Solving Cauchy reaction-diffusion equation by using Picard method

In this paper, Picard method is proposed to solve the Cauchy reaction-diffusion equation with fuzzy initial condition under generalized H-differentiability. The existence and uniqueness of the solution and convergence of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method. Also, we obtain the switching points in examples.

Definition 2.1. An arbitrary fuzzy number u in the parametric form is represented by an ordered pair of functions (u, u) which satisfy the following requirements: (i) u : r → u(r) ∈ R is a bounded left-continuous non-decreasing function over [ 0, 1], (ii) u : r → u(r) ∈ R is a bounded left-continuous non-increasing function over [ 0, 1], (iii) u(r) ≤ u(r), 0 ≤ r ≤ 1.
Definition 2.2. For arbitrary fuzzy numbersũ,ṽ ∈ E 1 , we use the distance (Hausdorff metric) (Goetschel and Voxman 1986) D(u(r), v(r)) = max{sup r∈[0,1] |u(r) − v(r)|, sup |u(r) − v(r)|}, and it is shown (Puri and Ralescu 1986) that (E 1 , D) is a complete metric space and the following properties are well known: Definition 2.3. Consider x, y ∈ E. If there exists z ∈ E such that x = y + z then z is called the Hdifference of x and y, and is denoted by x y (Bede and Gal 2005). (Bede and Gal 2005).
Definition 2.4. (see (Bede and Gal 2005) and the following limits hold: ) and the following limits hold: ) and the following limits hold: and the following limits hold: and similarly for (ii), (iii) and (iv) differentiability.

Definition 2.6. (see (Chalco-Cano and Romn-Flores
Definition 2.7. A triangular fuzzy number is defined as a fuzzy set in E 1 , that is specified by an ordered triple are the endpoints of r-level sets for all r ∈[ 0, 1], where The set of triangular fuzzy numbers will be denoted by E 1 . Definition 2.8. (see (Chalco-Cano and Romn-Flores 2006)) The mapping f : T → E n for some interval T is called a fuzzy process. Therefore, its r-level set can be written as follows:

Description of the method
To obtain the approximation solution of Eqs.(1,2), based on Definition (2.6) we have two cases as follows: Case (2) Now, we can write successive iterations (by using Picard method) as follows: Case (1): Case (2): Remark 1. For u xx we have cases as follows: Case (1): u and u be (i)-differentiable and u and u be (ii)-differentiable

Remark 2.
We discuss about switching points as follows:  u (x, t, r), u (x, t, r)] and x 0 is a switching point in the form (iv). u (x, t, r), u (x, t, r)] and x 1 is a switching point in the form (iv). u (x, t, r), u (x, t, r)] and x 0 is a switching point in the form (iii). u (x, t, r), u (x, t, r)] and x 1 is a switching point in the form (iii). u (x, t, r), u (x, t, r)].

Existence and convergence analysis
In this section we are going to prove the existence and uniqueness of the solution and convergence of the method by using the following assumptions.

Numerical examples
In this section, we solve the Cauchy reaction-diffusion equation by using the Picard method. The program has been provided with Mathematica 6 according to the following algorithm where ε is a given positive value.
Step 4. Print u n (x, t) as the approximate of the exact solution.
Example 5.1. Consider the Cauchy reaction-diffusion equation as follows: With initial condition: = 10 −4 . x = 0 is a switching point.
Case (1): α = 0.7546. Table 2 shows that, the approximation solution of the Cauchy reaction-diffusion equation is convergent with 17 iterations by using the Picard method when u is (i)-differentiable. Case (2): α = 0.7762. Table 3 shows that, the approximation solution of the Cauchy reaction-diffusion equation is convergent with 21 iterations by using the Picard method when u is (ii)-differentiable.

Conclusion
The Picard method has been shown to solve effectively, easily and accurately a large class of nonlinear problems with the approximations which convergent are rapidly to exact solutions. In this work, the Picard method has been successfully employed to obtain the approximate solution of the Cauchy reaction-diffusion equation under generalized H-differentiability.