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- Open Access
Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions
- Hammad Khalil^{1, 2}Email authorView ORCID ID profile,
- Rahmat Ali Khan^{3},
- Dumitru Baleanu^{4} and
- Samir H Saker^{5}
https://doi.org/10.1186/s13662-016-0910-7
© Khalil et al. 2016
Received: 11 April 2016
Accepted: 24 June 2016
Published: 7 July 2016
Abstract
This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrices that we will developed in this paper have the ability to convert fractional differential equations together with its nonlocal boundary conditions to a system of easily solvable algebraic equations. Some test problems are presented to illustrate the efficiency, accuracy, and applicability of the proposed method.
Keywords
- Bernstein polynomials
- operational matrices
- m-point boundary conditions
- fractional differential equations
MSC
- 35C11
- 65T99
1 Introduction
Recently the studies of fractional differential equations (FDEs) gained the attention of many scientists around the globe. This topic remains a central point in several special issues and books. Fractional-order operators are nonlocal in nature and due to this property, they are most nicely applicable to various systems of natural and physical phenomena. This property has motivated many scientists to develop fractional-order models by considering the ideas of fractional calculus. Examples of such systems can be found in many disciplines of science and engineering such as physics, biomathematics, chemistry, dynamics of earthquakes, dynamical processes in porous media, material viscoelastic theory, and control theory of dynamical systems. Furthermore, the outcome of certain observations indicates that fractional-order operators possess some properties related to systems having long memory. For details of applications and examples, we refer the reader to the work in [1–7].
The qualitative study of FDEs which discusses analytical investigation of certain properties like existence and uniqueness of solutions has been considered by several authors. In 1992 Guptta [8] studied the solvability of three point boundary value problems (BVPs). Since then many researchers have been working in this area and provided many useful results which guarantee the solvability and existence of a unique solution of such problems. For the reader interested in the existence theory of such problem we refer to work presented by Ruyun Ma [9] in which the author presents a detailed survey on the topic. In [10] the author derived an analytic relation which guarantees the existence of positive solution of a general third order multi-point BVPs. Also some analytic properties of solutions of FDEs are discussed by El-Sayed in [11]. Often it is impossible to arrive at the exact solution when FDEs has to be solved under some constraints in the form of boundary conditions. Therefore the development of approximation techniques remains a central and active area of research.
The spectral methods, which belong to the approximation techniques, are often used to find approximate solution of FDEs. The idea of the spectral method is to convert FDEs to a system of algebraic equations. However, different techniques are used for this conversion. Some of well-known techniques are the collocation method, the tau method, and the Galerkin method. The tau and Galerkin methods are analogous in the sense that the FDEs are enforced to satisfy some algebraic equations, then some supplementary set of equations are derived using the relations of boundary conditions (see, e.g., [12] and the references therein). The collocation method [13, 14], which is an analog of the spectral method, consists of two steps. First, a discrete representation of the solution is chosen and then FDEs are discretized to obtain a system of algebraic equations.
These techniques are extensively used to solve many scientific problems. Doha et al. [15] used collocation methods and employed Chebyshev polynomials to find an approximate solution of initial value problems of FDEs. Similarly, Bhrawy et al. [16] derived an explicit relation which relates the fractional-order derivatives of Legendre polynomials to its series representation, and they used it to solve some scientific problems. Saadatmandi and Dehghan [17] and Doha et al. [18] extended the operational matrices method and derived operational matrices of fractional derivatives for orthogonal polynomials and used it for solving different types of FDEs. Some recent and good results can be found in the articles like [15, 19, 20].
Some other methods have also been developed for the solution of FDEs. Among others, some of them are iterative techniques, reproducing kernel methods, finite difference methods etc. Esmaeili and Shamsi [21] developed a new procedure for obtaining an approximation to the solution of initial value problems of FDEs by employing a pseudo-spectral method, and Pedas and Tamme [22] studied the application of spline functions for solving FDEs. In [23], the author used a quadrature tau method for obtaining the numerical solution of multi-point boundary value problems. Also the authors in [24–31] extended the spectral method to find a smooth approximation to various classes of FDEs and FPDEs. Some recent results in which orthogonal polynomials are applied to solve various scientific problems can be found in [32–36].
Multi-point nonlocal boundary value problems appears widely in many important scientific phenomena like in elastic stability and in wave propagation. For the solution of such a problem Rehman and Khan [37, 38] introduced an efficient numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear multi-point boundary value problems for FDEs. FDEs subject to multi-point nonlocal boundary conditions are a little bit difficult. In this area of research a few articles are available. Some good results on solution of nonlocal boundary value problems can be found in [39–44].
Bernstein polynomials are frequently used in many numerical methods. Bernstein polynomials enjoy many useful properties, but they lack the important property of orthogonality. As the orthogonality property is one of more important properties in approximation theory and numerical simulations, these polynomials cannot be directly implemented in the current technique of numerical approximations. To overcome this difficulty, these non-orthogonal Bernstein polynomials are transformed into an orthogonal basis [45–48]. But as the degree of the polynomials increases the transformation matrix becomes ill conditioned [49, 50], which results some inaccuracies in numerical computations. Recently Bellucci introduced an explicit relation for normalized Bernstein polynomials [51]. One applied Gram-Schmidt orthonormalization process to some sets of Bernstein polynomials of different scale levels, identifying the pattern of polynomials, and generalizing the result. The main results presented in this article are based on these generalized bases.
The main aim in this paper is to find a smooth approximation to \(U(t)\), which satisfies a given set of m-point boundary conditions. We consider the following two types of boundary constraints.
We organized the rest of article as follows. In Section 2, we recall some basic concepts definition from fractional calculus, approximation theory, and matrix theory. Also we present some properties of normalized Bernstein polynomials which are helpful in our further investigation. In Section 3, we present a detailed procedure for the construction of the required operational matrices. In Section 4, the developed matrices are employed to solve FDEs by introducing a new algorithm. In Section 5, the proposed algorithm is applied to some test problems to show the efficiency of the proposed algorithm. The last section is devoted to a short conclusion.
2 Some basic definitions
In this section, we present some basic notation and definitions from fractional calculus and well-known results which are important for our further investigation. More details can be found in [5, 7].
Definition 1
Definition 2
The integral of the triple product of fractional-order Legendre polynomials over the domain of interest was recently used in [52]. There the author used this value to solve fractional differential equations with variable coefficients directly. We use a triple product for Bernstein polynomials to construct a new operational matrix which is of basic importance in solving FDEs with variable coefficients. The following theorem is of basic importance.
Theorem 1
Proof
Now, we present the inverse of the well-known Vandermonde matrix. The inverse of this matrix will be used when we use operational matrices to solve under local boundary conditions.
Theorem 2
3 Operational matrices of derivative and integral
Now we are in a position to construct new operational matrices. The operational matrices of derivatives and integrals are frequently used in the literature to solve fractional-order differential equations. In this section, we present the proofs of constructions of four new operational matrices. These matrices act as building blocks in the proposed method.
Theorem 3
Proof
Theorem 4
Proof
The operational matrices developed in the previous theorems can easily solve FDEs with initial conditions. Here we are interested in the approximate solution of FDEs under complicated types of boundary conditions. Therefore we need some more operational matrices such that we can easily handle the boundary conditions effectively.
The following matrix plays an important role in the numerical simulation of fractional differential equations with variable coefficients.
Theorem 5
Proof
Using (47) in (39) we get the desired result. The proof is complete. □
Since one of our aims in this paper is to solve FDEs under different types of local and non-local boundary conditions, we have to face some complicated situations, so to handle these situations we will use the operational matrix developed in the next theorem.
Theorem 6
Proof
4 Application of operational matrices
5 Examples
To show the applicability and efficiency of the proposed method, we solve some fractional differential equations. The numerical simulation is carried out using MatLab. However, we believe that the algorithm can be simulated using any simulation tool kit.
Example 1
Comparison of exact and approximate solution of Example 1 using the boundary condition as defined in \(\pmb{\mathbf{S_{1}}}\)
t | N | ||||
---|---|---|---|---|---|
Exact U ( t ) | N = 5 | N = 7 | N = 9 | N = 11 | |
t = 0.4 | 0.95105651629 | 0.9677391807 | 0.9499276887 | 0.9510565133 | 0.95105651554 |
t = 0.8 | 0.58778525229 | 0.5806739865 | 0.5877691649 | 0.5877785192 | 0.5877882463 |
t = 1.2 | −0.58778525229 | −0.5879798137 | −0.5877896195 | −0.5877852801 | −0.58778528235 |
t = 1.6 | −0.95105651629 | −0.9318025727 | −0.9510687377 | −0.9510565637 | −0.9510565256 |
t = 2.0 | −0.00000000000 | −0.0160927142 | −0.0000151536 | −0.0000001456 | −0.00000000975 |
t = 2.4 | 0.95105651629 | 0.9684572120 | 0.9510519403 | 0.9510565108 | 0.95105651183 |
t = 2.8 | 0.58778525229 | 0.5667990904 | 0.5879771600 | 0.5877852533 | 0.5877852523 |
t = 3 | 0.00000000000 | 0.0634719308 | −0.0001774663 | 0.0000000045 | 0.00000000125 |
Comparison of exact and approximate solution of Example 1 using the boundary condition as defined in \(\pmb{\mathbf{S_{1}}}\)
\(\boldsymbol {S_{1}}\) | \(\boldsymbol {S_{2}}\) | |||
---|---|---|---|---|
\(\boldsymbol {\|E_{N}\|_{2}}\) | \(\boldsymbol {\|E_{N}\|_{\infty}}\) | \(\boldsymbol {\|E_{N}\| _{2}}\) | \(\boldsymbol {\|E_{N}\|_{\infty}}\) | |
N = 5 | 7.23 × 10^{−2} | 6.53 × 10^{−1} | 4.72 × 10^{−1} | 3.68 × 10^{−1} |
N = 8 | 1.003 × 10^{−4} | 2.913 × 10^{−2} | 6.53 × 10^{−2} | 0.58 × 10^{−3} |
N = 10 | 4.82 × 10^{−6} | 3.21 × 10^{−5} | 0.587 × 10^{−3} | 0.778 × 10^{−4} |
N = 15 | 3.92 × 10^{−11} | 7.84 × 10^{−9} | 9.51 × 10^{−5} | 9.510 × 10^{−8} |
N = 20 | 6.22 × 10^{−16} | 5.61 × 10^{−14} | 1.51 × 10^{−10} | 1.456 × 10^{−12} |
N = 30 | 9.36 × 10^{−19} | 8.22 × 10^{−18} | 9.510 × 10^{−11} | 9.108 × 10^{−13} |
N = 40 | 9.28 × 10^{−21} | 0.781 × 10^{−18} | 5.87 × 10^{−16} | 5.33 × 10^{−18} |
Example 2
Example 3
Comparison of exact and approximate solution of Example 3 using the boundary condition as defined in \(\pmb{\mathbf{S_{2}}}\)
t | N | |||||
---|---|---|---|---|---|---|
Exact U ( t ) | N = 2 | N = 3 | N = 4 | N = 5 | N = 6 | |
t = 0.0 | 1.000000000 | 1.0152591045 | 0.9989268650 | 1.0000584609 | 0.9999973751 | 1.000000102 |
t = 0.1 | 1.105170918 | 1.1048778054 | 1.1055690936 | 1.1051373901 | 1.1051724714 | 1.105170872 |
t = 0.2 | 1.221402758 | 1.2115236915 | 1.2222416889 | 1.2213694032 | 1.2214034018 | 1.221402755 |
t = 0.3 | 1.3498588075 | 1.3351967629 | 1.3506284171 | 1.3498396744 | 1.3498594914 | 1.349858762 |
t = 0.4 | 1.4918246976 | 1.4758970196 | 1.4924130441 | 1.4918009400 | 1.4918267043 | 1.491824610 |
t = 0.5 | 1.6487212707 | 1.6336244616 | 1.6492793361 | 1.6486734983 | 1.6487243484 | 1.648721203 |
t = 0.6 | 1.8221188003 | 1.8083790888 | 1.8229110590 | 1.8220452095 | 1.8221217781 | 1.822118764 |
t = 0.7 | 2.0137527074 | 2.0001609012 | 2.0149919788 | 2.0136714962 | 2.0137550986 | 2.013752651 |
t = 0.8 | 2.2255409284 | 2.2089698990 | 2.2272058616 | 2.2254753425 | 2.2255438696 | 2.225540838 |
t = 0.9 | 2.4596031111 | 2.4348060820 | 2.4612364735 | 2.4595472949 | 2.4596078085 | 2.459603059 |
t = 1.0 | 2.7182818284 | 2.6776694503 | 2.7187675805 | 2.7181454617 | 2.7182834949 | 2.718281628 |
Example 4
Comparison of exact and approximate solution of Example 4 using the boundary condition as defined in \(\pmb{\mathbf{S_{2}}}\)
t | N | ||||
---|---|---|---|---|---|
Exact U ( t ) | First iteration | Second iteration | Third iteration | Fourth iteration | |
t = 0.0 | 1.0000000000 | 0.9993612934 | 1.0001381646 | 1.0000260271 | 1.000000020 |
t = 0.1 | 1.0168063300 | 1.1131408265 | 1.0322239559 | 1.0171510593 | 1.016806465 |
t = 0.2 | 1.0338951135 | 1.2060058045 | 1.0595571295 | 1.0344373486 | 1.033895324 |
t = 0.3 | 1.0512710963 | 1.2803801650 | 1.0835040499 | 1.0519291558 | 1.051271352 |
t = 0.4 | 1.0689391057 | 1.3388134647 | 1.1052061945 | 1.0696628704 | 1.068939387 |
t = 0.5 | 1.0869040495 | 1.3839808784 | 1.1255801546 | 1.0876670104 | 1.086904346 |
t = 0.6 | 1.1051709180 | 1.4186831997 | 1.1453176345 | 1.1059622221 | 1.105171226 |
t = 0.7 | 1.1237447856 | 1.4458468410 | 1.1648854519 | 1.1245612803 | 1.123745105 |
t = 0.8 | 1.1426308117 | 1.4685238330 | 1.1845255377 | 1.1434690886 | 1.142631142 |
t = 0.9 | 1.1618342427 | 1.4898918253 | 1.2042549363 | 1.1626826787 | 1.161834578 |
t = 1.0 | 1.1813604128 | 1.5132540859 | 1.2238658054 | 1.1821912111 | 1.181360731 |
6 Conclusion
From experimental results and analysis of the proposed method we conclude that the method is efficient in solving linear and nonlinear fractional-order differential equations under different types of boundary conditions. The method has the ability to solve both local and nonlocal boundary value problems. We use normalized Bernstein polynomials for our analysis. But the method can be used to generalize such types of operational matrices for almost all types of orthogonal polynomials. It is also possible to get a more approximate solution of such problems using other types of orthogonal polynomials like Legendre, Jacobi, Laguerre, Hermite etc. It is not clear to us which is the best set of orthogonal polynomials for this method. Further investigation is required to generalize the method to solve other types of scientific problems.
Declarations
Acknowledgements
The authors are thankful to the reviewers and the editor for carefully reading and useful suggestion which improved the quality of the article.
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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