Applications of New Iterative Method to Fractional Non Linear Coupled ITO System

Nonlinear fractional partial differential equations arise in modeling of complicated physical phenomena. Unfortunately in such situation we have no analytical method that solve these nonlinear fractional models exactly and therefore researchers knock the door of approximate methods for finding solution of these problems. Varieties of methods in literature have been used to find solutions of these types of problems. These methods include numerical and semi analytical techniques. Some of well-known semi analytical techniques such as Adomian decomposition method (ADM) [2,3], Variational Iteration Method (VIM) [4,5], Homotopy Perturbation Method (HPM) [6,7,8,9], Homotopy Analysis Method (HAM) [10,11] and Optimal Homotopy Asymptotic Method (OHAM) [12,13,14], etc. [15,16,17,18] New Iterative Method (NIM) proposed by Daftardar-Gejji and Jafari is one of the most realistic techniques for solving Differential equations, Integro-differential equations and Differential Difference equations of both classical and fractional order. By means of more reliable polynomials called Jafari polynomials (NIM) handles nonlinear partial differential equations in accurate way.


Introduction
Nonlinear fractional partial differential equations arise in modeling of complicated physical phenomena. Unfortunately in such situation we have no analytical method that solve these nonlinear fractional models exactly and therefore researchers knock the door of approximate methods for finding solution of these problems. Varieties of methods in literature have been used to find solutions of these types of problems. These methods include numerical and semi analytical techniques. Some of well-known semi analytical techniques such as Adomian decomposition method (ADM) [2,3], Variational Iteration Method (VIM) [4,5], Homotopy Perturbation Method (HPM) [6,7,8,9], Homotopy Analysis Method (HAM) [10,11] and Optimal Homotopy Asymptotic Method (OHAM) [12,13,14], etc. [15,16,17,18] New Iterative Method (NIM) proposed by Daftardar-Gejji and Jafari is one of the most realistic techniques for solving Differential equations, Integro-differential equations and Differential Difference equations of both classical and fractional order. By means of more reliable polynomials called Jafari polynomials (NIM) handles nonlinear partial differential equations in accurate way.
Our aim in this paper is to find reliable solutions of time fractional nonlinear coupled ITO systems by using (NIM). The concept of nonlinear generalized coupled ITO system was first introduced in the literature by Masaaki ito in 1980 [19]. This system has various applications in science and engineering i.e. coupled ITO equations are used for continuous quantum state measurement and estimation [20]. Time fractional Coupled ITO systems under consideration in this paper are as follows: R. Nawaz, S. Farid and S. Bushnaq with initial conditions: withe initial conditions:

Preliminaries
In this section we state some definitions and results from the literature which are relevant to our work.
When we formulate the model of real world problems with fractional calculus, the Riemann-Liouville have certain disadvantages. Caputo proposed a modified fractional differential operator D α a in his work on the theory of viscoelasticity Definition 2.3. The fractional derivative of f (x) in Caputo sense is defined as One can find the properties of the operator I α a in [24,25]. We mention the following:

Basic theory of (NIM) [26,27]
Consider the following nonlinear equation: Where f is the given function, ξ, ψ are given linear and nonlinear functions of u(x), v(x) respectively.
Since ξ is linear, we can write ξ The nonlinear function ψ can be decomposed as The general solution can be written as We write The recurrence relation is defined as Hence k-terms approximate solution of eq.

Proof:
See [22]. Sufficient condition for convergence is as follows: G m is absolutely convergent.
Proof: See [22]. For more detail of convergence we refer [23] where illustrative example is solved for convergence analysis.
where 0 < α ≤ 1, 0 < β ≤ 1, 0 < γ ≤ 1 with conditions: where a, c are any constants. For α = β = γ = 1, the exact solution og the above system is: Applying I α , I β and I γ to both sides of system (4.1) and using definition 2.4. we have By using (NIM) and eq. (3.2) we have: , R. Nawaz, S. Farid and S. Bushnaq Therefore four terms approximate solution of u(x, t), v(x, t) and w(x, t) can be written in the form:

Conclusion
Approximate solutions of ITO systems with time fractional derivatives have been obtained by successful application of (NIM). In the recent development of fractional order differential equations in some fields of applied mathematics, conceive it necessary to inspect methods of solutions for such types of equations and we anticipate that this work is a step in towards solutions of fractional problems. It has been observed that approximate solutions by extended formulation are in excellent agreement with the exact solutions. Consistent accuracy throughout the domain of the problems is also observed. The accuracy of proposed method can further be increased by taking higher iterations.