TENSOR PRODUCT TECHNIQUE AND FRACTIONAL DIFFERENTIAL EQUATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Some times it is not easy to find the exact solution of certain partial differential equations. In this paper we use tensor product technique of Banach spaces to find certain solutions of certain fractional differential equations.

If the limit exists then it is called the α− conformable fractional derivative of f at x. The new definition satisfies: 1.T α (a f + bg) = aT α ( f ) + bT α (g), for all a, b ∈ R.
Further, for α ∈ (0, 1] and and f , g be α−differentiable at a point t, with g(t) = 0. Then We list here the fractional derivatives of certain functions, On letting α = 1 in these derivatives, we get the corresponding ordinary derivatives.
One should notice that a function could be α−conformable differentiable at a point but not But T 1 ( f )(0) does not exist. This is not the case for the known classical fractional derivatives.
Through out this paper, if u is a function of two variables x and t, then we write D α x u to denote the α−conformable derivative of u with respect to x. Similarly for D α t u. In case the function h is a function of 1-variable, then we write h (α) to denote the α−derivative of h.
For more on fractional calculus and its applications we refer to [1], [8] and [9].

TENSOR PRODUCT
Let X and Y be two Banach spaces and X * be the dual of X. Assume x ∈ X and y ∈ Y. The operator T : X * → Y, defined by T (x * ) = x * (x)y is a bounded one rank linear operator. We write x ⊗ y for T. Such operators are called atoms. Atoms are among the main ingredient in the theory of tensor product. Atoms are used in theory of best approximation in Banach spaces, [6], and [7].
We need the following Lemma in our paper: If the sum of two atoms is an atom, then either the first components are dependent or the second components are dependent.
That is: if then we can assume with no loss of generality that either x 1 = x 2 , or y 1 = y 2

MAIN RESULTS
Consider the fractional partial differential equation If one tries to solve this equation via separation of variables, then it is not possible since the variables cant be separated.
Hence we use the tensor product technique to solve the equation.
Procedure. Let u(x,t) = P(x)Q(t). Substitute in equation (2), to get: Equation (3) has the form of equation (1). Then using Lemma 2.1 we have two cases to consider:

Solution of Case (i).
From P(x) = g(x)P (α) (x), we get where c is a constant of integration. Now take equation (4) and substitute in (3) to get: Now equation (5) is a linear fractional deferential equation of order α, [10]. The solution is One can find c 1 and c 2 whenever boundary or initial conditions are given or assumed in the equation.
Finally, the atomic solution is the product: P(x)Q(t), where P is given in (4) and Q is given in (6).

Solution of case (ii).
Q (α) (t) = Q(t). Then Now substitute (7) in (3) to get Notice that a(t) and h(t) are known and considered constants with respect to x. Further b(x) = 1 g(x) is given in the equation.
Again, equation (8) is a fractional linear equation, [10]. Hence The atomic solution of (3) for the case (ii) is P(x)q(t), where P(x) is given in (9) and Q(t) is given in (7).

FURTHER RESULTS
Let us consider the equation Here W 1 (x), R 1 (t), W 2 (x), R 2 (t) are all given.
Again, it is very difficult to use separation of variables to solve this equation.
Let us use Tensor product technique to find certain atomic solutions.
So put u(x,t) = P(x)Q(t). Substitute in equation (10) we get: In equation (11), we have the sum of two atoms is an atom. Hence we have two cases: As for case(i) here, it is very similar to case(i) in section III, and so Now substitute in (11) to get which is very similar to equation (5), and so So the atomic solution to this case is P(x).Q(t) from (12) and (14) respectively.
The second case is exactly the same as case (ii) in section III.
We end the paper with this problem: Problem. How can one get an atomic solution if the equation is of the form: where g and f are not atomic functions?
Remark. This work is part of the Ph.D. thesis of I. Kadiri under the supervision of M.
AlHorani and R. Khalil at the University of Jordan.