Mathematical Models and Computation in Discrete DynamicsView this Special Issue
Research Article | Open Access
Dunqian Cao, "An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 7056254, 16 pages, 2020. https://doi.org/10.1155/2020/7056254
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. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of was achieved with smooth functions.
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  and Raghavarao et al.  used cubic polynomial splines to solve such problems. Blue  used quintic polysplines to solve such problems. Caglar et al.  used cubic B-splines in their solving scheme. Chawla and Shiva Kumar 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 , Rao and Kumar , Tirmizi et al. , Ramadan et al. , Surla and Stojanović , Jha [11, 12], and Kadalbajoo and Patidar  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