The application of block pulse functions for solving higher-order differential equations with multi-point boundary conditions
- Zakieh Avazzadeh^{1}Email author and
- Mohammad Heydari^{2}
https://doi.org/10.1186/s13662-016-0822-6
© Avazzadeh and Heydari 2016
Received: 10 January 2016
Accepted: 30 March 2016
Published: 5 April 2016
Abstract
In this paper, the block pulse function method is proposed for solving high-order differential equations associated with multi-point boundary conditions. Although the orthogonal block pulse functions frequently have been applied to approximate the solution of ordinary differential equations associated with the initial conditions, the presented method provides the flexibility with respect to multi-point boundary conditions in separated or non-separated forms. This technique, which may be named the augmented block pulse function method, reduces a system of high-order boundary value problems of ordinary differential equations to a system of algebraic equations. The illustrated results confirm the computational efficiency, reliability, and simplicity of the presented method.
Keywords
MSC
1 Introduction
The systems of ordinary differential equations (ODEs) with different boundary conditions are well known for their applications in biology, chemistry, physics, engineering, and sciences [1–4]. There are many different reliable methods which can find the solution of ODEs for simple forms of boundary conditions. But the mathematical models of many phenomena in the real world are enforced by more difficult forms of boundary conditions such as multi-point boundary conditions in separated or non-separated forms.
Because of the importance, the boundary value problems have been solved several times by many different methods such as the finite difference method, the spline method, the radial basis functions, the wavelet method, and many other numerical and analytical methods; see [5–10] and the references therein. We recall that boundary conditions that are more difficult imply developing numerical methods to find the solution of the ordinary differential systems. However, some of these methods are reliable and applicable for solving ordinary differential equations; the most of these methods provide the solution only for a particular kind of differential equations or a particular kind of boundary conditions. In this study, we describe the application of the block pulse function method for solving arbitrary-order differential equations.
Recently, orthogonal block pulse functions have been widely discussed and applied to approximate the solutions of some difficult systems defined in engineering and science [11–22]. The most important properties of BPFs are disjointness, orthogonality, and completeness which cause the popularity among the computational methods.
To make the article self-contained in Section 2 a short description on block pulse functions is added. In Section 3 the description of the method shows BPFs how can be applied to solve the high-order differential equations with a different kind of boundary conditions. The numerical results are illustrated in Section 4 to clarify more details of the proposed method and expectedly confirm the convergence and applicability of the method. Finally, a brief conclusion is stated in Section 5.
2 Block pulse functions
Theorem 2.1
Remark 2.2
3 Description of the method
Remark 3.1
It is worth noting here that we can do a few simple modifications when some of \(f^{(i)}(a)\), \(i=0,1,\ldots,n-1\), are given. Particularly if \(f^{(i)}(a)\), \(i=0,1,\ldots,n-1\), all are given, the system becomes an initial value problem and there is no need to consider any \(c_{i}\), \(i=m,\ldots,m+n-1\). In addition, we can keep the structure of the algorithm and input the given initial value into the described scheme. Obviously, the first state considers the value of \(f^{(i)}(a)\), \(i=0,1,\ldots,n-1\), precisely and the second state find them approximately such that there are good agreement between precise and approximated values. In this paper, the reported results are based on the second assumption.
Remark 3.2
Definitely we need \((n+m)\) equations which are linear independent to find a unique solution including \((n+m)\) unknown coefficients. Note that the intersection of \(\{x_{0}, x_{1},\ldots,x_{k}\}\) defined in (3), and the collocation points defined (10) should be an empty set. If there exists any common point, we can simply change the collocation points non-uniformly such that every collocation point may be chosen from \([\frac{i}{m},\frac{i+1}{m} )\), \(i=0,1,2,\ldots, m-1\) in order to include all basis functions and keep the structure of constructed matrices demonstrated in (13), (32), and (33) and for higher order. Obviously, there are many sets of points that are appropriate candidates for leading to the independent algebraic equations.
4 Numerical examples
In order to assess the accuracy of block pulse function method for solving higher-order differential equations with multi-point boundary conditions we will consider the following examples. The associated computations with the examples were performed using MAPLE 17 with 64 digits precision on a personal computer.
Example 1
The observed maximum absolute error for different values of m for Example 1
\(\boldsymbol {y^{(j)}}\) | m = 6 | m = 10 | m = 16 | m = 32 |
---|---|---|---|---|
y | 1.8 × 10^{−4} | 7.1 × 10^{−5} | 2.8 × 10^{−5} | 7.1 × 10^{−6} |
\(y^{(1)}\) | 5.7 × 10^{−4} | 2.2 × 10^{−4} | 8.7 × 10^{−5} | 2.3 × 10^{−5} |
\(y^{(2)}\) | 1.8 × 10^{−3} | 6.8 × 10^{−4} | 2.7 × 10^{−4} | 6.8 × 10^{−5} |
\(y^{(3)}\) | 5.8 × 10^{−3} | 2.3 × 10^{−3} | 9.3 × 10^{−4} | 2.4 × 10^{−4} |
\(y^{(4)}\) | 2.6 × 10^{−2} | 8.8 × 10^{−3} | 3.3 × 10^{−3} | 7.5 × 10^{−4} |
\(y^{(5)}\) | 3.5 × 10^{−1} | 1.2 × 10^{−1} | 4.9 × 10^{−2} | 1.2 × 10^{−2} |
\(y^{(6)}\) | 7.2 | 4.3 | 2.7 | 1.3 |
Example 2
The observed maximum absolute error for different values of m for Example 2
x | m = 10 | m = 16 | RKM ( m = 151) [25] |
---|---|---|---|
0.0 | 2.9 × 10^{−7} | 9.8 × 10^{−8} | 1.1 × 10^{−6} |
0.1 | 6.8 × 10^{−8} | 1.6 × 10^{−8} | 2.0 × 10^{−7} |
0.2 | 7.8 × 10^{−9} | 6.9 × 10^{−10} | 5.9 × 10^{−9} |
0.3 | 7.5 × 10^{−9} | 7.7 × 10^{−10} | 4.4 × 10^{−9} |
0.4 | 4.4 × 10^{−8} | 7.6 × 10^{−9} | 8.5 × 10^{−8} |
0.5 | 1.2 × 10^{−7} | 3.1 × 10^{−8} | 2.6 × 10^{−7} |
0.6 | 1.9 × 10^{−7} | 5.8 × 10^{−8} | 4.3 × 10^{−7} |
0.7 | 1.9 × 10^{−7} | 5.7 × 10^{−8} | 4.1 × 10^{−7} |
0.8 | 2.3 × 10^{−8} | 2.1 × 10^{−8} | 2.7 × 10^{−8} |
0.9 | 6.1 × 10^{−7} | 2.4 × 10^{−7} | 1.2 × 10^{−6} |
1.0 | 1.8 × 10^{−6} | 6.9 × 10^{−7} | 3.4 × 10^{−6} |
Example 3
5 Conclusion
The block pulse functions provide the efficient method to solve high-order ODEs associated with the general type of multi-point boundary conditions. According to the presented method, the nth-order ODE defined in (1), which can be linear or nonlinear system with separated and non-separated boundary conditions, will be reduced to the algebraic equations by using block pulse functions and a polynomial function of degree \(n-1\). The most important privileges of the proposed method are computational efficiency due to sparse matrices, simplicity, and reliability, so one may increase the number of basis functions and consequently the accuracy will be improved.
Declarations
Acknowledgements
Z Avazzadeh wish to thank Natural Science Foundation of Jiangsu Province (Project No. BK20150964) and gratefully acknowledges ‘A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions’.
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
References
- Agarwal, RP: Boundary Value Problems for High Order Differential Equations. World Scientific, Singapore (1986) View ArticleGoogle Scholar
- Ang, WT, Park, YS: Ordinary Differential Equations: Methods and Applications. Universal-Publishers, Boca Raton (2008) View ArticleGoogle Scholar
- Hsu, S-B: Ordinary Differential Equations with Applications. World Scientific, Singapore (2006) MATHGoogle Scholar
- Roberts, C: Ordinary Differential Equations: Applications, Models, and Computing. CRC Press, Boca Raton (2011) Google Scholar
- Loghmani, GB, Ahmadinia, M: Numerical solution of sixth order boundary value problems with sixth degree B-spline functions. Appl. Math. Comput. 186, 992-999 (2007) MathSciNetMATHGoogle Scholar
- Loghmani, GB, Alavizadeh, SR: Numerical solution of fourth-order problems with separated boundary conditions. Appl. Math. Comput. 191, 571-581 (2007) MathSciNetMATHGoogle Scholar
- Siddiqi, SS, Akram, G: Septic spline solutions of sixth-order boundary value problems. J. Comput. Appl. Math. 215, 288-301 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Temimi, H, Ansari, AR: A new iterative technique for solving nonlinear second order multi-point boundary value problems. Appl. Math. Comput. 218, 1457-1466 (2011) MathSciNetMATHGoogle Scholar
- Vedat, SE: Solving nonlinear fifth-order boundary value problems by differential transformation method. Selçuk J. Appl. Math. 8(1), 45-49 (2007) MathSciNetMATHGoogle Scholar
- Wazwaz, AM: The numerical solution of fifth-order boundary value problems by the decomposition method. J. Comput. Appl. Math. 136, 259-270 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Datta, KB, Mohan, BM: Orthogonal Functions in Systems and Control. World Scientific, Singapore (1995) View ArticleMATHGoogle Scholar
- Deb, A, Sarkar, G, Bhattacharjee, M, Sen, SK: All-integrator approach to linear SISO control system analysis using block pulse functions (BPF). J. Franklin Inst. 334(2), 319-335 (1997) View ArticleMATHGoogle Scholar
- Deb, A, Sarkar, G, Sen, SK: Block pulse functions, the most fundamental of all piecewise constant basis functions. Int. J. Syst. Sci. 25(2), 351-363 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Harmuth, HF: Transmission of Information by Orthogonal Functions. Springer Science & Business Media. Springer, Berlin (2013) MATHGoogle Scholar
- Hatamzadeh-Varmazyar, S, Masouri, Z, Babolian, E: Numerical method for solving arbitrary linear differential equations using a set of orthogonal basis functions and operational matrix. Appl. Math. Model. (2015). doi:10.1016/j.apm.2015.04.048 MathSciNetMATHGoogle Scholar
- Hatamzadeh-Varmazyar, S, Masouri, Z: Numerical method for analysis of one- and two-dimensional electromagnetic scattering based on using linear Fredholm integral equation models. Math. Comput. Model. 54, 2199-2210 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Hatamzadeh-Varmazyar, S, Naser-Moghadasi, M, Masouri, Z: A moment method simulation of electromagnetic scattering from conducting bodies. Prog. Electromagn. Res. 81, 99-119 (2008) View ArticleGoogle Scholar
- Hatamzadeh-Varmazyar, S, Naser-Moghadasi, M, Babolian, E, Masouri, Z: Numerical approach to survey the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions. Prog. Electromagn. Res. 81, 393-412 (2008) View ArticleGoogle Scholar
- Jiang, ZH, Schaufelberger, W: Block Pulse Functions and Their Applications in Control Systems. Lecture Notes in Control and Information Sciences, vol. 179. Springer, Berlin (1992) View ArticleMATHGoogle Scholar
- Rao, CP: Piecewise Constant Orthogonal Functions and Their Application to Systems and Control. Springer, Berlin (1983) View ArticleMATHGoogle Scholar
- Sannuti, P: Analysis and synthesis of dynamic systems via block-pulse functions. Proc. Inst. Electr. Eng. 124(6), 569-571 (1977) View ArticleGoogle Scholar
- Wang, C-H: Generalized block-pulse operational matrices and their applications to operational calculus. Int. J. Control 36, 67-76 (1982) MathSciNetView ArticleMATHGoogle Scholar
- Maleknejad, K, Khodabin, M, Rostami, M: A numerical method for solving m-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix. Comput. Math. Appl. 63, 133-143 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Siddiqi, SS, Twizell, EH: Spline solutions of linear sixth order boundary value problems. Int. J. Comput. Math. 60, 295-304 (1996) View ArticleMATHGoogle Scholar
- Lin, YZ, Lin, JN: Numerical algorithm about a class of linear nonlocal boundary value problems. Appl. Math. Lett. 23, 997-1002 (2010) MathSciNetView ArticleMATHGoogle Scholar