Comparison of solution between Legendre wavelet method (LWM) [47], Jacobi polynomial method (JM) and Bernstein polynomials method (BM) for Example 4.5.
Abstract
In this chapter, we develop an efficient numerical scheme for the solution of boundary value problems of fractional order differential equations as well as their coupled systems by using Bernstein polynomials. On using the mentioned polynomial, we construct operational matrices for both fractional order derivatives and integrations. Also we construct a new matrix for the boundary condition. Based on the suggested method, we convert the considered problem to algebraic equation, which can be easily solved by using Matlab. In the last section, numerical examples are provided to illustrate our main results.
Keywords
 Bernstein polynomials
 coupled systems
 fractional order differential equations
 operational matrices of integration
 approximate solutions
 2010 MSC: 34L05
 65L05
 65T99
 34G10
1. Introduction
Generalization of classical calculus is known as fractional calculus, which is one of the fastest growing area of research, especially the theory of fractional order differential equations because this area has wide range of applications in reallife problems. Differential equations of fractional order provide an excellent tool for the description of many physical biological phenomena. The said equations play important roles for the description of hereditary characteristics of various materials and genetical problems in biological models as compared with integer order differential equations in the form of mathematical models. Nowadays, most of its applications are found in biomedical engineering as well as in other scientific and engineering disciplines such as mechanics, chemistry, viscoelasticity, control theory, signal and image processing phenomenon, economics, optimization theory, etc.; for details, we refer the reader to study [1, 2, 3, 4, 5, 6, 7, 8, 9] and the references there in. Due to these important applications of fractional order differential equations, mathematicians are taking interest in the study of these equations because their models are more realistic and practical. In the last decade, many researchers have studied the existence and uniqueness of solutions to boundary value problems and their coupled systems for fractional order differential equations (see [10, 11, 12, 13, 14, 15, 16, 17]). Hence the area devoted to existence theory has been very well explored. However, every fractional differential equation cannot be solved for its analytical solutions easily due to the complex nature of fractional derivative; so, in such a situation, approximate solutions to such a problem is most efficient and helpful. Recently, many methods such as finite difference method, Fourier series method, Adomian decomposition method (ADM), inverse Laplace technique (ILT), variational iteration methods (VIM), fractional transform method (FTM), differential transform method (DTM), homotopy analysis method (HAM), method of radial base function (MRBM), wavelet techniques (WT), spectral methods and many more (for more details, see [9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]) have been developed for obtaining numerical solutions of such differential equations. These methods have their own merits and demerits. Some of them provide a very good approximation. However, in the last few years, some operational matrices were constructed to achieve good approximation as in [39]. After this, a variety of operational matrices were developed for different wavelet methods. This method uses operational matrices, where every operation, for example differentiation and integration, involved in these equations is performed with the help of a matrix. A large variety of operational matrices are available in the literature for different orthogonal polynomials like Legendre, Laguerre, Jacobi and Bernstein polynomials [40, 41, 42, 43, 44, 45, 46, 47, 48]. Motivated by the above applications and uses of fractional differential equations, in this chapter, we developed a numerical scheme based on operational matrices via Bernstein polynomials. Our proof is more generalized and there is no need to convert the Bernstein polynomial function vector to another basis like block pulse function or Legendre polynomials. To the best of our knowledge, the method we consider provides a very good approximation to the solution. By the use of these operational matrices, we apply our scheme to a single fractional order differential equation with given boundary conditions as
where
where
This chapter is designed in five sections. In the first section of the chapter, we have cited some basic works related to the numerical and analytical solutions of differential equations of arbitrary order by various methods. The necessary definitions and results related to fractional calculus and Bernstein polynomials along with the construction of some operational matrices are given in Section 2. In Section 3, we have discussed the main theory for the numerical procedure. Section 4 contains some interesting practical examples and their images. Section 5 describes the conclusion of the chapter.
2. Basic definitions and results
In this section, we recall some fundamental definitions and results from the literature, which can be found in [10, 11, 12, 13, 14, 15, 16].
where
has a unique solution of the form
for arbitrary
Hence it follows that
2.1 The Bernstein polynomials
The Bernstein polynomials
where
where
Note that the sum of the Bernstein polynomials converges to 1.
where
from which we know that
Due the best approximation
Hence we have
Let
and
is a finite dimensional and closed subspace. So if
where coefficient
Let
where
where
where
where
taking fractional integration of both sides, we have
Now to approximate righthand sides of above
where
where entries of the vector
evaluating this result for i = 0,1,2....m, we have
further writing
we get
Let us represent
thus
□
where
where
It is to be noted in the polynomial function
Now approximating RHS of (25) as
further implies that
After careful simplification, we get
On further simplification, we have
Let
so
which is the desired result. □
is given by
where
Let us evaluate the general terms
Since by
taking inverse Laplace of both sides, we get
now Eq. (32) implies that
Now using the approximation
On further simplification, we get
So
and
which is the required result. □
3. Applications of operational matrices
In this section, we are going to approximate a boundary value problem of fractional order differential equation as well as a coupled system of fractional order boundary value problem. The application of obtained operational matrices is shown in the following procedure.
3.1 Fractional differential equations
Consider the following problem in generalized form of fractional order differential equation
where
Applying fractional integral of order
using boundary conditions, we have
Using the approximation and Lemma 2.2
Hence
Now
and
Putting Eqs. (38)–(41) in Eq. (37), we get
In simple form, we can write (42) as
where
Eq. (43) is an algebraic equation of Lyapunov type, which can be easily solved for the unknown coefficient vector
3.2 Coupled system of boundary value problem of fractional order differential equations
Consider a coupled system of a fractional order boundary value problem
subject to the boundary conditions
where
applying boundary conditions, we have
let us approximate
then
and
Thus system (44) implies that
Rearranging the terms in the above system and using the following notation for simplicity in Eq. (46)
the above system (46) becomes
which is an algebraic equation that can be easily solved by using Matlab functional solver or Mathematica for unknown matrix
we get the required approximate solution.
4. Applications of our method to some examples
subject to the boundary conditions
subject to the boundary conditions
subject to the boundary conditions
and
subject to the boundary conditions
From Table 1 , we see that Bernstein polynomials also provide excellent solutions to fractional differential equations [48].








0.5  10  2  1 



1.5  15  1.6  0.9 



2.0  20  1.8  0.8 



3.5  25  1.9  0.7 



5. Conclusion and future work
The above analysis and discussion take us to the conclusion that the new method is very efficient for the solution of boundary value problems as well as initial value problems including coupled systems of fractional differential equations. One can easily extend the method for obtaining the solution of such types of problems with other kinds of boundary and initial conditions. Bernstein polynomials also give best approximate solutions to fractional order differential equations like Legendre wavelet method (LWM), approximation by Jacobi polynomial method (JPM), etc. The new operational matrices obtained in this method can easily be extended to twodimensional and higher dimensional cases, which will help in the solution of fractional order partial differential equations. Also, we compare our result to that of approximate methods for different scale levels. We observed that the proposed method is also an accurate technique to handle numerical solutions.
Acknowledgments
This research work has been supported by Higher Education Department (HED) of Khyber Pakhtankhwa Government under grant No: HEREF46 and Higher Education Commission of Pakistan under grant No: 10039.
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