In the past several years ago, various methods have been proposed to obtain the numerical solution of partial differential-algebraic equations , , , , , , , . In this study, we consider the following system of partial differential-algebraic equations of fractional order
2 Basic Definitions
We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.
A real function f (x), x > 0 is said to be in the space Cµ, µεR if there exists a real number P > µ such that f (x) = xp f1(x), where f1(x)εC[(0, ∞). Clearly Cµ < Cβ if µ < β.
A function f (x), x > 0 is said to be in the space
The Riemann-Liouville fractional integral operator of the order α > 0 of a function, f ∈ Cµ, µ ≥ −1 is defined as:
Properties of the operator Jα can be found in (Caputo, 1967), we mention only the following:
For fCµ, µ ≥ −1, α, β ≥ 0, and γ > −1.
The fractional derivative of f (x) in the Caputo sense is defined as
If −1 < α < m, m ∈ N and µ ≥ −1, then
3 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 , , , , , , , , the function u(x, y) can be represented as:
The operations for the two-dimensional differential transform method
|Transformed function||Original function|
|Uα,β = Vα,β +Wα,β||u(x,y) = v(x,y)w(x,y)|
|Uα,β = λVα,β||u(x,y) = λv(x,y)|
|u(x,y) = v(x,y)w(x,y)|
|u(x,y) = (x − x0)mα (y − y0)nβ|
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 Uα,β (k, h), wα,β (k, h) and Vα,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 .
Then, the fractional differential transform (10) becomes;
4 Numerical example
Here, the fractional differential transform method will be applied for solving the fractional partial differential-algebraic equation.
Consider the fractional partial differential-algebraic equation
With initial condition
Equivalently, equation (12) can be written as
By using the DTM in equation (17) , we obtain
By using the differential inverse reduced transform of V1(k, h), V2(k, h) and V3(k, h), we get
For special case α = 1, the solution will be as follows:
Numerical solution of v1(x, t)
|x||t||v1FDTMfor α = 0.5||v1FDTMfor α = 0.75||v1FDTMfor α = 1||v1Exact|
Numerical solution of v2(x, t)
|x||t||v2FDTMfor α = 0.5||v2FDTMfor α = 0.75||v2FDTMfor α = 1||v2Exact|
Numerical solution of v3(x, t)
|x||t||v3FDTMfor α = 0.5||v3FDTMfor α = 0.75||v3FDTMfor α = 1||v3Exact|
|0.01||0.01||0.0000099833416||0.00000316175064||9.99983333 · 10−7||9.999833334 · 10−7|
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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