In this paper, we present a novel technique to obtain approximate analytical solution of fractional physical models. The new technique is a combination of a domain decomposition method and natural transform method called a domain decomposition natural transform method (ADNTM). The fractional derivatives are considered in Caputo sense. To illustrate the power and reliability of the method some applications are provided.
Fractional differential equations have gained importance and popularity, mainly due to its demonstrated applications in science and engineering. For example, these equations are increasingly used to model problems in research areas as diverse as dynamical systems, mechanical systems, control, chaos, chaos synchronization, continuous-time random walks, anomalous diffusive and sub diffusive systems, unification of diffusion and wave propagation phenomenon and others. The most important advantage of using fractional differential equations in these and other applications is their non-local property. It is well known that the integer order differential operator is a local operator but the fractional order differential operator is non-local. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. This is more realistic and it is one reason why fractional calculus has become more and more popular [
With reference to the articles [
Over the set of functions
The natural transform of f (t) is defined as:
where
If
If
If
where
If
of order
Let the function
Let the function
If
Then the natural transform of
The natural transform of T-periodic function
To illustrate the basic idea of a domain decomposition natural transform method (ADNTM), we consider the general inhomogeneous nonlinear equation with initial conditions given below [
where L is the lowest order derivative which is assumed to be easily invertible, R is a linear differential operator of order less than L, FU represents the nonlinear terms and
Using the differential property of natural transform and initial conditions we get:
By arrangement we have:
The second step in natural decomposition method is that we represent solution as an infinite series:
and the nonlinear term can be decomposed as:
where
Substitution of (14) and (15) into (13) yields:
On comparing both sides of (16) and using standard ADM we have:
Then it follows that:
In more general, we have:
On applying the inverse natural transform to (17) and (18), we get:
where
where the functions
The terms
where
The Fokker-Planck equation was first introduced by Fokker and Planck to describe the Brownian motion of particles [
with initial condition
where
Equation (20) is also well known as a forward Kolmogorov equation. There exists another type of this equation is called a backward one as [
A generalization of Equation (20) to N-variables of
with the initial condition
The nonlinear Fokker-Planck equation is a more general form of linear one which has also been applied in vast areas such as plasma physics, surface physics, and astrophysics the physics of polymer fluids and particle beams, nonlinear hydrodynamics, theory of electronic-circuitry and laser arrays, engineering, biophysics, population dynamics, human movement sciences, neurophysics, psychology and marketing [
The nonlinear form of the Fokker-Planck equation can be expressed in the following form:
A generalization of Equation (24) to N-variables of
Now, we consider the following nonlinear Fokker-Planck equation:
Then, Equation (26) becomes:
with initial condition
According to ADNTM, by applying natural transform of both sides of Equation (28) and using the initial condition we get:
The second step in natural decomposition method is that we represent solution as an infinite series:
i.e.
Then, recursive relations are:
and nonlinear terms can be decomposed as:
are domain polynomials of [
According to ADM we have
Hence
At special case, when
which is the exact solution and is same as obtain by ADM [
Nonlinearly interacting waves are often described by asymptotic equations [
with initial condition
The exact solution at
By applying natural transform of Equation (36), we obtain:
Using the differential property of natural transform and initial conditions we get:
The second step in natural decomposition method is that we represent solution as an infinite series:
and the nonlinear term can be decomposed as:
where
Substitution of (39) and (38) into (37) yields
where nonlinear term is given by
In view of (41), and following the formal techniques used before to derive the a domain polynomials, we can easily derive that F(u) has the following polynomials representation:
On comparing both sides of (40) and using standard ADM we have:
According to ADM we have
Hence,
At special case, when
which is the exact solution of (36) by [
The Klein-Gordon equation [
with the initial conditions
As the previous, by applying ADNTM method, we have:
According to ADM we have
Hence
which is the exact solution as obtained by VIM [
As shown in the three examples of this paper, the domain decomposition natural analytical solutions of time- fractional Fokker-Planck equation, time-fractional Schrödinger equation, and time fractional Kelin-Gorden equation were in excellent agreement with the exact solutions. Finally, generally speaking, the proposed method can be further implemented to solve other physical models in fractional calculus field.
We thank Directors, Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt and Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt for their cooperation during this work.
Ahmed SafwatAbdel-Rady,Saad ZagloulRida,Anas Ahmed MohamedArafa,Hamdy RagabAbedl-Rahim, (2015) Natural Transform for Solving Fractional Models. Journal of Applied Mathematics and Physics,03,1633-1644. doi: 10.4236/jamp.2015.312188