Solving the Interval Riccati differential equation by Wavelet operational matrix method

Riccati differential equation is an important equation, in many fields of engineering and applied sciences, so recently lots of methods have been proposed to solve this equation. Haar Wavelet operational matrix,is one of the effective methods to solve this equation, that is very simple and easy, compared to other orders. In this paper, we want to solve the nonlinear riccati differential equation in interval initial condition. first we simplify it by using the block pulse function to expand the Haar wavelet one. we have three cases for each interval, but now it can be solved for positive interval Haar coefficients. The results reveal that the proposed method is very effective and simple.


Introduction
Riccati differential equation is an important equation in the transmission line phenomena, theory of random processes, optimal control theory and diffusion problems.Since the beginning of the 1980s, the Adomian decomposition method (ADM) has been applied to a wide class of functional equations [4,5].In this method the solution is given as an infinite series usually converging to an accurate solution.El-Tawil et al. applied the multistage Adomian's decomposition method for solving Riccati differential equation and compared the results with standard ADM [8] Abbasbandy solved quadratic Riccati differential equation using He's VIM [1], homotopy perturbation method (HPM) [2] and iterated He's HPM [3] and compared the accuracy of the obtained solution with that derived by Adomian decomposition method.Moreover, there are other methods for solving Riccati differential equations, which are the Homotopy analysis method (HAM) [17] and the piecewise variational iteration method (VIM) [10].Limitations and demerits of the abovementioned methods have been commented by Mohammadi [14].In 2013, Wavelet operational matrix method, which was a very effective and convenient method, was proposed to solve nonlinear fractional Riccati differential equations.Compared to ADM, HPM, and VIM, HWOMM was very simple and easy to http://www.ispacs.com/journals/jfsva/2016/jfsva-00288/International Scientific Publications and Consulting Services implement.Moreover it was able to approximate the solution more accurate in a bigger interval.[13] In this paper we want to change the initial conditions, and solve interval Riccati equation.The Riccatti differential equation is defined as 2 ( ) ( ) ( ) ( ) ( )[ ( )] , 1 , 0, D y t u t v t y t w t y t n n t subject to the interval initial conditions () (0) , 0,1, , 1, where α is a parameter describing the order of fractional derivative, n is an integer, u(t), v(t) and w(t) are given functions, and k g is an interval.When α is a positive integer, the fractional equation becomes the Classical Riccati differential equation.The article is organized as follows: first we introduce some necessary definitions and mathematical preliminaries of the fractional calculus which are required for establishing our results.Then, in section 3 we define three cases for our three different kinds of intervals, then we generalize the Haar wavelet operational matrix.Section 4 is for applications, so we approximate the functions and use them in our riccati differential equation.In the fifth section we have a numerical example.Also a conclusion is given in section 6.

Preliminaries and notations
The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notations of integer-order differentiation of the fractional calculus which are used in this paper.The Riemann-Liouville fractional integration, for a function f(t),of order α>0 is defined as [7] 11 and its fractional derivative of order α > 0 is normally used: Where n is an integer.For Riemann-Liouvilles definition,one has In this paper,we shall introduce a modified fractional differential operator D α proposed by caputo [6]  Where n is an integer.caputosintegral operator has a useful property: Where n is an integer.For more details on fractional calculus you can see [7].

Generalized Haar wavelet operational matrix to fractional integration
Haar wavelets is the oldest and most basic of the wavelet systems.it is a group of square waves with magnitudes of ±1 in certain intervals and zero outside of them, i.e.

 
Because we are solving the equation in interval initial conditions,most of the coefficients and functions will be intervals.Now we define three cases: The wavelet series representation of the function f(t)ϵL 2 ([0,1)) in terms of an orthonormal basis is given by We define "^" sign for interval functions or vectors, then we have while the interval Haar coefficients ˆi c , i = 0,1,2, …, are defined as follows:   n = 2 j + k for j ≥ 0 and 0 ≤ k ≤ 2 j − 1 as standard and non-standard difference, the following integral square error ε is minimized Standard difference: Non-standard difference: by applying the orthogonal property of Haar wavelet Since it is not realistic to use an infinite number of wavelets to represent the function f(t), Eq.(3.12) terminates at a finite term and we use the following wavelet representation of 1 ˆ() ft of the function f(t): Where m = 2 j+1 , the superscript T indicates transposition,we define Haar coefficient vector C m , Haar function vector H m (t) and m-square Haar matrix .
Taking the collocation points as following Then we have T the coefficient vectors have been proposed in ref [13] The integration of the   m Ht defined in Eq. (3.19) can be approximated by Haar series with Haar coefficient matrix P [11].
As the Haar functions are piecewise constant, it may be expanded into an m-term BPF as [12] ,kilicman and zhour have given Block Pulse operational matrix of the fractional order integration then, the Haar wavelet operational matrix of the fractional integration P m×m α is given by 1 .

Applications of the generalized Haar wavelet operational matrix
In the previous section we generalized Haar wavelet operational matrix ,then,in this part we will use it to solve nonlinear Riccati differential equation Eq. (1.1) in the interval initial condition expressed by Eq. (1.2).Because are in the Haar wavelet region, we can approximate them as follows: where U, V, W, C and H m (t), are given in Eqs.(3.19).
Then we can rewrite equation (1.1) : And suppose 0, C 0 ( )  then, the equations are the same as 1 st 4 th : suppose  then, the equations are the same as 2 nd 5 th : suppose  then, the equations are the same as 6 th 8 th : suppose now we want to expand Eq (4.28).First making the linear term: using Eq (3.24) we have then by using different positions of X we have: As definition of Block-Pulse Function (BPF) in Ref [13] we have Then, we deal with the nonlinear term of the Eq.(4.28).From Eq. (3.24), we can rewrite which together with Eq. (4.42) or (4.43), leads to   Where U, V,W and ϕ m×m are known.Noting that the Haar coefficients of vector C are included in the two diagonal matrices of Eq. (51-52), the latter represents a nonlinear system of equation for unknown vector C.This nonlinear system of equation can be solved by the Newton method.After finding the unknown vector C we can get the approximation solution by inserting C into Eq.(4.29).

Conclusion
We proposed the Haar wavelet operational matrix method, to solve nonlinear fractional differential equations, in interval initial conditions.first we simplified the nonlinear term by using Block-Pulse function to expand Haar wavelet one.Then we generalized the Haar wavelet operational matrix to fractional