NUMERICAL SOLUTION OF FRACTIONAL BOUNDARY VALUE PROBLEMS BY USING CHEBYSHEV WAVELET METHOD

In this paper Chebyshev Wavelets Method (CWM) is applied to obtain the numerical solutions of fractional fourth, sixth and eighth order linear and nonlinear boundary value problems. The solutions of the fractional order problems are shown to be convergent to the integer order solution of that problem. The computational work is done successfully with the help of the proposed algorithm and hence this algorithm can be extended to other physical problems. High level of accuracy is obtained by the present method.


INTRODUCTION
Fractional calculus has a number of applications in science and technology [1][2][3].The study of fractional calculus initially started by Gemant and Scot-Blair, they were the first, who proposed a fractional derivative model for Viscoelasticity and anomalous strain and stress [4,5].Fractional calculus is applied to many other physical phenomena such as frequency dependent damping behavior of many viscoelastic materials, oscillation of earth quakes, fluid-dynamic traffic, control theory and signal processing [6][7][8][9][10].
The different numerical methods are developed for the numerical solutions of different problems in various branches of sciences and engineering.In this regard, a relatively new numerical technique based on Wavelets is being developed.The most common Wavelets schemes are Haar Wavelets (HW), Harmonic Wavelets of successive approximation, Legendre Wavelets and CWM [11][12][13][14][15][16][17][18][19][20].In the present research work, the CWM is fully compatible with the complexity of the problems and has shown extremely accurate results, especially in case of fractional linear and nonlinear boundary problems of fourth, sixth and eighth order [21][22][23][24][25][26][27].Some other well -known methods for the solution of fractional differential equations are given in [28][29][30][31][32][33].

DEFINITIONS AND PRELIMINARIES CONCEPTS
In this section, we give some important definitions and preliminaries concepts about fractional calculus theory, which is the foundation for this paper [28].

Definition 2.1 The Riemann-Liouville fractional integral operator
This operator has the following properties: n is an integer.The derivative of this type has certain disadvantages dealing with the fractional differential equation.There after Caputo proposed a modified fractional differential operator.

Definition 2.3
Caputo proposed fractional differential operator is given by ( ) The Caputo operator has the following two properties:

CHEBYSHEV WAVELET METHOD (CWM)
Wavelets generally constitute a family of functions constructed from dilation and translation of single function

( )
x  which is called the mother wavelet.For different continuous parameters a and b of dilation and translation respectively, we obtain the following family of continuous wavelet [15].m is the degree of the second Chebyshev polynomials, and the normalized time.This CW family is defined on the interval

CHEBYSHEV WAVELET METHOD (CWM)
In this section, we consider the following fractional boundary value problems with the boundary conditions Assume that Equation (4.4) is exact at linear system or the nonlinear equations as the case may be occur for the problem.Same procedure can be extended to fractional differential equations of order sixth and eight.

METHOD IMPLEMENTATION
The analytical solution for this problem is    solutions The error associated with the present method and that of Optimal Homotopy Asymptotic Method (OHAM) method is compared.The table shows that the accuracy of the current method is higher than Optimal Homotopy Asymptotic Method (OHAM).
  1, 1→ , and satisfy the following recursive formula: Equation (4.1) can be expressed as a CW series of the form this, five conditions are given by the following boundary conditions: conditions can be obtained by putting Equation (4.2) in Equation (4.1) as

Problem 1 .
Consider the following fractional nonlinear boundary value problem of fourth order

. 4 y
The analysis of the absolute error between exact solution and approximate solution is done successfully.The numerical solutions obtained by CWM are compared with Optimal Homotopy Asymptotic Method (OHAM).In the table exacty represent the exact solution for Problem 1.The approximate solutions are obtained by Chebyshev Wavelet Method for different order  , that is for 25 and OHAM, shows the respective errors given by the CWM and Optimal Homotopy Asymptotic Method.

Figure 1 :Problem 2 .
Figure 1: The solution graph, by Chebyshev method for different fractional order  Problem 2. Given fractional order BVP

Figure 2 :Table 2 :.Problem 3 .Table 3 :
Figure 2: The Chebyshev solutions graph for the fractional differential equations of different order . Table 2: The numerical results for Problem 2 for different fractional order

Figure 3 :
Figure 3: The Chebyshev solutions graph for the fractional differential equations given in Problem 3 of different order  .

Table 1 :
Numerical results of Problem 1

Table 1 ,
shows the solutions given by Chebyshev method when