Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices of Bernstein Polynomials

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.


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 real-life 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 bio-medical 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,) 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 f t ð Þ is the source term, A, B are any real numbers; then we extend our method to solve a boundary value problem of coupled system of fractional order differential equations of the form where f t ð Þ, g t ð Þ are source terms of the system, Þare any real constants. Also we compare our approximations to exact values and approximations of other methods like Jacobi polynomial approximations and Haar wavelets methods to evaluate the efficiency of the proposed method. We also provide some examples for the illustration of our main results.
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.
Hence it follows that

The Bernstein polynomials
The Bernstein polynomials B i,m t ð Þ on 0, 1 ½ can be defined as Þ!i! , which on further simplification can be written in the most simplified form as Note that the sum of the Bernstein polynomials converges to 1. Lemma 2.3. Convergence Analysis: Assume that the function g ∈ C mþ1 0, 1 ½ that is m þ 1 times continuously differentiable function and let ð Þ is the best approximation of g out of X, then the error bound is presented as In view of Taylor polynomials, we have from which we know that Hence we have be a Hilbert space, then the inner product can be defined as is a finite dimensional and closed subspace. So if f ∈ H is an arbitrary element then its best approximation is unique in Y: This terminology can be achieved by using y 0 ∈ Y and for all y ∈ Y, we have ∥ f À y 0 ∥ ≤ ∥ f À y∥: Thus any function can be approximated in terms of Bernstein polynomials as where coefficient c i can easily be calculated by multiplying (4) by B j,m ð Þ t ð Þ, j ¼ 0, 1, 2, … m and integrating over 0, 1 ½ by using inner product and which implies that where P α MÂM is the fractional integration's operational matrix defined as taking fractional integration of both sides, we have Now to approximate right-hand sides of above where C i ð Þ M can be approximated by using (3) as where entries of the vector X i ð Þ M can be calculated in generalized form as further writing Let us representP  (3), then fractional order derivative is defined as where G α MÂM is the operational matrix of fractional order derivative given by where Φ MÂM is the dual matrix given in (3) and where Ψ i,j ð Þ is defined for two different cases as Proof. Consider the general element as It is to be noted in the polynomial function B i,m the power of the variable 't' is an ascending order and the lowest power is 'i' therefore the first α À 1 ½ terms becomes zero when we take derivative of order α: ÞBy the use of definition of fractional derivative Now approximating RHS of (25) as further implies that After careful simplification, we get On further simplification, we have where Since by taking inverse Laplace of both sides, we get Now using the approximation Δ i,m ϕ t On further simplification, we get : which is the required result. □

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.

Fractional differential equations
Consider the following problem in generalized form of fractional order differential equation where f t ð Þ is a source term; A, B are any real constants and y t ð Þ is an unknown solution which we want to determine. To obtain a numerical solution of the above problem in terms of Bernstein polynomials, we proceed as Applying fractional integral of order α we have using boundary conditions, we have Using the approximation and Lemma 2.2 Now Putting Eqs. (38)- (41) in Eq. (37), we get In simple form, we can write (42) as Eq. (43) is an algebraic equation of Lyapunov type, which can be easily solved for the unknown coefficient vector K M . When we find K M , then putting this in Eq. (39), we get the required approximate solution of the problem.

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 Þare any real constants, f t ð Þ, g t ð Þ are given source terms. We approximate the solution of the above system in terms of Bernstein polynomials such as applying boundary conditions, we have which is an algebraic equation that can be easily solved by using Matlab functional solver or Mathematica for unknown matrix K M L M ½ . Calculating the coefficient matrix K M , L M and putting it in equations : The exact solution of the above problem is We solve this problem with the proposed method under different sets of parameters as defined in S 1 , S 2 , S 3 . The observation and simulation demonstrate that the solution obtained with the proposed method is highly accurate. The comparison of exact solution with approximate solution obtained using the parameters set S 1 is displayed in Figure 1 subplot (a), while in Figure 1 subplot (b) we plot the absolute difference between the exact and approximate solutions using different scale levels. One can easily observe that the absolute error is much less than 10 À12 : The order of derivatives in this set is an integer.
By solving the problem under parameters set S 2 and S 3 , we observe the same phenomena. The approximate solution matches very well with the exact solution. See  Figures 2 and 3    proposed method, the graph of exact and approximate solutions for different values of M and at α ¼ 1:7 is shown in Figure 4. From the plot, we observe that the approximate solution becomes equal to the exact solution at α ¼ 2. We approximate the error of the method at different scale levels and record that when scale level increases the absolute error decreases as shown in Figure 4 subplot (b) and accuracy approaches 10 À9 , which is a highly acceptable figure. For convergence of our proposed method, we examined the quantity Ð 1 0 |y exact À y approx |dt for different values of M and observed that the norm of error decreases with a high speed with the increase of scale level M as shown in Figure 4b.  x t ð Þ ¼ t 5 À t 4 À 1, and y t ð Þ ¼ t 4 À t 3 À 2: The source terms are given by À 4t 3 þ 6t 4 À 2t 5 À 2: Approximating the solution with the proposed method, we observe that our scheme gives high accuracy of approximate solution. In Figure 7, we plot the exact solutions together with the approximate solutions in Figure 7(a) and (b) for x t ð Þ and y t ð Þ , respectively. We see from the subplots (a) and (b) that our approximations have close agreement to that of exact solutions. This accuracy may be made better by increasing scale level. Further, one can observe that absolute error is below 10 À10 in Figure 8, which indicates better accuracy of our proposed method for such types of practical problems of applied sciences.  In Figure 8, the subplot (a) represents absolute error for x t ð Þ while subplot (b) represents the same quantity for y t ð Þ. From the subplots, we see that maximum absolute error for our proposed method for the given problem (4.4) is below 10 À10 : This is very small and justifies the efficiency of our constructed method. Example 4.5. Consider the boundary value problem Taking α ¼ 2, ν ¼ 1 and ω ¼ 1, 3, 5, … , the exact solution is given by We plot the comparison between exact and approximate solutions to the given example at M ¼ 10 and corresponding to ω ¼ 3:5, α ¼ 2, β ¼ 1. Further, we approximate the solution through Legendre wavelet method (LWM) [47], Jacobi polynomial method (JM) and Bernstein polynomials method (BM), as shown in Figure 9.
From Table 1, we see that Bernstein polynomials also provide excellent solutions to fractional differential equations [48].

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 two-dimensional 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.