1. Introduction
The Fokker-Planck equation (FPE) was primarily introduced by Fokker and Planck to design the Brownian type motion of particles (Risken, 1989) [
1], that is, it articulates the transformation of likelihood estimates of a random function in time and space, therefore Fokker-Planck model is employed to demonstrate and elucidate solute transport [
2]. Fokker-Planck model particularized the time progression of density function of location and speed of the particle. Scientific manifestations such as uncharacteristic dispersion, constant random motion, wavy promulgation, polymer-macro-molecular architectures, electric-charge transporter in non-crystalline semiconductors, Biological code of DNA and RNA molecules, and arrangement materialization are demonstrated by [FPPDEs] with time and space fractional differential expressions as given in Heinsalu et al. [
2], 2006; Yan et al. [
3], Yang et al. [
4], 2009; Zhuang et al., 2006/07 [
5]. These applications of FPPDE with time and space-fractional differential equations have involved relevant researchers to invistigate the problem. Brownian motion [
6] and the material diffusivity based approaches in reaction-kinetics of reactive fluids [
7] are now deliberated, in numerous technologies; in physicochemical systems and biological synthesis [
1]. The FPE is aspired in reaction-kinetics [
8]. Some other technical aspects of these equations have been studied by He and Wu [
9], Jumarie [
10], Kamitani and Matsuba [
11], Xu et al. [
12], and Zak [
13].
The general form of FPE for the motion of a focus field
with space and time variables
s,
t is defined as [
1]
Initial Condition
where
and
are drift and diffusion coefficients respectively. This equation is also called the forward Kolmogorove equation. The diffusion and drift coefficients may also be time dependent such as
There is an alternate form of FPE, which is called the nonlinear FPE. The nonlinear FPE has significant solicitations in numerous capacities such as modeling the combustion and the interaction between fluid dynamics and chemistry. To model many real-life problems in fields such as plasma physics, chemical particle dynamics, hydrodynamics, solid state physics, chemical particle-dynamics, hydrodynamics, solid state physics, and additional course of studies, nonlinear fractional equations are used [
14].
In the case of single variable, FPE is expressed as:
Due to number of applications of the FPE, some investigations have been carried out to obtain its solution numerically. In this regard, the works of Buet et al. [
15], Harrison [
16], Palleschi et al. [
17], Vanaja [
18], and Zorzano et al. [
19] are important conntributions.Here, we take classical integer order of FPE (4), and the nonlinear FPE with time fractional derivative
where
denotes the Caputo or Riemann-Liouville non-integral order derivative a FPE with time and space derivatives in fractional form. The function
is taken as fundamental function of space and time. Specifically, for
, the time-fractional order FPE (5) reduces to the classical non-linear FPE given by (4) in the case
.
In the fractional calculus, the concept of non-integral ordered differentiation and integration has been used. Fractional calculus is derived from classical calculus. In historical perspective, fractional calculus is seen as classical calculus. However, in the current era, fractional calculus has more attentiveness because of its wide-ranging solicitations in many technological fields. The theoretical explanation of the subject has been studied in detail and developed by Oldham and Spanier [
20] Miller and Rose [
21] and Podlubny [
22] and provide us with a lot of knowledge. In the meanwtime, it has been noted by the many mathematicians, researchers and scientists who have observed that the role of non-integer operators is very important in expressing the properties of physical phenomena. Many procedures have been proficiently expounded by fractional differentiation and integrals. Additional relative study has been done between classical models and fractional models. It was concluded that fractional models are more efficient than classical models. Different types of modeling such as traffic flow, fluid flow, signal processing etc., belonging to real world problems, results in FPDE’s. The use of NLEE’s in physical circumstances is very important. The areas in which we can use NLEE’s including Plasma physics, quantum field theory, chemical reactions and biological applications. Recently, many researchers has introduced many methods to obtain analytical solutions of NLEEs such as generalized Kudryashov method [
23], modified extended tanh function method [
24], exp function method [
25], extended trial equation method [
26], sine-cosine method [
27], and G’/G-expansion method [
28].
To solve nonlinear problems, the idea of homotopy has been combined with perturbation method. In his investigation Lio [
29] did the fundamental work by using the homotopy analysis method. For the first time in 1998, the homotopy perturbation method was presented by He [
30]. A novel technique which is known as OHAM was created by Marinca et al. [
31,
32,
33]. The benefit of OHAM is that it establishs its convergence criteria similar to HAM but more pliable. In various research papers S. Iqbal et al. [
34,
35,
36], Sarwar et al. [
37,
38] and Alkhalaf [
39] have proved sufficient generalization and trust of this method, achieved well approximate solutions, and presented important applications in science and engineering. The concept of OHAM has been articulated in this paper. It provides logical, trust worthy solution to linear, non-linear, time dependent, time fractional and space fractional differential equations and PDEs. The arrangement of the paper is as follows. In
Section 2, basic definitions of fractional calculus are given,
Section 3 is dedicated to the scheme of method,
Section 4 includes model problems, results and discussions,
Section 5 includes conclusions.
3. OHAM Scheme for Time Fractional Parabolic Partial Differential Equation
According to the {OHAM} algorithm [
36,
37], we shall extend this scheme for time fractional Fokker-Planck partial differential equations {tFFPPDEs} in the following steps.
Step-1:
Compose the time fractional order Fokker-Planck governing equation as
is domain. Now Equation (
10) is decomposed in to
Where
Q is a fractional part and
T is a non-fractional part.
Step-2:
Make an optimal homotopy for time fractional order partial differential equation,
:
which satisfies
where
and
is an embedding parameter, for
is a nonzero auxiliary function and
when
P increases in the interval
the solution
certifies a rapidly Convergence to the exact solution.
where
are auxiliary convergence control parameters, and
can be function on the variables. The selections of
may be polynomial, exponential and soon. The selection of functions is very important, because the rate of convergence of the solution really depends on the functions.
Step-3:
Expand
in Taylor’s series about
to improve an approximate results as
It has been cleared that the rate of convergence of the (
13) depends upon auxiliary constants
Step-4:
Compare the coefficients of like powers
p after replacing Equation (
13) in (
11), we can get (Zeroth, First, Second and higher-order) problems if needed.
and so on
Step-5:
Put Equation (
14) in to Equation (
10), outcomes the bellow expression for residual.
R is a residual of the problem Fokker-Planck
Auxiliary constants
can be calculated as follows:
If then will be the exact solution. Normally it doesn’t happen, likely in non-linear problems.
Step-6:
The use of auxiliary constants in Equation (
14), we can get the rapidly convergent approximate solutions.
Step-7:
Accuracy of the method by