Communications in Nonlinear Science and Numerical Simulation
The fractional q-differential transformation and its application
Highlights
► A new generalization of the differential transform method is developed to solve q-fractional differential equations. ► The definitions and operations of the one-dimensional and two-dimensional fractional q-differential transform is proposed. ► Several concrete problems were tested by applying the new technique and the results have shown remarkable performance.
Introduction
Fractional calculus, which deals with differentiation and integration to an arbitrary order, has gained considerable interest through the pioneering works of Leibniz, Bernoulli, Euler, Lagrange, Abel, Fourier, Riemann, Liouville and many others. In the present decade notable contributions to both theory and applications of this subject have been carried out [21], [15]. Recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation. Fractional differential equations, therefore find numerous applications in the field of visco-elasticity, feedback amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles, neuron modeling encompassing different branches of Physics, Chemistry and Biological sciences [1], [5], [6], [12], [21], [25], [26].
At the last quarter of 20th century, q-calculus appears as a connection between mathematics and physics (see [9], [13], [16], [17], [18]). It has a lot of applications in different mathematical areas, such as number theory, Combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences: quantum theory, mechanics and theory of relativity.
The differential transform method was first applied in the engineering domain in [27]. In general, the differential transform method is applied to the solution of electric circuit problems. The differential transform method is a numerical 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 differential transform method obtains a polynomial series solution by means of an iterative procedure. The application of differential transform method is successfully extended to obtain analytical approximate solutions to linear and nonlinear ordinary differential equations of fractional order [3]. Liu [19] extended the use of the differential transformation method to the case of the nonlinear q-difference equations. Recently, El-shahed and Gaber [8] developed an analytical technique for solving linear and nonlinear two-dimensional q-difference equations. This technique is based on classical differential transform method and q-Taylor’s formula. They tested their approach on several examples and the results obtained are in good agreement with the existing ones in open literature. In this paper we present a new generalization of the differential transform method. The technique is based on classical differential transform, q-generalized Taylor’s formula and Caputo fractional derivative. The purpose of this paper is to employ the Fractional q-differential transformation method to solve fractional q-differential equations which are often encounter in many branches of physics, Combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences: quantum theory, mechanics, theory of relativity, chemicals and engineering (see for instance [9], [13], [16], [17], [18]). To the authors knowledge, this paper represents the first application of differential transform to solve q-fractional differential equations.
The present paper has been organized as following: In Section 2, basic definitions and notations of the fractional q-calculus are introduced. The operational properties of the fractional q-differential transformation that never existed before are introduced with their proofs in Sections 3 One-dimensional fractional, 4 Two-dimensional fractional. Numerical examples have been presented in Section 5, to illustrate the effectiveness of the proposed method.
Section snippets
Preliminaries
In this section, we summarize the basic definitions and properties of q-calculus and fractional q-integrals and derivatives. For more details on the theory of q-calculus we refer to [9], [14] and for theory of q-fractional calculus to [2], [23], [24]. Definition 1 Consider an arbitrary function . Its q-differential isand
Note that, if is differentiable then Definition 2 Let . For any functions and , we have
One-dimensional fractional q-differential transforms
In this section, we shall derive the one-dimensional fractional q-differential transform that we have developed for the numerical solution of q-differential equations of fractional order. The proposed method is based on generalized q-fractional Taylor’s formula. Definition 8 At , the one-dimensional fractional q-differential transform of is defined asIn Eq. (6), is the original function and is the transformed function. Definition 9 The one-dimensional
Two-dimensional fractional q-differential transforms
In this section, we shall derive the two-dimensional fractional q-differential transform. Definition 20 Two-dimensional fractional q-differential transform of function at is defined as followIn Eq. (10), is the original function and is the transformed function. Definition 21 The two-dimensional fractional q-differential inverse transform of is defined as followIn fact from
Applications
In this section, we apply the fractional q-differential transforms method to solve some types of fractional q-differential equations.
Conclusion
In this work, we developed a new generalization of the differential transform method that will extend the application of the method to q-differential equations of fractional order. Moreover, this method does not require the problem to be changed into a form suitable for the use of linear theory or perturbation. It is also noteworthy that no symbolic computation is required, which can be difficult especially in non-linear cases. On the other hand the results are quite reliable. Several concrete
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