- Research
- 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
References
- Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston (2012) MATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
- Lazarevic, MP, Spasic, AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math. Comput. Model. 49(3), 475-481 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Magin, RL: Fractional Calculus in Bioengineering. Begell House, Redding (2006) Google Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
- Sabatier, J, Agrawal, OP, Machado, JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007) MATHGoogle Scholar
- Zaslavsky, GM: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2008) MATHGoogle Scholar
- Gupta, CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. Math. Anal. Appl. 168, 540-551 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Ma, R: A survey on nonlocal boundary value problems. Appl. Math. E-Notes 7, 257-279 (2007) MathSciNetMATHGoogle Scholar
- Guezane-Lakoud, A, Zenkoufi, L: Existence of positive solutions for a third-order multi-point boundary value problem. Appl. Math. 3, 1008-1013 (2012) View ArticleMATHGoogle Scholar
- El Sayed, AMA, Bin-Tahir, EO: Positive solutions for a nonlocal boundary-value problem of a class of arbitrary (fractional) orders differential equations. Int. J. Nonlinear Sci. Numer. Simul. 14(4), 398-404 (2012) MathSciNetGoogle Scholar
- Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA: Spectral Methods in Fluid Dynamics. Springer, New York (1988) View ArticleMATHGoogle Scholar
- Bhrawy, AH, Alofi, AS: A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations. Commun. Nonlinear Sci. Numer. Simul. 17, 62-70 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Funaro, D: Polynomial Approximation of Differential Equations. Lecturer Notes in Physics. Springer, Berlin (1992) MATHGoogle Scholar
- Doha, EH, Bhrawy, AH, Ezz-Eldien, SS: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35, 5662-5672 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, AH, Alofi, AS, Ezz-Eldien, SS: A quadrature tau method for variable coefficients fractional differential equations. Appl. Math. Lett. 24, 2146-2152 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Saadatmandi, A, Dehghan, M: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59, 1326-1336 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Doha, EH, Bhrawy, AH, Ezz-Eldien, SS: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364-2373 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Ghoreishi, F, Yazdani, S: An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis. Comput. Math. Appl. 61, 30-43 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Vanani, SK, Aminataei, A: A Tau approximate solution of fractional partial differential equations. Comput. Math. Appl. 62, 1075-1083 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Esmaeili, S, Shamsi, M: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3646-3654 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Pedas, AA, Tamme, E: On the convergence of spline collocation methods for solving fractional differential equations. J. Comput. Appl. Math. 235, 3502-3514 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, AH, Al-Shomrani, MM: A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Differ. Equ. 2012 8 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Khalil, H, Khan, RA: The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations. Int. J. Comput. Math. (2014). doi:10.1080/00207160.2014.945919 MATHGoogle Scholar
- Shah, K, Ali, A, Khan, RA: Numerical solutions of fractional order system of Bagley-Torvik equation using operational matrices. Sindh Univ. Res. J. (Sci. Ser.) 47(4), 757-762 (2015) Google Scholar
- Khalil, H, Khan, RA: A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation. Comput. Math. Appl. (2014). doi:10.1016/j.camwa.2014.03.008 MathSciNetGoogle Scholar
- Khalil, H, Khan, RA: New operational matrix of integration and coupled system of Fredholm integral equations. Chin. J. Math. 2014, Article ID 146013 (2014). MathSciNetView ArticleMATHGoogle Scholar
- Khalil, H, Khan, RA: A new method based on Legendre polynomials for solution of system of fractional order partial differential equation. Int. J. Comput. Math. 91, 2554-2567 (2014). doi:10.1080/00207160.2014.880781 MathSciNetView ArticleMATHGoogle Scholar
- Khalil, H, Khan, RA, Al Smadi, MH, Freihat, A: Approximation of solution of time fractional order three-dimensional heat conduction problems with Jacobi polynomials. J. Math. 47(1), 35-56 (2015) MathSciNetMATHGoogle Scholar
- Khalil, H, Rashidi, MM, Khan, RA: Application of fractional order Legendre polynomials: a new procedure for solution of linear and nonlinear fractional differential equations under m-point nonlocal boundary conditions. Commun. Numer. Anal. 2016(2), 144-166 (2016) Google Scholar
- Khalil, H, Khan, RA, Baleanu, D, Rashidi, MM: Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains. Comput. Appl. Math. (2016). doi:10.1016/j.camwa.2016.04.014 Google Scholar
- Bhrawy, AH, Zaky, MA: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comp. Physiol. 281, 876-895 (2015) MathSciNetView ArticleGoogle Scholar
- Bhrawy, AH, Zaky, MA: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn. 80(1-2), 101-116 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, AH, Zaky, MA: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67(2), 340-349 (2015) Google Scholar
- Zaky, MA, Bhrawy, AH, Van Gorder, RA: A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer. Algorithms 71(1), 151-180 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy, AH, Zaky, M: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl. Math. Model. 40(2), 832-845 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Rehman, M, Khan, RA: A numerical method for solving boundary value problems for fractional differential equations. Appl. Math. Model. 36, 894-907 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Rehman, M, Khan, RA: The Legendre wavelet method for solving fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 4163-4173 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Y: Numerical solution of the heat equation with nonlocal boundary condition. J. Comput. Appl. Math. 110, 115-127 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Ang, W: A method of solution for the one-dimensional heat equation subject to nonlocal condition. Southeast Asian Bull. Math. 26, 185-191 (2002) MathSciNetView ArticleGoogle Scholar
- Dehghan, M: The one-dimensional heat equation subject to a boundary integral specification. Chaos Solitons Fractals 32, 661-675 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Noye, KHBJ: Explicit two-level finite difference methods for the two-dimensional diffusion equation. Int. J. Comput. Math. 42, 223-236 (1992) View ArticleMATHGoogle Scholar
- Avalishvili, G, Avalishvili, M, Gordeziani, D: On integral nonlocal boundary value problems for some partial differential equations. Bull. Georgian. Natl. Acad. Sci. (N. S.) 5, 31-37 (2011) MathSciNetMATHGoogle Scholar
- Sajavicius, S: Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions, a case of two-dimensional Poisson equation. Eng. Anal. Bound. Elem. 37, 788-804 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Yousefi, SA, Behroozifar, M: Operational matrices of Bernstein polynomials and their applications. Int. J. Inf. Syst. Sci. 41(6), 709-716 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Doha, EH, Bhrawy, AH, Saker, MA: On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equations. Bound. Value Probl. 2011, 829543 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Doha, EH, Bhrawy, AH, Saker, MA: Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations. Appl. Math. Lett. 24(1), 559-565 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Juttler, B: The dual basis functions for the Bernstein polynomials. Adv. Comput. Math. 8(4), 345-352 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Farouki, RT: Legendre Bernstein basis transformations. J. Comput. Math. 119(1), 145-160 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Hermann, T: On the stability of polynomial transformations between Taylor, Bernstein, and Hermite forms. Numer. Algorithms 13(2), 307-320 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Bellucci, MA: On the explicit representation of orthonormal Bernstein polynomials. http://arxiv.org/abs/1404.2293v2
- Chen, Y, Sun, Y, Liu, L: Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions. Appl. Comput. Math. 244, 847-858 (2014) MathSciNetMATHGoogle Scholar
- Knuth, DE: The Art of Computer Programming. Fundamental Algorithms, vol. 1. Addison-Wesley, Reading (1968) MATHGoogle Scholar
- https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix
- Bellman, RE, Kalaba, RE: Quasilinearization and Non-linear Boundary Value Problems. Elsevier, New York (1965) MATHGoogle Scholar
- Stanley, EL: Quasilinearization and Invariant Imbedding. Academic Press, New York (1968) Google Scholar
- Agarwal, RP, Chow, YM: Iterative methods for a fourth order boundary value problem. J. Comput. Appl. Math. 10(2), 203-217 (1984) MathSciNetView ArticleMATHGoogle Scholar
- Akyuz Dascioglu, A, Isler, N: Bernstein collocation method for solving nonlinear differential equations. Math. Comput. Appl. 18(3), 293-300 (2013) MathSciNetGoogle Scholar
- Charles, A, Baird, J: Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations. J. Optim. Theory Appl. 3(4), 227-242 (1969) MathSciNetView ArticleMATHGoogle Scholar
- Mandelzweig, VB, Tabakin, F: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Commun. Comput. Phys. 141(2), 268-281 (2001) MathSciNetView ArticleMATHGoogle Scholar