Abstract
The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.
1. Introduction
The numerical methods of PDE play an important role in many applications, which have been frequently applied in many areas such as communication engineering, fluid dynamics, economics, quantum mechanics, fluid mechanics, chemical reactions, physics, and electrical networks [1]. Many researchers have utilized various numerical methods to solve PDE [2–9]. In [2–9], Allan used the optimal auxiliary function method to solve PDEs, which was extended to general PDEs [2]. Jornet used the differential transform method for solving Burgers’ PDE [3]. Radek solved PDEs using the lattice Boltzmann method [4]. Wang employed localized Chebyshev collocation method for solving elliptic PDEs in arbitrary 2D domains [5]. Young adopted the two-step MPS-MFS ghost point method for solving PDEs [6]. Han used a derivative-free method for solving elliptic PDEs with deep neural networks [7]. Ren used a higher order nonlocal operator method for solving PDEs [8]. Stephen solved the conservation equation by using the symmetric multiple reduction method [9]. In the previous works [10, 11], the author presented the TRKM for solving all kinds of PDEs. Nevertheless, the TRKM cannot get accurate numerical solutions. The TRKM must be improved. The main purpose of this paper is to find an effective and time-saving computational method through adaptive technology. In order to present the APIRKM, we mainly consider the PDE arising from communication engineering as follows:where and are two small parameters; , , , and are given, , .
2. Reproducing Kernel Space (RKS)
The RKSs and their reproducing kernels used in this paper are given as follows:(1)The RKS is one-variable absolutely continuous function, . An inner product is The reproducing kernel is(2)The RKS are one-variable absolutely continuous functions, . An inner product is The reproducing kernel is(3)The RKS are one-variable absolutely continuous functions, . An inner product: The reproducing kernel: where and are given in [10].
3. Representation of the Solution
Before using the TRKM to solve equation (1), it is necessary to homogenize the initial and boundary conditions. Then, we get a new equation and denote the exact solution of the new equation by , , and we get
The reproducing kernel space , , , , , and have been defined in [11], the exact solution of equation (8) is , and the approximate solution is . The error estimate and convergence analysis are given in [12, 13], which will not be elaborated in this paper. However, satisfactory results cannot be obtained by directly using the above method, so the adaptive piecewise interpolation is used. The main idea is as follows:
Firstly, the region is divided into small regions, , , …, , and then the numerical solution of the first small region is solved by using the TRKM, so the boundary value of the first small region is the initial value condition of the second small region ; by parity of reasoning, the boundary value of the last small region is given.
Secondly, a maximum absolute error is set in each region , ; when the absolute errors are greater than , the region will continue to divide until the absolute errors are smaller than .
Finally, when the absolute errors are smaller than , the dividing is stopped, and the optimal small region is obtained, and then the TRKM is used to solve the problem in each region to obtain the numerical solution of the whole region .
4. Numerical Examples
Example 1. Considering the following PDEwith the initial condition and boundary condition,The exact solution of equation (9) is ; is obtained by . By Mathematica 7.0, let and . Figure 1 shows the exact solution. Figure 2 shows the numerical solutions by the TRKM (red line). Figure 3 shows the numerical solutions by the APIRKM (red line) for and . Figure 4 shows the numerical solutions by the APIRKM (red line) for and ; the blue line in Figures 2–4 is the exact solution. Figure 5 provides the absolute errors by the TRKM. Figure 6 provides the absolute errors by the APIRKM for and . Figure 7 provides the absolute errors by the APIRKM for and .
Example 2. Considering the following PDEwith the initial condition and boundary condition,The exact solution of equation (11) is ; is given by . By Mathematica 7.0, let and . Figure 8 shows the absolute errors by the TRKM. Figure 9 provides the absolute errors by the APIRKM for and . Figure 10 provides the absolute errors by the APIRKM for and .
Example 3. Considering the following Black–Scholes equation in financial economicswith the initial and boundary conditions, , , , in which is the risk-free rate, is the red rate, is the maturity date, is the volatility, is the strike price, and the exact solution is . Let , , , and . By Mathematica 7.0, and . Figure 11 shows the absolute errors by TRKM. Figure 12 shows the absolute errors by APIRKM.
5. Conclusions
In this paper, the APIRKM is used to successfully solve a glass of PDE, which is widely used in solving science and engineering problems. The three numerical examples show that the more the piece regions we devide, the more accurate the results we get. Moreover, the APIRKM presented in this paper can solve more complex PDEs.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Inner Mongolia (No. 2019BS01011), the Science Research Project of Higher Education in Inner Mongolia Autonomous Region (No. NJZY20158), and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (No. NJYT-20-B18).