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

By Kamal Shah, Thabet Abdeljawad, Hammad Khalil and Rahmat Ali Khan

Submitted: April 26th 2019Reviewed: October 28th 2019Published: January 27th 2020

DOI: 10.5772/intechopen.90302

Downloaded: 339


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.


  • 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 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, 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 ftis the source term, A,Bare 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 ft,gtare source terms of the system, Ai,Bi,Ci,Dii=12are 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.


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].

Definition 2.1.The fractional integral of orderγR+of a functionyL101Ris defined as


Definition 2.2.The Caputo fractional order derivative of a function y on the interval01is defined by


where n=γ+1and γrepresents the integer part of γ.

Lemma 2.1.The fractional differential equation of orderγ>0


has a unique solution of the form yt=d0+d1t+d2t2++dn1tn1, where dkRand k=0,1,2,3,.,n1..

Lemma 2.2.The following result holds for fractional differential equations


for arbitrary dkR, k=0,1,2,,n1.

Hence it follows that


2.1 The Bernstein polynomials

The Bernstein polynomials Bi,mton 01can be defined as


where mi=m!mi!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 functiongCm+101that ism+1times continuously differentiable function and letX=B0,mB1,mBm,m. IfCTΨxis the best approximation ofgout ofX, then the error bound is presented as


where M=maxx01gm+1x,S=max1x0x0.

Proof.In view of Taylor polynomials, we have


from which we know that

gFx=gm+1ηxx0m+1Γm+2,there existη01.

Due the best approximation CTΨxof g, we have


Hence we have


Let H=L201be a Hilbert space, then the inner product can be defined as




is a finite dimensional and closed subspace. So if fHis an arbitrary element then its best approximation is unique in Y.This terminology can be achieved by using y0Yand for all yY,we have fy0fy.Thus any function can be approximated in terms of Bernstein polynomials as


where coefficient cican easily be calculated by multiplying (4) by Bjmt,j=0,1,2,mand integrating over 01by using inner product and di=01Bimtftdt,θij=01BimtBjmtdt,i,j=0,1,2.m,we have

01ftBjmtdt=01i=0mciBimt.Bjmtdt,j=0,1,2m,which implies that01ftBjmtdt=i=0mci01Bimt.Bjmtdt,j=0,1,2mwhich impliesθ00θ01θ0rθ0mθ10θ11θ1rθ1mθr0θr1θrrθrmθm0θm1θmrθmm.E5

Let,,where M=m+1where Mis the scale level and ΦM×M=θ00θ01θ0rθ0mθ10θ11θ1rθ1mθr0θr1θrrθrmθm0θm1θmrθmm,so


where Φm×Mis called the dual matrix of the Bernstein polynomials. After calculating ci, (4) can be written as

ft=CMBMTt,CMis coefficient matrix



Lemma 2.4.LetBMTtbe the function vector defined in(3), then the fractional order integration ofBMTtis given by


where PM×Mαis the fractional integration’s operational matrix defined as


andΦM×M1is given in(3) andPM×Mαis given by






taking fractional integration of both sides, we have


Now to approximate right-hand sides of above


where CMican be approximated by using (3) as


where entries of the vector XMican be calculated in generalized form as


evaluating this result for i = 0,1,2....m, we have


further writing


we get


Let us represent




Lemma 2.5.LetBMTtbe the function vector as defined in(3), then fractional order derivative is defined as


where GM×Mαis the operational matrix of fractional order derivative given by


whereΦM×Mis the dual matrix given in(3) and


where Ψijis defined for two different cases as

Case I:i<α


Case II:iα


Proof.Consider the general element as


It is to be noted in the polynomial function Bi,mthe power of the variable ‘t’ is an ascending order and the lowest power is ‘i’ therefore the first α1terms becomes zero when we take derivative of order α.

Case I: i<αBy the use of definition of fractional derivative


Now approximating RHS of (25) as


further implies that


Case II: iαif iαthen


After careful simplification, we get


On further simplification, we have

Ψi,j=k=αmil=0mjθikm.θjlmΓk+i+1Γk+iα+11k+l+j+iα+1i<αΨi,j=k=0mil=0mjθikm.θjlmΓk+i+1Γk+iα+11k+l+j+iα+1we getDαBMTt=ĜM×MαΦM×M1.BMTt.E30





which is the desired result. □

Lemma 2.6.An operational matrix corresponding to the boundary condition by takingBMTtis function vector andKis coefficient vector by taking the approximation


is given by




Proof.Let us take ut=KB̂t, then


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 Δi,mϕt=i=0mĉiBit=CMiBMT,and using Eq. (3) we get CMi=KMiΦM×M1BMTand using cj=01ϕtBjtdt,


On further simplification, we get






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

Dαyt+ADμyt+Byt=ft,1<α2,0<μ1,subject to the boundary conditionsy0=a,y1=b,E37

where ftis a source term; A,Bare any real constants and ytis 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 αwe have


using boundary conditions, we have


Using the approximation and Lemma 2.2



yt=KMPM×MαBMTt+a+tbatKMPM×MαBMTtt=1,which givesyt=KMPM×MαBMTt+FM1BMTtQM×Mα,ϕBMTt=KMPM×MαQM×Mα,ϕBMTt+FM1BMTt.E39





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 KM. When we find KM, then putting this in Eq. (39), we get the required approximate solution of the problem.

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 Ai,Bi,Ci,Dii=12are any real constants, ft,gtare given source terms. We approximate the solution of the above system in terms of Bernstein polynomials such as


applying boundary conditions, we have


let us approximate






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 KMLM. Calculating the coefficient matrix KM,LMand putting it in equations


we get the required approximate solution.

4. Applications of our method to some examples

Example 4.1.Consider


subject to the boundary conditions


Solution:We solve this problem under the following parameters sets defined as

S1=α=2ν=1c1=1c2=1,S2=α=1.8ν=0.8c1=10c2=100,S3=α=1.5ν=0.5c1=1/10c2=1/100,and select source term forS1as


The exact solution of the above problem is


We solve this problem with the proposed method under different sets of parameters as defined inS1,S2,S3. 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 S1 is displayed inFigure 1 subplot (a), while inFigure 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 than1012.The order of derivatives in this set is an integer.

Figure 1.

(a) Comparison of exact and approximate solution of Example 4.1, under parameters setS1. (b) Absolute error in the approximate solution of Example 4.1, under parameters setS1.

By solving the problem under parameters set S2 and S3, we observe the same phenomena. The approximate solution matches very well with the exact solution. SeeFigures 2 and3 respectively.

Figure 2.

(a) Comparison of exact and approximate solution of Example 4.1, under parameters setS2. (b) Absolute error in the approximate solution of Example 4.1, under parameters setS2.

Figure 3.

(a) Comparison of exact and approximate solution of Example 4.1, under parameters setS3. (b) Absolute error in the approximate solution of Example 4.1, under parameters setS3.

Example 4.2.Consider


subject to the boundary conditions


Solution:The exact solution of the above problem isyt=sin2t,whenα=2. However the exact solution at fractional order is not known. We use the well-known property of FDEs that whenα2,the approximate solution approaches the exact solution for the evaluations of approximate solutions and check the accuracy by using different scale levels. By increasing the scale level M, the accuracy is also increased. By the proposed method, the graph of exact and approximate solutions for different values of M and atα=1.7is shown inFigure 4 . From the plot, we observe that the approximate solution becomes equal to the exact solution atα=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 inFigure 4 subplot (b) and accuracy approaches 10−9, which is a highly acceptable figure. For convergence of our proposed method, we examined the quantity01yexactyapproxdtfor 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 inFigure 4b .

Figure 4.

(a) Comparison of exact and approximate solution of Example 4.2. (b) Absolute error for different scale level M of Example 4.2.

Example 4.3.Consider the following coupled system of fractional differential equations


subject to the boundary conditions


Solution:The exact solution is


We approximate the solution of this problem with this new method. The source terms are given by


In the givenFigure 5 , we have shown the comparison of exactxt,ytand approximatext,ytin subplot (a) and (b) respectively.

Figure 5.

Comparison of exact and approximate solution of Example 4.3 for different scale levelM.

As expected, the method provides a very good approximation to the solution of the problem. At first, we approximate the solutions of the problem atα=2because the exact solution atα=2is known. We observe that at very small scale levels, the method provides a very good approximation to the solution. We approximate the absolute error by the formula




We approximate the absolute error at different scale level of M, and observe that the absolute error is much less than1010at scale levelM=7, seeFigure 6 . We also approximate the solution at some fractional value ofαand observe that asα2the approximate solution approaches the exact solution, which guarantees the accuracy of the solution at fractional value ofα. Figure 6shows this phenomenon. InFigure 6 , the subplot (a) represents the absolute error ofxtand subplot (b) represents the absolute error ofyt.

Figure 6.

Absolute error in approximate solutions at different scale levelM=3:7 for Example 4.3.

Example 4.4.Consider the following coupled system


subject to the boundary conditions


Solution:The exact solution forα=β=2is


The source terms are given by


Approximating the solution with the proposed method, we observe that our scheme gives high accuracy of approximate solution. InFigure 7 , we plot the exact solutions together with the approximate solutions inFigure 7(a) and (b) forxtandyt, 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 below1010inFigure 8 , which indicates better accuracy of our proposed method for such types of practical problems of applied sciences.

Figure 7.

Comparison of exact and approximate solution at scale levelM=3, 7 for Example 4.4.

Figure 8.

Absolute error for different scale levelM=3:7 for Example 4.4.

InFigure 8 , the subplot (a) represents absolute error forxtwhile subplot (b) represents the same quantity foryt. From the subplots, we see that maximum absolute error for our proposed method for the given problem (4.4) is below1010.This is very small and justifies the efficiency of our constructed method.

Example 4.5.Consider the boundary value problem


Takingα=2,ν=1andω=1,3,5,,the exact solution is given by


We plot the comparison between exact and approximate solutions to the given example atM=10and corresponding toω=3.5,α=2,β=1. Further, we approximate the solution through Legendre wavelet method (LWM)[47], Jacobi polynomial method (JM) and Bernstein polynomials method (BM), as shown inFigure 9 .

Figure 9.

(a) Comparison of exact and approximate solution at scale levelM=10,ω=3.5,α=2,ν=1for Example 4.5. (b) Absolute error atM=10.

From Table 1 , we see that Bernstein polynomials also provide excellent solutions to fractional differential equations [48].

ωMανxappxexat BMxappxexat WMxappxexat JM

Table 1.

Comparison of solution between Legendre wavelet method (LWM) [47], Jacobi polynomial method (JM) and Bernstein polynomials method (BM) for Example 4.5.

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 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.



This research work has been supported by Higher Education Department (HED) of Khyber Pakhtankhwa Government under grant No: HEREF-46 and Higher Education Commission of Pakistan under grant No: 10039.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kamal Shah, Thabet Abdeljawad, Hammad Khalil and Rahmat Ali Khan (January 27th 2020). Approximate Solutions of Some Boundary Value Problems by Using Operational Matrices of Bernstein Polynomials, Functional Calculus, Kamal Shah and Baver Okutmuştur, IntechOpen, DOI: 10.5772/intechopen.90302. Available from:

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