A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach
Introduction
Fractional derivative operators (FDOs) have successfully been observed to provide mathematical analytic tools with potential for applications in mathematical sciences, physics and mechanics (see [1], [2]). Fractional partial differential equations (FDEs) describing anomalous phenomena in real world have been witnessed over the past 50 years (see [3], [4], [5]). There are some methods for finding analytical, numerical and numerical-analytical solutions to FDEs. Several analytical techniques, such as the Adomian decomposition [6], variational iteration [7] and homotopy analysis [8] methods, were successfully investigated for solving various families of FDEs. Furthermore, several approaches for finding numerical solutions for FDEs, such as the Kansa method [9], discontinuous spectral element [10] and Chebyshev spectral methods [11], and other methods [12], [13], [14] were also discussed. A semi-numerical technique for solving FDEs, called differential transform (DT) (see [15], [16]), was proposed and developed to deal with modified KdV [17], oscillator [18], and coupled Burgers’ [19] equations with FDOs.
The theory of the local FDOs [20], [21], [22] was adopted to describe non-differentiable problems from fractal physical phenomena, which are not dealt with the above non-local operator [1], [2], [3], [4], [5]. Hence, several analytical and numerical approaches for local fractional partial differential equations (LFPDEs) [23], [24], [25], [26], [27], [28] were formulated. In [23], the one-dimensional extended DT (ODEDT) was proposed. However, the two-dimensional extended differential transform (TDEDT) is not discussed. The main target of this article is to point out a semi-numerical technique, called the TDEDT, in order to find the non-differentiable solution for the local fractional diffusion equation (LFDE). The layout of this article is presented as follows. In Section 2, the notations and conceptions of LFDOs are given. In Section 3, the TDEDT via LFDO and its associated basic theorems and properties are discussed. In Section 4, the non-differentiable solution to the LFDE is investigated. Finally, the conclusions are presented in Section 5.
Section snippets
Preliminaries
Let Cκ(ξ, ζ) be a set of the non-differentiable functions with the fractal dimension κ(κ ∈ (0, 1]) (see [20], [28]).
For Φ(λ) ∈ Cκ(ξ, ζ), the LFDO of Φ(λ) of order κ(κ ∈ (0, 1]) at the point is defined as follows (see [20], [23], [24], [25], [26], [27], [28], [29], [30]): where The LFPDE of Φ(λ, ω) of order κ (κ ∈ (0, 1]) with respect to λ at the point (λ0, ω) is defined as follows (see [20]):
The TDEDT technique via LFDO
In this section, the conceptions and theorems of the TDEDT via LFDO are presented.
Definition 1 If is a local fractional analytic function [20] in the domain Ξ, then the TDEDT of the function via LFDO is defined as follows:
where κ ∈ (0, 1], and the two-dimensional differential inverse transform (TDDIT) of in the domain Ξ via LFDO is defined by
Here Θ(μ, η) is
Applications
Let us investigate the following LFDE (see [29]): subject to the initial condition given by From (8), the Eq. (38) changes into the following form: Now, making use of (8) and (39), we get Adopting (40) and (41), and applying the process recursively, the simulating results are listed in Table 2 .
Therefore, the numerical solution of (42) in non-differentiable
Conclusions
In this work, the TDEDT is first introduced and suggested for solving partial differential equations with local fractional derivatives. The LFDE with non-differentiable initial value is chosen to test the proposed technique. The comparison between numerical solution with non-differentiable terms and analytical solution with non-differentiable terms is also presented. The obtained results show the efficiency and accuracy of the non-differentiable algorithm to deal with LFPDEs.
References (30)
- et al.
Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition
Appl. Math. Comput.
(2006) Analytical solution of a fractional diffusion equation by variational iteration method
Comput. Math. Appl.
(2009)- et al.
Fractional diffusion equations by the Kansa method
Comput. Math. Appl.
(2010) - et al.
A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order
Comput. Math. Appl.
(2011) - et al.
Algorithms for the fractional calculus: a selection of numerical methods
Comput. Methods Appl. Mech. Eng.
(2005) - et al.
Solution of fractional differential equations by using differential transform method
Chaos Solitons Fractals
(2007) - et al.
A generalized differential transform method for linear partial differential equations of fractional order
Appl. Math. Lett.
(2008) - et al.
Approximate analytical solution for the fractional modified KdV by differential transform method
Commun. Nonlinear Sci. Numer. Simul.
(2010) - et al.
Solutions of a fractional oscillator by using differential transform method
Comput. Math. Appl.
(2010) - et al.
Numerical solutions of the space- and time-fractional coupled Burger’s equations by generalized differential transform method
Appl. Math. Comput.
(2011)
Static-kinematic duality and the principle of virtual work in the mechanics of fractal media
Comput. Methods Appl. Mech. Eng.
A tutorial review on fractal space-time and fractional calculus
Int. J. Theor. Phys.
An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives
Commun. Nonlinear Sci. Numer. Simul.
Theory and Applications of Fractional Differential Equations
Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media
Cited by (143)
A New Analytical Method for Solving Nonlinear Burger's and Coupled Burger's Equations
2023, Materials Today: ProceedingsCitation Excerpt :In recent years, many researchers have paid attention to study the behavior of physical problems by using various analytical and numerical techniques which are not described by the common observations, such as the FVIM [1,2,3,4,5], FDTM [6,7,8], FSEM [9,10], FSVIM [11,12], FLTM [13], FHPM [14], FSDM [15,16,17], FFSM [18], FRDTM [19,20,21], FADM [22,23,24], FLDM [25], FLHPM [26], and FLVIM [27,28,29,30,31].
A new approach in handling one-dimensional time-fractional Schrödinger equations
2024, AIMS MathematicsFractal dynamics and computational analysis of local fractional Poisson equations arising in electrostatics
2023, Communications in Theoretical PhysicsA new fractional derivative operator and its application to diffusion equation
2023, Mathematical Methods in the Applied SciencesSumudu adm on time-fractional 2d coupled burgers' equation: An analytical aspect
2023, Advance Numerical Techniques to Solve Linear and Nonlinear Differential Equations