Block pulse functions for solving fractional Poisson type equations with Dirichlet and Neumann boundary conditions
- Jiaquan Xie^{1, 3},
- Qingxue Huang^{1, 2}Email author,
- Fuqiang Zhao^{1, 3} and
- Hailian Gui^{1, 3}
Received: 2 December 2016
Accepted: 2 March 2017
Published: 14 March 2017
Abstract
In this study, the numerical technique based on two-dimensional block pulse functions (2D-BPFs) has been developed to approximate the solution of fractional Poisson type equations with Dirichlet and Neumann boundary conditions. These functions are orthonormal and have compact support on \([ 0,1 ]\). The proposed method reduces the original problems to a system of linear algebra equations that can be solved easily by any usual numerical method. The obtained numerical results have been compared with those obtained by the Legendre and CAS wavelet methods. In addition an error analysis of the method is discussed. Illustrative examples are included to demonstrate the validity and robustness of the technique.
Keywords
1 Introduction
Fractional calculus is an important theoretical branch of mathematical theories [1], which has been widely applied in various fields such as the complex physical, mechanical, biological, and engineering fields. For example, fractional calculus has been applied to model the nonlinear oscillations of earthquake [2], fluid-dynamic traffic [3], continuum and statistical mechanics [4], signal processing [5], control theory [6] and dynamics of interfaces between nanoparticles and subtracts [7]. In these practical applications, fractional calculus has a certain geometric and physical meaning. Due to its historical dependence and globally related characteristics, fractional differentiation can be approximately represented and vigorously developed in the anomalous diffusion. In view of the fact that fractional calculus has great practical significance, it is very important to study fractional differential equations. In general, the exact solutions of many fractional partial differential equations cannot be obtained, so scholars are committed to obtaining their numerical solutions to reflect the exact solutions. In recent years, the theory of fractional calculus has been greatly developed, and lots of articles about fractional calculus have been published. Numerical algorithms for different types of fractional differential equations have been presented. These algorithms include the Chebyshev and Legendre polynomials method [8, 9], the wavelet method [10, 11], the piecewise constant orthogonal functions method [12], the differential transform method [13], the collocation method [14], the Adomian decomposition method [15], and so on. In [16], the authors proposed a compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients. In [11], the authors acquired the numerical solution of fractional Poisson equations using a two-dimensional Legendre wavelet. In [17], Maleknejad and Mahdiani proposed to solve nonlinear mixed Volterra-Fredholm integral equations with two-dimensional block pulse functions using a direct method. Reference [18] gave the numerical methods for solving two-dimensional nonlinear integral equations of fractional order by using a two-dimensional block pulse operational matrix. In this study, we applied two-dimensional block pulse functions to obtain the numerical solutions of fractional Poisson type equations with Dirichlet and Neumann boundary conditions.
The current paper is organized as follows. In Section 2, the model with respect to Poisson type equations is given. In Section 3, some basic definitions and mathematical preliminaries of fractional calculus are introduced. Section 4 introduces the definitions and properties of two-dimensional block pulse functions. In Section 5, we discussed the error analysis of our approach. Section 6 introduces the method for solving fractional Poisson type equations. In Section 7, the proposed approach is tested through several numerical examples. Finally, a conclusion is given in Section 8.
2 Illustration of the proposed model
In the following, we introduce the propagation of an elastic wave for a one-dimensional rod model under the action of impact load.
3 Preliminaries and notations
In this section, we gave some necessary definitions and preliminaries of the fractional calculus theory which will be used in the article [1].
Definition 1
Definition 2
4 Two-dimensional block pulse functions
One-dimensional block pulse functions have been widely used for differential and integral equations. More details for block pulse functions of the one-dimensional case are given in [19]. These conclusions can be extended to the two-dimensional block pulse functions.
4.1 Definitions and properties
- 1.Disjointness:$$ \phi_{i_{1},i_{2}} ( x,t )\phi_{j_{1},j_{2}} ( x,t ) = \left \{ \textstyle\begin{array}{l@{\quad}l} \phi_{i_{1},i_{2}} ( x,t ), &i_{1} = j_{1} \mbox{ and } i_{2} = j_{2}, \\ 0,& \mbox{otherwise}. \end{array}\displaystyle \right . $$(11)
- 2.Orthogonality:In the region of \(x \in [ 0,T_{1} )\) and \(t \in [ 0,T_{2} )\), where \(i_{1},j_{1} = 1,2, \ldots,m_{1}\) and \(i_{2},j_{2} = 1,2, \ldots,m_{2}\).$$ \int_{0}^{T_{1}} \int_{0}^{T_{2}} \phi_{i_{1},i_{2}} ( x,t ) \phi_{j_{1,}j_{2}} ( x,t )\,\mathrm{d}t\,\mathrm{d}x = \left \{ \textstyle\begin{array}{l@{\quad}l} h_{1}h_{2},& i_{1} = i_{2} \mbox{ and } j_{1} = j_{2}, \\ 0, &\mbox{otherwise}. \end{array}\displaystyle \right . $$(12)
- 3.Completeness: For every \(u \in L^{2} ( [ 0,T_{1} ) \times [ 0,T_{2} ) )\) when \(m_{1}\) and \(m_{2}\) approach infinity, Parseval’s identity holds:where$$ \int_{0}^{T_{1}} \int_{0}^{T_{2}} u^{2} ( x,t )\,\mathrm{d}t\, \mathrm{d}x = \sum_{i_{1} = 1}^{\infty} \sum _{i_{2} = 1}^{\infty} u_{i_{1},i_{2}}^{2} \bigl\Vert \phi_{i_{1},i_{2}} ( x,t ) \bigr\Vert ^{2}, $$(13)$$u_{i_{1},i_{2}} = \frac{1}{h_{1}h_{2}} \int_{0}^{T_{1}} \int_{0}^{T_{2}} u ( x,t )\phi_{i_{1},i_{2}} ( x,t )\, \mathrm{d}t\,\mathrm{d}x. $$
4.2 BPFs expansions
4.3 Operational matrix
In this part, we may simply introduce the operational matrix of fractional integration of block pulse functions; more details can be found in [20].
Let \(D_{\alpha}\) is the block pulse operational matrix for the fractional differentiation. According to the property of fractional calculus, \(D_{\alpha} F_{\alpha} = I\), we can easily obtain the matrix \(D_{\alpha}\) by inverting the \(F_{\alpha}\) matrix.
5 Error analysis
The error can be achieved when a function \(u ( x,t )\) is represented by 2D-BPFs over the region \(D = [ 0,T ) \times [ 0,T )\). Let \(m_{1} = m_{2} = m\), so \(h_{1} = h_{2} = \frac{T}{m}\).
From equation (24), the conclusion is drawn that the method in this paper is convergent when it is used to solve the numerical solutions of fractional partial differential equations. For more details see [21].
6 Description of the proposed method
- (i)Dirichlet boundary conditions: For equation (4) with boundary conditions of equation (5), we haveThe entries of vector \(\Phi^{T} ( x )\) and \(\Phi ( t )\) are independent, so from equation (30) we can obtain$$ \begin{aligned} &\Phi^{T} ( x ) U\Phi ( 0 ) \approx \Phi^{T} ( x ) C_{1},\qquad \Phi^{T} ( 0 ) U\Phi ( t ) \approx C_{3}^{T}\Phi ( t ), \\ &\Phi^{T} ( x ) U\Phi ( \tau ) \approx \Phi^{T} ( x ) C_{2}, \qquad\Phi^{T} ( L ) U\Phi ( t ) \approx C_{4}^{T}\Phi ( t ). \end{aligned} $$(30)$$ \begin{aligned} &\Lambda_{1} = U\Phi ( 0 ) - C_{1} \approx 0,\qquad \Lambda_{3} = \Phi^{T} ( 0 ) U - C_{3}^{T} \approx 0, \\ &\Lambda_{2} = U\Phi ( \tau ) - C_{2} \approx 0,\qquad \Lambda_{4} = \Phi^{T} ( L ) U - C_{4}^{T} \approx 0, \end{aligned} $$(31)
- (ii)Neumann boundary conditions: Similar to equation (31), we can obtainequation (29) together with equation (31) or equation (32) gives a system of linear equations, we apply the least-square-method to solve the system, then the unknown function can be found.$$ \begin{aligned} &\Lambda_{1} = U\Phi ( 0 ) - C_{1} \approx 0,\quad\quad \Lambda_{3} = \Phi^{T} ( 0 ) U - C_{3}^{T} \approx 0, \\ &\Lambda_{2} = U\mathbf{D}_{1}\Phi ( \tau ) - C_{2} \approx 0,\qquad \Lambda_{4} = \Phi^{T} ( L )\mathbf{D}_{1}^{T} U - C_{4}^{T} \approx 0, \end{aligned} $$(32)
7 Numerical experiments
In this section, we applied several numerical examples to test our proposed method, and compared the obtained numerical results with those obtained by the Legendre and the CAS wavelet methods.
Example 1
Example 2
Example 3
Numerical results of some values of v , γ and m for Examples 3
( x , t ) | v = 2, γ = 1.8 | v = 2, γ = 1.9 | Exact solution | ||
---|---|---|---|---|---|
m = 32 | m = 64 | m = 32 | m = 64 | ||
(0.2,0.2) | 0.0080999 | 0.0078027 | 0.0072661 | 0.0068916 | 0.0065498 |
(0.4,0.4) | 0.0451791 | 0.0443244 | 0.0442091 | 0.0440639 | 0.0429004 |
(0.6,0.6) | 0.1254325 | 0.1229019 | 0.1221991 | 0.1208139 | 0.1185433 |
(0.8,0.8) | 0.2421242 | 0.2382542 | 0.2363440 | 0.2339913 | 0.2300564 |
(1.0,1.0) | 0.4533293 | 0.3819264 | 0.3791327 | 0.3750676 | 0.3678794 |
Example 4
Numerical results of different m for Examples 4
( x , t ) | Exact solution | m = 16 | m = 32 | m = 64 | m = 128 |
---|---|---|---|---|---|
(0,0) | 0 | 0 | 0 | 0 | 0 |
(1/4,1/4) | 0.0039 | 0.0036 | 0.0037 | 0.0039 | 0.0039 |
(2/4,2/4) | 0.0625 | 0.0620 | 0.0623 | 0.0625 | 0.0625 |
(3/4,3/4) | 0.3164 | 0.3159 | 0.3160 | 0.0362 | 0.3164 |
(1,1) | 1.0000 | 0.9996 | 0.9998 | 0.9999 | 1.0000 |
(5/4,5/4) | 2.4414 | 2.4406 | 2.4409 | 2.4413 | 2.4414 |
(6/4,6/4) | 5.0625 | 5.0618 | 5.0620 | 5.0624 | 5.0625 |
(7/4,7/4) | 9.3789 | 9.3776 | 9.3780 | 9.3785 | 9.3789 |
(2,2) | 16.0000 | 15.9987 | 15.9990 | 15.9996 | 15.9999 |
8 Conclusion
In the present analysis, we have applied two-dimensional block pulse functions to approximate the solutions of Poisson type equations with two kinds of boundary conditions. Using this method, the system of fractional partial differential equations has been reduced to solving a system of algebraic equations. The accuracy of the solutions of the original problems will be improved using suitable \(m_{1}\) and \(m_{2}\). The block pulse functions are orthogonal piecewise continuous functions which prompts flexibility for applications. Compared with other numerical methods, our proposed method has the great advantages of easy theoretical construction and being less time consuming. Additionally, the illustrated examples analyze and justify the ability and the reliability of the proposed method.
Declarations
Acknowledgements
This work was supported by the Natural Science Foundation of Shanxi Province (201601D011051), and the Dr. Startup funds of Taiyuan University of Science and Technology (20122054).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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