Abstract
In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.
1. Introduction
In this paper, we consider the following time-fractional order diffusion equation:where , are known functions, and is the variable order Caputo fractional derivative of order :
The time-fractional order equation [1–4] has wide applications in viscoelastic materials, signal processing, fluid mechanics, and dynamic of viscoelastic materials. The analytic solution to this equation is almost impossible to obtain. In recent years, several numerical methods [5–11] have been proposed. In previous work, the author used Taylor’s formula or Delta function to construct reproducing kernel space [12–17]. In this paper, we structure some new reproductive kernel spaces based on Jacobi polynomials and give a numerical solution of a class of time-space fractional order diffusion equation using piecewise reproducing kernel method.
Definition 1. Let be a real Hilbert spaces of functions . A function is called reproducing kernel for if(i) for all ,(ii) for all and all .
2. Structing Reproductive Kernel Space Based on the Shifted Jacobi Polynomials
The shifted Jacobi polynomials of degree is given [18] bywhere
The shifted Jacobi polynomials on the interval are orthogonal with the orthogonality conditionwhere is a weight function and
Definition 2. Letis the weighted inner product space of the shifted Jacobi polynomials on . The inner product and norm are defined aswhereFrom [17, 19], we can prove that is a reproducing kernel Hilbert space. Its reproducing kernel iswhere . Using Ref. [5] and the reproducing kernel of , we can get following reproducing kernel spaces.(1)Space , with the same inner product as and is a reproducing kernel space and its reproducing kernel is(2)Space , with the same inner product as and is a reproducing kernel space and its reproducing kernel is where .(3)Space , and its reproducing kernel iswhere from (11) and (12). Reproducing kernels with different are shown in Figures 1–8.
3. Piecewise Reproducing Kernel Method
After homogenization, equation (1) is converted to the following form:where is a operator, and
, , .
Let be nodes in interval , .where the are the coefficients resulting from Gram–Schmidt orthonormalization.
Theorem 1. (see [11, 19–23]). If is existing and is denumerable dense points in , thenis an analytical solution of (14).
Deriving from the form of (17), we get the approximate solution of (14) asHowever, the direct application of (17) could not have a good numerical simulation effect possibly for (1). The focus of this paper is to fill this gap, so we use the piecewise reproducing kernel method. The main technique of the piecewise reproducing kernel method see Ref. [16, 21, 24, 25]. More about convergence theorem and error estimation, those detailed proof can be seen in [23–26].
4. Numerical Experiments
In this section, some numerical experiments are studied to demonstrate the accuracy of the present method.
Experiment 1. We consider the following time fractional reaction-diffusion equation:where , the exact solution . Numerical solution of Experiment 1 is shown in Figures 9–11 and Table 1. From Table 1, we can see that the absolute error is getting smaller and smaller when is smaller. Figure 10 shows the relationship between absolute error and reproducing kernel. Figure 11 shows the relationship between absolute error and .
Experiment 2. We consider the following time fractional reaction-diffusion equation [7]where , the exact solution . Numerical solution of Experiment 2 is shown in Table 2. From Table 2, we can see that the absolute error obtained by the present method is smaller than the absolute error obtained by Ref. [7].
Experiment 3. We consider the following time-space fractional advection-reaction-diffusion equationwhere , the exact solution . Numerical solution of Experiment 3 is shown in Figure 12 and Table 3.
(a)
(b)
Experiment 4. We consider the following time-space fractional advection-reaction-diffusion equation:where , the exact solution . By mathematical 8.0, the numerical comparison of absolute errors by present method are given in Figures 13–20 at . The reproducing kernel of Experiment 4 with different is shown in Table 4 by present method at . Figures 13 and 14 show the relationship between absolute error and . From Figures 15 and 16, we can see that the absolute error is small. Figures 17–20 show the relationship between absolute error and .
5. Conclusions
In this paper, some new reproductive kernels are given. The numerical results of some models show that the present method has high precision compared with traditional RKM, and has a better convergence for this kind of model. Besides, the method can also be used to study other time variable fractional order advection-dispersion model.
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 there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The authors would like to express our deep thanks to the anonymous reviewers whose valuable comments and suggestions helped us improve this article greatly. This paper was supported by the Natural Science Foundation of Inner Mongolia (2017MS0103).