Haar Wavelet-picard Technique for Fractional Order Nonlinear Initial and Boundary Value Problems

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution.


INTRODUCTION
Haar wavelet is the lowest member of the Daubechies family of wavelets and is convenient for computer implementations due to the availability of explicit expression for the Haar scaling and wavelet functions (Daubechies, 1990).Operational approach is pioneered by Chen (Chen and Hsiao, 1997) for uniform grids.The basic idea of Haar wavelet technique is to convert differential equations into a system of algebraic equations of finite variables.
Boundary value problems are considerably more difficult to deal with than the initial value problems.The Haar wavelet method for boundary value problems is more complicated than for initial value problems.Secondorder boundary value problems are solved in Siraj-ul-Islam et al. (2010) by the Haar wavelets; they considered the six sets of different boundary conditions for the solution.Boundary value problems for fractional differential equations are solved in Mujeeb and Khan (2012) which considers the numerical solution by the Haar wavelet for different boundary value problems of fractional order.
The Picard approach (Bellman and Kalaba, 1965) is used to linearize the individual or system of nonlinear ordinary and partial differential equations.In this paper, we consider the case of fractional order nonlinear ordinary differential equations which contain various forms of nonlinearity.The main aim of the present paper is to get the numerical solutions of nonlinear fractional order initial ad boundary value problems over a uniform grids with a simple method based on the Haar wavelets and Picard technique.

THE HAAR WAVELETS
The Haar functions contain just one wavelet during some

Convergence of Picard technique
Consider the nonlinear second order differential equation ( 5) Application of Picard technique to (5) yields ( 6) Let be some initial approximation.Each function is a solution of the linear equation ( 6), where is always considered known and is obtained from the previous iteration.
According to Picard iteration: , where .( 7) This shows that there is linear convergence, if there is convergence at all.

Convergence of Haar wavelet method
Let be a differentiable function and assume that have bounded first derivative on , that is, there exists ; for all Haar wavelet approximation for the function is given by Babolian and Shahsavaran (2009) gave -error norm for Haar wavelet approximation, which is , or where and is the maximal level of resolution.From inequality (8), we conclude that error is inversely proportional to the level of resolution.Equation ( 8) ensures the convergence of Haar wavelet approximation at higher level of resolution, that is, when M is increased.

APPLICATIONS
Here, we solve nonlinear differential equations of fractional order by the Haar wavelets-Picard technique and compare the results with the exact solution.Throughout this work we use Caputo derivatives and for the details of fractional derivatives and integrals we refer the readers to Podlubny (1999).We fix the order of the differential Equation ( 9), and the level of resolution, .The graph in Figure 1 shows the exact and approximate solutions by proposed method at four iterations.The absolute error reduces with the increasing iterations.
Results at fifth iteration of proposed method at fixed level of resolution, and at different values of are shown in Figure 2 with exact solution at . Figure 2 showed that numerical solutions converge to the exact solution when approaches to 2.    with the initial approximation .Here we consider and .We fix the order of the differential Equation 28, , and level of resolution, .The graph in Figure 7 shows the exact and approximate solutions by proposed method at six iterations.The absolute error reduces with increasing iterations.
Results at sixth iteration of proposed method at fixed level of resolution, , and at different values of are shown in Figure 8 with the exact solution at Figure 8 showed that the numerical solutions converge to the exact solution when approaches to 2.

Conclusion
This study showed that Haar wavelet-Picard technique gives excellent results when applied to different fractional order nonlinear initial and boundary value problems.
The solution of the fractional order, nonlinear differential equation converge to the solution of the integer order differential equation as shown in Figures 2,  4, 6 and 8.
Other Figures shows that approximate solution converge to the exact solution while iterations are increased and absolute error goes down.
Different type of nonlinearities can easily be handled by the Haar wavelet-Picard technique.

Fractional
subinterval of time and remains zero elsewhere and are orthogonal.The Haar wavelets are useful for the treatment of solution of differential equations (Chen and Hsiao, 1997).The ith Haar wavelet is interval [a, b], is the Haar scaling function.For the Haar wavelet, the wavelet collocation method is applied.The collocation points are usually taken as our work is based on Picard technique and Haar wavelet method, we thus analyze the convergence of both schemes.
iteration to the Equation 9, we get (11) with the initial condition Now we apply the Haar wavelet method to Equation 11; we approximate the higher order derivative term by the Haar wavelet series as (12) Lower order derivatives are obtained by integrating Equation 12 and using the initial condition (13) Substituting Equations 12 and 13 in Equation 11, we get (14) with the initial approximation

Example 2 :Figure 1 .
Figure 1.Comparison of exact solution and solutions by Haar wavelet-Picard technique at J = 5; for different iterations, and α=2.

Figure 7 .
Figure 7.Comparison of exact solution and solutions by Haar wavelet-Picard technique at J = 5, for different iterations, and α = 2.
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