In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.
1.
Introduction
Fractional calculus plays a substantial role in different branches of physics, fluid mechanics, diffusive transport, electrical networks, electromagnetic theory, biological sciences and groundwater problems etc. [1,2,3,4]. Many researchers have modelled several physical phenomena using the fractional-order differential equations. As we know that, solving a linear differential equation is easier than that of nonlinear differential equations; therefore, numerous methods have been recommended for solving such type of equations. Some of the numerical and analytical methods for solving linear and nonlinear FDE are, Finite Element Method(FEM), time-space spectral method, compact numerical method [7,18,19], Adomian Decomposition Method (ADM) [5,6], Variational Iteration Method (VIM) [8,9], Homotopy Analysis Method (HAM) [10] and HPM [11]. Each of the above methods has its own advantages and disadvantages.
Daftardar-Gejji and Jafari presented a powerful method, namely called the new iterative method which works without any small or large parameter in the equation like other perturbation methods. The proposed method has been used in literature for the solution of different nonlinear differential equations [12,13].
In the present work, NIM has been extended to the solution of fractional order Roseau-Hyman equation and system of Inhomogeneous fractional order partial differential equations. The fractional order Roseau-Hyman equation has the following form [14,15,16].
where, q is the parameter describes order the fractional derivative such that 0 < q≤1, t time, and r represents spatial coordinate. Eq (1) has appeared in the study of the formation of patterns in liquid drops.
The system of inhomogeneous fractional order partial differential equations has the following form [17].
The whole paper is divided into six sections. The introduction and literature survey are given in section 1, while section 2 is devoted to the basic definitions from the fractional calculus. The third section contains the fundamental theory of a new iterative method for general fractional order PDE's. In section 4, the proposed method is tested upon fractional order Roseau-Hyman equation and system of inhomogeneous fractional order partial differential equations. In section 5, the listed results are compared with HPM, VIM, LVIM, and LADM solution, which show the precision of the planned method. The conclusions of the paper are presented in the last section.
2.
Preliminaries
To investigate our problems with the help of NIM, we need some basic definitions from fractional calculus.
Definition 1. The Riemann-Liouville's (R-L) fractional integral is defined as
where Γ denotes the gamma function,
Definition 2. The Riemann-Liouville fractional derivative of function f(t)f with order α is defined as
where n is a positive integer which satisfies n−1<α⩽n.
Definition 3. Fractional derivative of order α in the Caputo sense, is defined as:
For n∈N,r>0,t≥−1andϕ∈Ct:
Definition 4. If n∈N,n−1<α≤n and f∈Cμα,μ≥−1, then
Remarks: Basic properties of fractional integration are, whenf∈Cλ,λ≥−1,α,β≥0 and ξ≥−1:
● IαtIβtf(t)=Iα+βtf(t),
● (IαtIβt)f(t)=(IβtIαt)f(t),
● Iαttγ=Γ(γ+1)Γ(γ+α+1)tγ+α.
3.
New iterative method
The new iterative method is presented for fractional order partial differential equation as follows:
Let us consider fractional order PDE's
subject to the initial condition
where Ψ denotes nonlinear functions of Υ. According to the fundamental idea of NIM, and using Eq (8) above Eq (7) takes the form
Assuming that the solution of Eq (7) has series form
The decomposition of nonlinear operator ψ(Υ(r,t))=1Γ(α)r∫0Ψ(Υ(r,t))α−1dr as follows
Hence general equation of (7) takes the following form
from the above we have
4.
Convergence
Let Υ is the series solution achieved by NIM and E the error of the solution of (8). Clearly E, satisfies (8), so, we can write
The recurrence relation is given as,
If ‖N(r)−N(y)‖⩽k‖x−y‖,0<k<1, then
Thus, En+1→0, as n→∞, which proves the convergence of the new iterative method.
5.
Numerical examples
To illustrate efficiency and precision of the NIM, the following fractional order differential equations are taken as test examples. All computational work has been done with the help of Mathematica 10.
5.1. Fractional order Roseau-Hyman equation
First, we consider time fractional Roseau-Hyman equation given as
Subject to initial condition
For special case q=1, exact solution for Eq (17) can be found in [14] and ε is an arbitrary constant
By applying Iq to both sides Eq (17), we get the equivalent integral form of (16) is
The nonlinear term is
Using NIM formulation discussed in section 3, we get the approximations as
The expression for the solution of Υ(r,t) is given as
For q=1.0, the seventh order NIM solution for Roseau-Hyman equation is
Similarly for q=0.7, the seventh order NIM solution is
5.2. Fractional Order inhomogeneous system
Consider time fractional inhomogeneous system given as [17]
With initial condition as
For q=1, exact solution for system (23) is [17]
By applying Iq to both sides of Eq (24), we get the equivalent integral form of (23) given as
Here nonlinear terms are in Eq (25)
Using NIM formulation discussed in section 3, we get the approximations as
The expression for the solution of and is given as
6.
Results and discussion
We implemented NIM for finding the approximate solutions of Roseau-Hyman and fractional order inhomogeneous partial differential equations. The results obtained by NIM for Roseau-Hyman and fractional order inhomogeneous partial differential equations with VIM, HPM and LADM in the form of tables and figures in section 3.
Table 1 shows the approximate solution obtain by NIM for Roseau-Hyman equation at different values of q Table 2 shows the comparison of absolute errors of NIM with VIM and HPM for Roseau-Hyman equation. Table 3 shows the numerical solution obtained by the proposed method for γ(r, t) and γ(r, t) inhomogeneous partial differential equations at q = 0.5. Similarly Table 4 shows the residuals obtain by the NIM γ(r, t) and ψ(r, t) inhomogeneous partial differential equations at q = 0.5. It is clear that for the fractional order inhomogeneous system, NIM has the same solution like LADM and LVIM, mention in [17]. That is why, we listed only NIM solution as well as residual obtain by NIM in Tables 3 and 4.
Figures 1–4 show the 3D plots obtain by NIM for γ(r, t) and exact solution for γ(r, t) of Roseau-Hyman equation respectively. Figures 5 and 6 shows the 2D plot of exact verses NIM solution and convergence of NIM for different values of q for ψ(r, t) respectively. Similarly, Figures 7 and 8 show, 2D graph for residual obtained by the NIM for Roseau-Hyman equation for q = 0.5 and 0.7 respectively.
Figures 9–12 show the 3D plots obtain by NIM for γ(r, t) and exact solution of inhomogeneous partial differential equations respectively at different values of q. Figures 13 and 14 show the 2D plot of exact verses NIM solution and convergence of NIM for different values of q for γ(r, t) part of inhomogeneous partial differential equation respectively.
Figures 15–18 show the 3D plots obtain by NIM for ψ(r, t) and exact solution of inhomogeneous partial differential equations respectively at different values of q. Figures. 19 and 20 shows the 2D plot of exact verses NIM solution and convergence of NIM for different values of q for ψ(r, t) part of inhomogeneous partial differential equation respectively.
From the numerical values and graphs, it is clear that NIM is very powerful tool for solution of coupled fractional order system of partial differential equations. The accuracy of the NIM can further be increased by taking higher order approximations.
7.
Conclusion
In the current article, we presented some FDE's arising in modern sciences. A novel and classy technique which is identified as NIM, is applied for fractional order problems. For pertinence and unwavering quality of the proposed method, the fractional order Roseau-Hyman equation and the system of fractional order non-homogeneous equations. It has been explored through graphical and tabulated results that the current method gives a precise and meriting investigation about the physical occurring of the problems. Also, the current method is favored when contrasted with other technique in light of its better pace of convergence. This course rouses the scientists towards the execution of the present method for other non-linear FDE's.
Acknowledgements
Authors acknowledge help and support from the Department of Mathematics of AWKUM for completion of this work.
Conflict of interest
The authors declare no conflict of interest.