Open access peer-reviewed chapter - ONLINE FIRST

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: 50

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 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 [8, 9, 11, 12, 13, 23, 24, 25, 26] 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 [1, 2, 3, 4, 5, 6, 7, 10]). 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 [15, 16, 17, 18, 19, 20, 21, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40]) 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 [14]. 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]. 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

Dαyt+ADμyt+Byt=ft,1<α2,0<μ1,y0=a,y1=b,E1

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

Dαxt+A1Dμ1xt+B1Dν1yt+C1xt+D1yt=ft,1<α2,0<μ1,ν11,Dβyt+A2Dμ2xt+B2Dν2yt+C2xt+D2yt=gt,1<β2,0<μ2,ν21,y0=a,y1=b,y0=c,y1=d,E2

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 [1, 2, 3, 4, 5, 6, 7].

Definition 2.1. The fractional integral of order γR+of a function yL101Ris defined as

I0+γyt=1Γγ0ttτγ1yτ.

Definition 2.2. The Caputo fractional order derivative of a function y on the interval 01is defined by

D0+γyt=1Γnγ0ttτnγ1ynτ,

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

Lemma 2.1. The fractional differential equation of order γ>0

Dγyt=0,n1<γn,

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

IγDγyt=yt+d0+c1t+d2t2++dn1tn1,

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

Hence it follows that

Dγtk=Γk+1Γkγ+1tkγ,Iγtk=Γk+1Γk+γ+1tk+γandDγconstant=0.

2.1 The Bernstein polynomials

The Bernstein polynomials Bi,mton 01can be defined as

Bi,mt=miti1tmi,fori=0,1,2m,

where mi=m!mi!i!,which on further simplification can be written in the most simplified form as

Bi,mt=k=0miΘikmtk+i,i=0,1,2m,E3

where

Θikm=1kmimik.

Note that the sum of the Bernstein polynomials converges to 1.

Lemma 2.3. Convergence Analysis: Assume that the function gCm+101that is m+1times continuously differentiable function and let X=B0,mB1,mBm,m. If CTΨxis the best approximation of gout of X, then the error bound is presented as

gCTΨ22MS2m+32Γm+22m+3,

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

Proof. In view of Taylor polynomials, we have

Fx=gx0+xx0g1x0+xx02Γ3g2++xx0mΓm+1gm,

from which we know that

gFx=gm+1ηxx0m+1Γm+2,there existη01.

Due the best approximation CTΨxof g, we have

gCTΨx22gF22=01gxFx2dx=01gm+1ηxx0m+1Γm+22dxM2Γ2m+201xx02m+2dx2M2S2m+3Γ2m+22m+3.

Hence we have

gCTΨx22MS2m+32Γm+22m+3.

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

<f,g>=01fx.gxdx

and

Y=spanB0,mB1,mBm,m

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

ft=i=0mciBim,E4

where coefficient ci can 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 implies thatd0d1.dm=c0c1.cmθ00θ01θ0rθ0mθ10θ11θ1rθ1mθr0θr1θrrθrmθm0θm1θmrθmm.E5

Let XM=d0d1.dm,CM=c0c1.cm,where M=m+1where M is the scale level and ΦM×M=θ00θ01θ0rθ0mθ10θ11θ1rθ1mθr0θr1θrrθrmθm0θm1θmrθmm,so

XM=CMΦM×MCM=XMΦM×M1.E6

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

ft=CMBMTt,CMis coefficient matrix

where

BMt=B0mB1m.Bmm.E7

Lemma 2.4. Let BMTtbe the function vector defined in (3), then the fractional order integration of BMTtis given by

IαBMTt=PM×MαBMTt,E8

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

PM×Mα=P̂M×MαΦM×M1

and ΦM×M1is given in (3) and PM×Mαis given by

P̂M×Mα=Ψ00Ψ01Ψ0rΨ0mΨ10Ψ11Ψ1rΨ1mΨr0Ψr1ΨrrΨrmΨm0Ψm1ΨmrΨmm,E9

where

Ψi,j=k=0mil=0mjθikmθjlmΓk+i+1i+j+k+l+α+1Γk+i+α+1.E10

Proof. Consider

Bi,mt=k=0miθikmtk+iE11

taking fractional integration of both sides, we have

IαBi,mt=k=0miθikmIαtk+i=k=0miθikmΓk+i+αΓk+i+α+1tk+i+α.E12

Now to approximate right-hand sides of above

k=0miθikmΓk+i+αΓk+i+α+1tk+i+α=CMiBMTtE13

where CMican be approximated by using (3) as

CMi=XMiΦM×M1,E14

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

XMj=01k=0miθikmΓk+i+αΓk+i+α+1tk+i+αBj,mtdt,j=0,1,2.mXMj=01k=0miθikmΓk+i+αΓk+i+α+1l=0mjθjlmtk+i+α.tl+jdt,j=0,1,2.m=k=0miθikml=0mjθjlmΓk+i+αΓk+i+α+11k+l+j+i+α+1,j=0,1,2,.mE15

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

IαB0,mtIαB1,mtIαBm,mt=XM0ΦM×M1BMTtXM1ΦM×M1BMTtXMmΦM×M1BMTt)E16

further writing

Ψi,j=k=0mil=0mjθikm.θjlmΓk+i+αΓk+i+α+11k+l+j+i+α+1

we get

IαBMTt=P̂M×Mα.ΦM×M1.BMTt.E17

Let us represent

P̂M×Mα.ΦM×M1=PM×Mα

thus

IαBMTt=PM×Mα.BMTt.E18

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

DαBMTt=GM×Mα.BMTtE19

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

GM×Mα=ĜM×MαΦM×M1,E20

where ΦM×Mis the dual matrix given in (3) and

ĜM×Mα=Ψ00Ψ01Ψ0rΨ0mΨ10Ψ11Ψ1rΨ1mΨr0Ψr1ΨrrΨrmΨm0Ψm1ΨmrΨmm,E21

where Ψijis defined for two different cases as

Case I: i<α

Ψi,j=k=αmil=0mjθikm.θjlmΓk+iαΓk+iα+11k+l+j+iα+1E22

Case II: iα

Ψi,j=k=0mil=0mjθikm.θjlmΓk+iαΓk+iα+11k+l+j+iα+1.E23

Proof. Consider the general element as

DαBi,mt=Dαk=0miθikm.tk+i=k=0miθikmDαtk+i.E24

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

DαBi,mt=k=αmiθikmΓk+i+1Γk+iα+1tk+iα.E25

Now approximating RHS of (25) as

k=αmiθikmΓk+i+1Γk+iα+1tk+iα=CMiBMTtE26

further implies that

XMj=01k=αmiθikmΓk+i+1Γk+iα+1tk+iαBj,mtdt,j=0,1,2.mXMj=k=αmiθikml=0mjθjlmΓk+i+1Γk+iα+1k+i+l+jα+1,j=0,1,2.mE27

Case II: iαif iαthen

XMj=k=0miθikml=0mjθjlmΓk+i+1Γk+iα+1k+i+l+jα+1,j=0,1,2.m.E28

After careful simplification, we get

DαB0,mtDαB1,mtDαBm,mt=XM0ΦM×M1BMTtXM1ΦM×M1BMTtXMmΦM×M1BMTt).E29

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

Let

ĜM×MαΦM×M1=GM×Mα

so

DαBMTt=GM×MαBMTt

which is the desired result. □

Lemma 2.6. An operational matrix corresponding to the boundary condition by taking BMTtis function vector and Kis coefficient vector by taking the approximation

ut=KB̂t

is given by

QM×Mα,ϕ=Ω00Ω01Ω0rΩ0mΩ10Ω11Ω1rΩ1mΩr0Ωr1ΩrrΩrmΩm0Ωm1ΩmrΩmm,E31

where

Ωi,j=01Δi,mϕtBjtdt,i,j=0,1,2.m.

Proof. Let us take ut=KB̂t, then

0I1αKB̂t=K0I1αB̂t=K0I1αB0t0I1αB1t0I1αBmt.

Let us evaluate the general terms

0I1αBitdt=1Γα011sα1Bi,msds=1Γαk=0miΘi,k,m011sα1sk+ids.E32

Since by

L011sα1sk+ids=ΓαΓk+i+1τk+α+i

taking inverse Laplace of both sides, we get

011sα1.sk+ids=L1Γα.Γk+i+1τk+α+i=Γα.Γk+i+1Γk+i+α+1

now Eq. (32) implies that

0I1αBitdt=k=0miΘi,k,mΓk+i+1Γk+i+α+1=Δi,m.E33

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

ϕtKIαB̂t=KΔ0,mϕtΔ1,mϕtΔm,mϕt=KCM0ΦM×M1BMTtCM1ΦM×M1BMTtCMmΦM×M1BMTt=Kc00c10cr0cm0c01c11cr1cm1c0rc1rcrrcmrc0mc1mcrmcmmΦM×M1BMTtΦM×M1BMTtΦM×M1BMTt.E34

On further simplification, we get

ϕtKIαB̂t=KΩ00Ω01Ω0rΩ0mΩ10Ω11Ω1rΩ1mΩr0Ωr1ΩrrΩrmΩm0Ωm1ΩmrΩmmB0tB1tBmt.E35

So

ϕt0I1αut=KQM×Mα,ϕBMTt,

and

Ωi,j=01Δi,mϕtBjtdt,i,j=0,1,2.m.E36

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

LetDαyt=KMBMTt.E38

Applying fractional integral of order αwe have

yt=KMPM×MαBMTtc0+c1t

using boundary conditions, we have

c0=ac1=baKMPM×MαBMTtt=1.

Using the approximation and Lemma 2.2

a+tba=FM1BMTttPM×MαBMTtt=1=QM×Mα,ϕBMTt.

Hence

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

Now

Dμyt=DμKMPM×MαQM×Mα,ϕBMTt+FM1BMTt=KMPM×MαQM×Mα,ϕGM×MμBMTt+FM1GM×MμBMTtE40

and

ft=FM2BMTt.E41

Putting Eqs. (38)(41) in Eq. (37), we get

KMBMT(t)+AKM(PM×MαQM×Mα,ϕ)GM×MμBMT(t)+AFM(1)GM×MμBMT(t)+BKM(PM×MαQM×Mα,ϕ)BMT(t)+BFM(1)BMT(t)=FM(2)BMT(t).E42

In simple form, we can write (42) as

KMBMTt+AKMPM×MαQM×Mα,ϕGM×MμBMTt+AFM1GM×MμBMTt+ BKMPM×MαQM×Mα,ϕBMTt+BFM1BMTtFM2BMTt=0KM+KMAP̂M×MαGM×Mμ+BP̂M×Mα+AFM1GM×Mμ+BFM1FM2,E43

where

P̂M×Mα=PM×MαQM×Mα,ϕ.

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

Dαxt+A1Dμ1xt+B1Dν1yt+C1xt+D1yt=ft,1<α2,0<μ1,ν11,Dβyt+A2Dμ2xt+B2Dν2yt+C2xt+D2yt=gt,1<β2,0<μ2,ν21,E44

subject to the boundary conditions

x0=a,x1=by0=c,y1=d,E45

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

Dαxt=KMBMTt,Dβyt=LMBMTtxt=KMPM×MαBMTt+c0+c1t,yt=LM(PM×MβBMTt+d0+d1t

applying boundary conditions, we have

xt=KM(PM×MαBMTt+a+tbatKMPM×MαBMTtt=1,yt=KM(PM×MβBMTt+c+tdctKMPM×MβBMTtt=1.

let us approximate

a+tba=FM1BMTt,c+tdc=FM2BMTttPM×MαBMTtt=1=QM×Mα,ϕBMTttPM×MβBMTtt=1=QM×Mβ,ϕBMTt

then

xt=KMPM×MαBMTt+FM1BMTtKMQM×Mα,ϕBMTtyt=LMPM×MβBMTt+FM2BMTtLMQM×Mβ,ϕBMTtDμ1xt=KMPM×MαBMTt+FM1BMTtKMQM×Mα,ϕBMTt=KMPM×MαQM×Mα,ϕGM×Mμ1+FM1GM×Mμ1BMTtDν1yt=Dν1LMPM×MβBMTt+FM2BMTtLMQM×Mβ,ϕBMTt=LMPM×MβQM×Mβ,ϕGM×Mν1+FM2GM×Mν1BMTtDμ2xt=Dμ2KMPM×MαBMTt+FM1BMTtKMQM×Mα,ϕBMTt=KMPM×MαQM×Mα,ϕGM×Mμ2+FM1GM×Mμ2BMTt

and

Dν2yt=Dν2KMPM×MβBMTt+FM2BMTtKMQM×Mβ,ϕBMTt=LMPM×MβQM×Mβ,ϕGM×Mν2+FM2GM×Mν2BMTtft=F3BMTt,gt=F4BMTt.

Thus system (44) implies that

KMBMTt+A1KMPM×MαQM×Mα,ϕGM×Mμ1+A1FM1GM×Mμ1BMTt+ B1LMPM×MβQM×Mβ,ϕGM×Mν1+B1FM2GM×Mν1BMTt+C1KMPM×MαBMTt+ C1FM1BMTtC1KMQM×Mα,ϕBMTt+D1LMPM×MβBMTt+D1FM2BMTt D1LMQM×Mβ,ϕBMTt=F3BMTtLMBMTt+A2KMPM×MαQM×Mα,ϕGM×Mμ2+A2FM1GM×Mμ2BMTt+ B2LMPM×MβQM×Mβ,ϕGM×Mν2+B2FM2GM×Mν2BMTt+C2KMPM×MαBMTt+ C2FM1BMTtC2KMQM×Mα,ϕBMTt+D2LMPM×MβBMTt+D2FM2BMTt D2LMQM×Mβ,ϕBMTt=F4BMTt.E46

Rearranging the terms in the above system and using the following notation for simplicity in Eq. (46)

Q̂M×Mα=A1PM×MαQM×Mα,ϕGM×Mμ1+C1PM×MαQM×Mα,ϕQ̂M×Mβ=B1PM×MβQM×Mβ,ϕGM×Mν1+D1PM×MβQM×Mβ,ϕR̂M×Mα=A2PM×MαQM×Mα,ϕGM×Mμ2+C2PM×MαQM×Mα,ϕR̂M×Mβ=B2PM×MβQM×Mβ,ϕGM×Mν2+D2PM×MβQM×Mβ,ϕFM=A1FM1GM×Mμ1+B1FM2GM×Mν1+C1FM1+FM2D1FM3GM=A2FM1GM×Mμ2+B2FM2GM×Mν2+C2FM1+D2FM2FM4,

the above system (46) becomes

KMBMT(t)+KMQ̂M×MαBMT(t)+LMQ̂M×MβBMT(t)+FMBMT(t)=0LMBMT(t)+KMR̂M×MαBMT(t)+LMR̂M×MβBMT(t)+GMBMT(t)=0[KMLM][BMT(t)00BMT(t)]+[KMLM][Q̂M×Mα00R̂M×Mβ][BMT(t)00BMT(t)]+[KMLM][0R̂M×MαQ̂M×Mβ0][BMT(t)00BMT(t)]+[FMGM][BMT(t)00BMT(t)]=0[KMLM]+[KMLM][Q̂M×MαR̂M×MαQ̂M×MβR̂M×Mβ]+[FMGM]=0,E47

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

xt=KMPM×MαBMTt+FM1BMTtKMQM×Mα,ϕBMTtyt=LMPM×MβBMTt+FM2BMTtLMQM×Mβ,ϕBMTt,

we get the required approximate solution.

4. Applications of our method to some examples

Example 4.1. Consider

Dαyt+c1Dνyt+c2yt=ft,1<α<2E48

subject to the boundary conditions

y0=0,y1=0.

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 for S1as

f1t=t6t13+t672t168+12630t4+3t53t2t12E49
f2t=11147682583723703125t2151750t34200t2+3255t806406548945561989414912+278692064593092578125t2655250t314350t2+12915t381325002760152062349017088+100t6t13,E50
f3t=5081767996463981t921344t33360t2+2730t715264146673456906240+5081767996463981t1121344t33808t2+3570t110522452467243837030400+t6t13100.E51

The exact solution of the above problem is

yt=t6t13.

We solve this problem with the proposed method under different sets of parameters as defined in S1,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 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 1012.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 set S1. (b) Absolute error in the approximate solution of Example 4.1, under parameters set S1.

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. See Figures 2 and 3 respectively.

Figure 2.

(a) Comparison of exact and approximate solution of Example 4.1, under parameters set S2. (b) Absolute error in the approximate solution of Example 4.1, under parameters set S2.

Figure 3.

(a) Comparison of exact and approximate solution of Example 4.1, under parameters set S3. (b) Absolute error in the approximate solution of Example 4.1, under parameters set S3.

Example 4.2. Consider

Dαyt2D0.9yt3yt=4cos2t7sin2tE52

subject to the boundary conditions

y0=0,y1=sin2.

Solution: The exact solution of the above problem is yt=sin2t,when α=2. However the exact solution at fractional order is not known. We use the well-known property of FDEs that when α2,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 at α=1.7is shown in Figure 4. From the plot, we observe that the approximate solution becomes equal to the exact solution at α=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 10−9, which is a highly acceptable figure. For convergence of our proposed method, we examined the quantity 01yexactyapproxdtfor 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.

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

D1.8xt+Dxt+9D0.8yt+2xtyt=ftD1.8yt6D0.8xt+Dytxt=gtE53

subject to the boundary conditions

x0=1,x1=2andy0=2,y1=2.

Solution: The exact solution is

xt=t51t,yt=t41t.

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

ft=2t5t1t4t1+t46t52229536516744740625t16510t71008806316530991104+1337721910046844375t16525t211008806316530991104E54
gt=t35t4t5t111147682583723703125t21515t13655724105745144217689181460669789625t11525t16144115188075855872.E55

In the given Figure 5, we have shown the comparison of exact xt,ytand approximate xt,ytin subplot (a) and (b) respectively.

Figure 5.

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

As expected, the method provides a very good approximation to the solution of the problem. At first, we approximate the solutions of the problem at α=2because the exact solution at α=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

Xerror=xexactxapprox.

and

Yerror=yexactyapprox.

We approximate the absolute error at different scale level of M, and observe that the absolute error is much less than 1010at scale level M=7, see Figure 6. We also approximate the solution at some fractional value of αand observe that as α2the approximate solution approaches the exact solution, which guarantees the accuracy of the solution at fractional value of α. Figure 6 shows this phenomenon. In Figure 6, the subplot (a) represents the absolute error of xtand subplot (b) represents the absolute error of yt.

Figure 6.

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

Example 4.4. Consider the following coupled system

D1.8xtxt+3yt=ftD1.8yt+4xt2yt=gt,E56

subject to the boundary conditions

x0=1,x1=1andy0=1,y1=1.

Solution: The exact solution for α=β=2is

xt=t5t41,andyt=t4t32.E57

The source terms are given by

ft=445907303348948125t3.225t213026418949592973312+t34t4+3t52,gt=89181460669789625t2.55t4144115188075858724t3+6t42t52.

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 xtand yt, 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 1010in Figure 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 level M = 3, 7 for Example 4.4.

Figure 8.

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

In Figure 8, the subplot (a) represents absolute error for xtwhile subplot (b) represents the same quantity for yt. From the subplots, we see that maximum absolute error for our proposed method for the given problem (4.4) is below 1010.This is very small and justifies the efficiency of our constructed method.

Example 4.5. Consider the boundary value problem

Dαxt+ωπ2Dνxt+xt=ωπsinωπt+ωπx0=0,x1=2.E58

Taking α=2,ν=1and ω=1,3,5,,the exact solution is given by

xt=cosωπt1.

We plot the comparison between exact and approximate solutions to the given example at M=10and corresponding to ω=3.5,α=2,β=1. Further, we approximate the solution through Legendre wavelet method (LWM) [48], Jacobi polynomial method (JM) and Bernstein polynomials method (BM), as shown in Figure 9.

Figure 9.

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

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

ωMανxappxexat BMxappxexat WMxappxexat JM
0.510217.00032.96611.5002
1.5151.60.96.09134.91821.6231
2.0201.80.81.23732.10822.7232
3.5251.90.71.00835.79521.8133

Table 1.

Comparison of solution between Legendre wavelet method (LWM) [48], 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.

Acknowledgments

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.

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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 [Online First], IntechOpen, DOI: 10.5772/intechopen.90302. Available from:

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