A Parabolic Transform and Averaging Methods for General Partial Differential Equations

Averaging method of the fractional partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.

Let C b ( n ) be the set of all bounded continuous functions on n . Consider the following Cauchy problem [8]: ∂u(x, t) ∂t = (D 2 1 + ... + D 2 n ) 2N +1 u(x, t), u(x, 0) = ϕ(x) ∈ C( n ), (4) where C( n ) is the set of all continuous functions on n , N is a sufficiently large positive integer.
The solution of the Cauchy problem (3), (4) is given by where dy = dy 1 ...dy n and the function G is the fundamental solution of the Cauchy problem (3), (4).
A parabolic transform of a function W is a functionW defined by [8] W (x, t 1 , ..., t r , where By using the equations (1), (2), we get where Γ is Gamma function.
By using the equations (15), (16), we get Another straightforward analysis shows the existence and uniqueness of the solutions of the problems (1), (2), (7), (15), (16) and (17) on the time-scale 1 ε . The norm . ∞ is defined by the supremum norm on the spatial domain and on the time-scale 1 ε and denoted by u ∞ = Sup x∈ n |u(x)|.
Notice that there exist a dense set E ∈ C b ( n ). Proof. We introducev(x, t) by the near-identity transformation: and suppose that the derivatives of v * (x, t),L(x, t, c 1 t, D) andL(x, D) are bounded.
To complete the proof we use the barrier functions and the Phragmèn-Lïndelöf principle see [12].
Suppose that the barrier function: and the functions (we omit the arguments) We obtain Z 1 (x, t) and Z 2 (x, t) are bounded so we apply the Phragmèn-Lïndelöf principle, resulting in Z 1 (x, t) ≤ 0 and on the time-scale 1 ε . We can use the triangle inequality to obtain nT , 1 n ) are the solutions of the equations (7), (17) with c 1 = 1 nT and c 2 = 1 n . Let By using the semi-group property and the equations (7), (17), we find that u n and u * n satisfy the following equations where ϕ n , a q,n ,ā q,n are defined as in Hence the required result.
Notice that if the coefficients a q 's andā q 's do not depend on x for all |q| ≤ m, then a q,n = a q andā q,n =ā q .
We have We can use the barrier functions and the Phragmèn-Lïndelöf principle see [12].
Z 3 (x, t) and Z 4 (x, t) are bounded so we apply the Phragmèn-Lïndelöf principle, resulting in Z 3 (x, t) ≤ 0 and Z 4 (x, t) ≥ 0. We get Similar to section (2), we have the required result.

Conclusion
A fractional partial differential equation can be solved without any restrictions on the characteristic forms by using the parabolic transform and the averaging methods. As a special case Cauchy problem is solved for a fractional partial differential equation.