Analytical and approximate solutions of Fractional Partial Differential-Algebraic Equa- tions

In this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.


Introduction
In the past several years ago, various methods have been proposed to obtain the numerical solution of partial differential-algebraic equations [2], [7], [11]- [16]. In this study, we consider the following system of partial differential-algebraic equations of fractional order

Basic Definitions
We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 1.
A real function f (x), x > 0 is said to be in the space C µ , µεR if there exists a real number P > µ such that f (x) = x p f 1 (x), where f 1 (x)εC[(0, ∞). Clearly C µ < C β if µ < β .
Definition 4. The fractional derivative of f (x) in the Caputo sense is defined as for m − 1 < α < m, m ∈ N, x > 0,.

fractional Two-Dimensional Differential Transform Method
Differential Transform Method (DTM) is an analytic method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional high order Taylor series method requires symbolic computation. However, the FDTM obtains a polynomial series solution using an iterative procedure. The proposed method is based on the combination of the classical two-dimensional FDTM and generalized Taylor's Table 1 formula. Consider a function of two variables u(x, y) and suppose that it can be represented as a product of two single-variable functions, that is, u(x, y) = f (x)g(y) based on the properties of fractional two-dimensional differential transform [1], [3]- [6], [8]- [10], the functionu(x, y) can be represented as: Where0 < α,β ≤ 1,U α,β (k, h) = F α (k)G β (h), is called the spectrum of u(x, y). The fractional two-dimensional differential transform of the function u(x, y) is given by Where(D α In the case of α = 1 and β = 1 the Fractional two-dimensional differential transform (9) reduces to the classical two-dimensional differential transform. Let U α,β (k, h), w α,β (k, h) and V α,β (k, h) are the differential transformations of the functions u(x, y), w(x, y) and v(x, y), from Equations (9) and (10), some basic properties of the two-dimensional differential transform are introduced in Table 1.

Numerical example
Here, the fractional differential transform method will be applied for solving the fractional partial differentialalgebraic equation.
Example 2. Consider the fractional partial differential-algebraic equation where with the exact solution Equivalently, equation (12) can be written as By using the DTM in equation (17), we obtain from the initial condition (13), we have By using the differential inverse reduced transform of   Table 3: Numerical solution of v 2 (x,t)

Conclusions
The generalized differential transformation method displayed in this work is an effective method for the numerical solution of a fractional partial differential-algebraic equation system. With full solutions, approximate solutions collected by the GDTM were compared to shapes and charts. On the other hand, the results are quite reliable for solving this problem. The exact closed-form solution was obtained for all the examples presented in this paper. FDTM offers an excellent opportunity for future research. As a result of this comparison, it is seen that the solutions obtained by the generalized differential transformation method are harmonious with the full solutions.
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