strong diagonally dominant matrix en, B is nonsingular and ρ(I − D − 1 B) < 1, where D � diag(B) and I is the identity matrix.
By (14), If A is reversible, combining (50), we obtain where E � (e i ) � U − S, and B, W, and T are given by (51)-(53). e reversibility of B can be proved by 1. From the boundary condition (2), we can get e 0 � e n � 0. us, discussing the convergence of ‖E‖ ∞ is consistent with that of ‖(e i ) n− 1 i�1 ‖ ∞ . 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 A, B, and W to obtain the n − 1-order ones, keeping the subscript value unchanged. If then To calculate B − 1 , the following lemma is needed: Lemma 2 (see [15]). Let the square matrix A be an n-order tridiagonal one, with the following expression: and b i c i ≠ 0, i � 1, 2, . . . , n − 1, a i ≠ 0, i � 1, 2, . . . , n; then, the expression of A − 1 � (a (− 1) ij ) can be written as where Using (2), (65) can be written as (with       Table 1 lists the results of the exponential spline difference method for ε � 0.1 and that of the method proposed in [7]. Figure 2 shows the results for ε � 0.1, and n � 5 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]): e computational results are shown in Table 2 for ε � 0.01 and various values of n ( Figure 3).

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. e 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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.