Application of Fractional Order Legendre Polynomials: a New Procedure for Solution of Linear and Nonlinear Fractional Differential Equations under m-point Nonlocal Boundary Conditions

In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear) under m̂-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the boundary value problem to a system of easily solvable algebraic equations. We show the applicability of the scheme by solving some test problems. The scheme is computer oriented.


Introduction
It is well known that fractional order differential equations provide more accurate results as compare to integer order differential equations.Fractional calculus is generalization of integer order integration and differentiation to its non-integer (fractional) order counterpart.Fractional calculus has proved to be a valuable tool in modeling various phenomena of physics, chemistry, engineering, aerodynamics, electrodynamics of complex medium, polymer rheology, etc. [1][2][3].In the last three decades, a tremendous amount of work is devoted to the study of fractional order differential equations and partial differential equations.In most cases it is necessary to solve fractional order differential equations, and the difficulty arise when the system has to be solved under complicated types of boundary conditions.The aim of this paper is to establish a scheme for the approximate solution of fractional order differential equations (linear and non linear) under m-point nonlocal boundary conditions.m-point boundary value problems appear in wave propagation and in elastic stability.For example, the vibrations of a guy wire of a uniform cross-section, composed of m sections of different densities can be molded as a m-point boundary value problem ( see [4,5] and the reference quoted there).The basic problem to be discussed in this paper is to find the approximate solution of the following classes of fractional differential equations (1.1) where the order of derivatives are defined as ) is given source function.In (1.1) b i are real constants.b i (t) in (1.2) are coefficients well defined on [0, τ].In (1.3) f is nonlinear function of U(t) and its fractional derivatives.In the above equations U(t) is the unknown solution to be determined under the following nonlocal m-point boundary conditions (1.4) Where ζ i are not all equal to zero, and η i are defined as 0 To approximate the solution of the above mentioned problems we use Generalized fractional order Legendre polynomials.These polynomials are recently applied by Yiming Chen [6] to approximate solution of fractional order partial differential equations with variable coefficients.We develop some new operational matrices and use them along with the matrices developed in [6] to develop a new efficient and simple method to approximate solution of non local boundary value problems.The method we have proposed is a spectral method.Spectral methods are extensively used in the solution of differential equations, function approximation, and variational problems (see, e.g., [7,8] and the references therein).
Recently spectral method gained attention of many authors.Among others, some of the well known mathematicians who successfully applied spectral method are A. H. Bhrawy [9,10], E.H. Doha [11][12][13][14][15][16], A. Saadatmandi [17].In these papers the authors solved many scientific problems using spectral methods.Actually we are motivated by the work of A. H Bhrawy [18], in which the author solved fractional order differential equations (including linear and nonlinear differential equations) with m-point boundary conditions (local) using shifted Legendre polynomials and some collocation technique.The method they proposed can handle local boundary conditions, however nonlocal boundary conditions are difficult to handle.This motivated us to use orthogonal polynomials for the numerical simulation of linear and nonlinear fractional differential equations with m-point nonlocal boundary conditions.The operational matrices that we introduced has the ability to convert the fractional order non local boundary value problem to a system of easily solvable algebraic equations.The resultant algebraic equations are linear and can be easily solved.Also we introduce the application of operational matrices together with Quasilinearization to solve the nonlinear problems under the same nonlocal boundary value problems.
The Quasilinearization method was introduced by Bellman and Kalaba [19] to solve nonlinear ordinary or partial differential equations as a generalization of the Newton-Raphson method.The origin of this method lies in the theory of dynamic programming.In this method, the nonlinear equations are expressed as a sequence of linear equations and these equations are solved recursively.The main advantage of this method is that it converges monotonically and quadratically to the exact solution of the original equations [20].Also some other interesting works in which qasi-linearization method is applied to scientific problems are [21][22][23][24].In our previous work we construct some efficient methods for the numerical simulation of couple system of fractional differential equations and fractional order partial differential equations [25][26][27][28][29][30][31][32][33][34][35].In these papers we only solve linear fractional differential equations subject to initial conditions.For the readers, new to the field, we recommend to study our previous work in order to get a better understanding.
Many authors are working on the establishment of basic results like existence and uniqueness of solution.In 1992 Guptta [35] studied the solvability of three point boundary value problem.Since then many researchers are working in this area and provide many useful results which guarantee the solvability and existence of solution of such problems.For interested reader in the existence theory of the problem we refer to study the survey paper presented by Ruyun Ma [36] in which the author presented a detail survey on this topic.In [37] the author showed the existence of positive solution of a general third order multi point boundary value problem.The existence of solution of such types of problems are also discussed by El-Sayed in [38].
The article is organized as follows: In section 2 we recall some definition and basic properties of Generalized fractional order Legendre polynomials and approximations theory.In section 3 we derive new operational matrices.We also recall some previously derived operational matrices which are helpful in our further investigation.In section 4 the new matrices are used to provide a theoretical treatment to the corresponding problem.In section 5 some test problems are solved.Results are displayed in tables and figures.Finally in the last section a short conclusion is made.

Preliminaries and Notation
In this section, we recall some basic definitions and known results from fractional calculus which are important for our further investigation.we refer to [1,2] for more detail.
provided the integral on right hand side exists.
Definition 2.2.For a given function ϕ (t) ∈ C n [a, b], the Caputo fractional order derivative of order σ is defined as provided the right side is point wise defined on (a, ∞), where n = [σ ] + 1 in case σ not an integer and n = σ in case σ is an integer.

The Generalized fractional-order Legendre polynomials
The GFL polynomials on the interval [0, τ] is defined as (see for example [6]) where (2.8) The GFL polynomials are orthogonal on the interval [0, τ].
Theorem 2.1.The GFL polynomials are orthogonal on the interval [0, τ] with respect to the weight function w α (t) = t α−1 .Then the orthogonality condition is (2.9) Proof.For the proof of this theorem see [6] 2

.2 Function Approximation with GFL polynomials
The orthogonality condition of the GFl polynomials allows us to expand any function f (t) ∈ L 2 [0, τ] in terms of GFL polynomials as follows: where the value of c i are obtained by the relation.
In practice we are interested in the truncated series of (2.10), so we can write the infinite series as where and is the best approximation to v(t) from P α m , then the error bound is presented as follows Proof.For the proof of this theorem see [6].
Theorem 2.3.The definite integral of product of three different GFL polynomials and weight function is constant, and the generalized constant is given by relation where (2.16) Proof.Consider the expression (2.17) The integral in the above relation derived as Then using (2.19) and (2.18) in (2.17) we get the desire proof.
The constant derived in the previous theorem is also used in [6], where the author use the relation to derive operational matrix for product.Here we use this constant to derive operational matrix for the fractional differential equations with variable coefficients.

Operational Matrices
In this section we generalize some new operational matrices.These matrices are used to convert the linear and nonlinear fractional differential equations to a system of easily solvable algebraic equations.Some of these proof are also available in different papers, but to make the paper self contained we gave a detail proof of these results.Lemma 3.1.Let Φ(t) be the function vector as defined in (2.13), then the fractional derivative of order σ of Φ(t) is generalized as where is the operational matrix of derivative of order σ , and is defined as Where the entries are defined as Proof.The proof of this lemma is available in [6].
Lemma 3.2.Let Φ(t) be the function vector as defined in (2.13), then the fractional integral of order σ of Φ(t) is generalized as where is the operational matrix of integration of order σ defined by the relation where the entries are defined as

.25)
Proof.Consider the general term Fl (α,τ) i (t), then we may write We can write where ĉ j can be easily calculate as which implies that Using relation (3.29) in (3.26), and after a short simplification we get Using the notation , we can also write it in simplified form as Evaluating (3.31) The following two operational matrices are of basic importance in our further analysis.
Lemma 3.3.Let U(t) and ϕ (t) be any two functions defined in [0, τ], also U(t) = KΦ(t) Then Where R (σ ,ϕ ) (M×M) is the operational matrix related to function ϕ (t) and σ , and is defined as The matrix Z (σ ,τ) (M×M) is the operational matrix of derivative as defined in Lemma 3.1 and where Where Θ α,τ (i,r,s) is similar as defined in Theorem 2.3, and Proof.Consider U(t) ≃ KΦ(t), then by the implication of Lemma 3.1 we can easily write Now in order to write the product in the form of matrix, we may write where ϕ (t) can be easily approximated with GFL polynomials, Using (3.38) in (3.37) we may write, (using more generalized notation) where Now consider the general term ℑ r (t), we can approximate it with GFL polynomials as where d (r,s) can be calculated using the relation Now repeating the procedure for r = 0, 1, In simplified notation we can write the above equation as M×M completes the proof of the Lemma.
The operational matrix developed in the previous lemma is of basic importance in the solution of Fractional differential equations(FDEs) with variable coefficients.In solving m-point boundary value problem the following matrix will play important role.
Lemma 3.4.Let ϕ c n = ct n , where c and n are real constants.Assume U(t) = KΦ(t) , then for where (M×M) is operational matrix related to η and ϕ c n , and is defined as where the entries d (i, j) are defined by the relation Then using the definition of fractional integral (2.5) By evaluating the integral and simplification we can write For simplicity of notation let We may also write Where d (i, j) can be derived as Which can be simplified as Or on further simplification we get (3.59) Now using (3.55) in (3.54) we may write Which can be written in matrix form as Where the entries of the matrix (M×M) are as defined in (3.59).And hence the proof of the lemma is complete.
The matrices derived in this section are of basic importance in the proposed method.

Application of the Operational Matrices
Now, we are in the position to state our main result.The operational matrices are used to convert fractional differential equations to system of easily solvable algebraic equations.We gave a detail procedure for solution of three classes of fractional differential equations.These equations are solved subject to m-point boundary conditions.

Linear Fractional Differential equations
Consider the following linear system of fractional differential equations subject to nonlocal m-point boundary conditions as where ζ i are all real constant and 0 We seek the solution of the problem in terms of shifted GFL polynomials such that the following holds.
By the application of fractional integral of order σ , and making use of Lemma 3.2 we get where d n ′ s are constants of integration.On using the n − 1 initial conditions we get the first n − 1 constants.
∑ l=0 u l t l + d n t n , (4.66) d n is up to now unknown.Using the m-point boundary condition we get From (4.63), we see that left sides of (4.67) and (4.68) are equal, therefore for the sack to obtain the value of d n , we can write On further simplification we get where l=0 ζ i u l η l i }.Now, using (4.70) in (4.66) we get Now, in view of Lemma 3.4 we can write (4.71) as where ∑ (n−1)

On further simplification we can write
(4.73) Where S = P (σ ,τ) (M×M) }.Now using (4.73) and Lemma 3.1 we may write where F 2 Φ(t) = f (t).On further simplification we can write Now the fractional order differential equations is converted to linear matrix equation.This equation is solvable for the unknown vector K. Using the value of K in (4.73) will lead us to the approximate solution of the problem.

FDEs with variable coefficients
Consider the following class of fractional order differential equations With the following m-point conditions where ζ i are all real constant and 0 Then repeating the same steps as in the previous section from (4.64) to (4.73) we get Here S is defined analogously as in previous section.Now, using (4.81) and Lemma 3.3 we can write

.83)
After simplification and cancelling out the common term as in previous part we can get system of algebraic equations as Equation (4.84) is system of easily solvable matrix equation.Which can easily solved for the unknown K. Using the value of K in (4.81) will lead us to the approximate solution of the problem.

Non linear FDEs with m-point boundary conditions
The nonlinear fractional order differential equations can be solved by using operational matrix method combined with qasilinearization method.The general procedure of qasilinearization method is as given in the following steps.
• Solve the linear part of nonlinear differential equations under the m−point nonlocal boundary conditions using the method developed in section 4.1.And label the solution of this linear part as U o (t).
• Linearize the nonlinear part of the differential equation at U o (t) with the help of multivariate Taylor series expansion, the linearized equation is now a linear differential equation with variable equation.Solve this linearized equation by the method developed in section 4.2.Label the solution as U 1 (t), and is the solution of the nonlinear problem at first iteration.
• Linearize the equations at U 1 (t), and again solve the linearized problem,and obtain the solution at 2nd iteration.
In this way a recurrence relation starts.
• The solution of the qasilinearization method converges to the exact solution of the problem in the sense that the difference between any two consecutive solution becomes smaller and smaller as we proceed the iteration.
The whole process can be seen as a recurrence relation.Consider the following nonlinear fractional differential equation.
First solve the linear part under the given nonlocal conditions This equation can be easily solved using the method developed in section 4.1.The solution of this part is labelled as U 0 (t).The next step is to linearize the nonlinear part with multi variate Taylor series expansion.So after linearizing and simplification we get (4.87) The above equation is a fractional order differential equation with variable coefficients.And can be easily solved with the method developed in section 4.2.The solution at this stage will be labelled as U 1 (t) and is the solution of the problem at first iteration.Again we have to linearize the problem about U 1 (t) to obtain the solution at second iteration.
The whole process can be seen as a recurrence relation like

.88)
And the boundary conditions becomes and It can be easily noted that (4.88) is fractional differential equation with variable coefficients.Using the initial solution X 0 (t) we may start the ittertions.The coefficients c i (t) and the source term F(t) can be updated at every iteration r to get the next solution at r + 1.At every step we may solve the problem at given nonlocal boundary conditions.

Test Problem
In order to show the efficiency of the method we solve some test problems whose exact solution in known .The results are displayed graphically.We solve the problems with different scale level.To show the accuracy of the method we measure the absolute difference of exact and approximate solution.The following notation is used in the figures.U M represents the quantity |U(t) − U M (t)|, where U M (t) is the approximate solution at scale level M. We calculate the quantity |dt at different scale level to show the convergence of the approximate solution with the increase of scale level.

Note:
The above calculation is obtained using computational software MatLab, and maybe the results are more accurate by simulating the algorithm using more powerful softwares.   .We solve this problem with the proposed method.We found that the method works well, and the approximate solution is in good agreement with the exact solution.The results of this problem are displayed in     36 , the exact solution of the problem is U(t) = e (t/6) We approximate the solution of this problem with the iterative method developed in section 4.3.We observed that as the iteration, N increases the solution becomes more and more accurate.Similarly the increase of scale level affects the speed of convergence of the iteration.This phenomena is shown in Fig 4, We calculate the value E U for this problem at every stage of iteration, we repeat this process using different scale level.It is clear that at high scale level the solution converge more faster as compare to low scale level.From Fig 4(d) it is seen that at scale level M = 7 and at iteration N = 6 the value E U is much more less than 10 −14 , which guarantees the high accuracy of the method.) .
The exact solution of this problem is U(t) = t 3 (t − 1) 2 .We approximate the solution of this problem with the proposed method.The iteration converges more rapidly by using high scale level.We calculate the value E U at different stage of iteration using different value of M. The results are displayed in Fig 5. We see that at high scale level the speed of convergence in more faster.And at scale level M = 6 the value E U is much more less than 10 −9 at seventh iteration.

Conclusion
From analysis and experimental results we observe that the proposed method is simple and provides a very high accurate estimate of the solution.The method can solve nonlocal boundary value problems very easily.In our test problems we consider up to 9-point boundary conditions and observe that the results obtained are satisfactory.The method also works well in solution of nonlinear FDEs with nonlocal conditions.We observe that by using high scale level the iteration converges more rapidly.As shown above at fifth iteration the norm of error is less than 10 − 8. However by using high scale level, much more accurate results can be achieved.Our future work is related to the extension of method in solution of fractional order partial differential equations with nonlocal boundary conditions.

Figure 1 :Example 5 . 3 .
Figure 1: (a) The comparison of approximate solution with exact solution of example 5.2 at different scale levels.(b) Absolute error at different scale levels.(c) E U vs M, convergence of the scheme.

Fig 2 .
In Fig 2 (a), the comparison of exact solution with the approximate solution at different scale levels is displayed.In Fig 2 (b) the absolute error is displayed at scale levels ranging from 4 to 16.We observed that the absolute error decrease with increasing of scale level.We also calculate the quantity E U for this example and the results are displayed in Fig 2 (c).

Figure 2 :Example 5 . 4 .
Figure 2: (a) The comparison of approximate solution with exact solution of example 5.3 at different scale levels.(b) Absolute error at different scale levels.(c) E U vs M, convergence of the scheme.

Figure 3 :
Figure 3: (a) The comparison of approximate solution with exact solution of example 5.4 at different scale levels.(b) Absolute error at different scale levels.(c) E U vs M, convergence of the scheme.

Table 1 :
[40]arison of absolute error of Example 5.1 obtained with proposed method(PM) and its comparison with absolute error obtained using Haar wavelets reported in[40].
We approximate the solution of this problem with the proposed method.The same conclusion is made.The approximate solution matches very well with the exact solution.The results are shown in Fig 3.From Fig3(c) one can see that the method is convergent.Also in (a) we see that the approximate solution is in good agrement with the exact solution.