Dynamical Study of Fokker-Planck Equations by Using Optimal Homotopy Asymptotic Method

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated.


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 v(s, t) with space and time variables s, t is defined as [1] ∂v (s, t) ∂t = − ∂A (s) ∂s where A (s) and B (s) > 0 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 ∂v (s, t) ∂t = − ∂A (s, t) ∂s 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 ∂ α ∂t α denotes the Caputo or Riemann-Liouville non-integral order derivative a FPE with time and space derivatives in fractional form.The function v(s, t) is taken as fundamental function of space and time.Specifically, for α = 1, the time-fractional order FPE (5) reduces to the classical non-linear FPE given by (4) in the case s > 0.
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.

Mathematical Preliminaries
A function of real value f (s), s > 0, assumed to be in space c µ , µ ∈ R, is very important for the study of fractional calculus.If it exists, p > µ is a real number such that f (s) = s p f 1 (s) , where f 1 (s) ∈ c (0, ∞) , assumed to be in space Definition 2. Riemann-Liouville sense integral operator of a function f (s) , of fractional order α > 0 is defined as Definition 3. Caputo sense derivative operator of a function f (s) of fractional order α > 0 is defined as , s > 0 (9)

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 Ω is domain.Now Equation (10) is decomposed in to Where Q is a fractional part and T is a non-fractional part.
where c i ; i = 1, 2, 3, ..., m are auxiliary convergence control parameters, and k i (x) , i = 1, 2, 3, ..., m can be function on the variables.The selections of Km (s, c i ) 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 Ψ (s, t; p, c) in Taylor's series about p, to improve an approximate results as It has been cleared that the rate of convergence of the ( 13) depends upon auxiliary constants c i . 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 (ii) Auxiliary constants c i can be calculated as follows: If R (s; k i ) = 0, then v (s; k i ) 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

Fractional Models of Time-Dependent PDE's and Results with Discussions
In this section, consider the three examples for the (OHAM) algorithm.which represents the accuracy, validity and effectiveness of the extended (OHAM) algorithm.The OHAM algorithm presented in Section 3 for time-fractional order FPEs and explanation of the formulation in the examples of Section 4, provides extremely valid results for the problems without any spatial discretization.While applying OHAM, there is no need to compute higher-order solutions.

Example
Let us assume the following FPE with time-fractional order.Where A (s) and B (s) are drift and diffusion coefficients.
Problem series of time-fractional order FPE with (zeroth, first, second and third-order).
Solution series of time fractional order Fokker-Planck equation with (zeroth, first, second and third-order).
In the following v (s, t) is the approximate result of time fractional order Fokker-Planck equation.

Example
Let us assume the following Fokker-Planck equation with time fractional order.Where A (s) and B (s) are drift and diffusion coefficients.
Problem series of time fractional order Fokker-Planck equation with (zeroth, first, second and third-order).
∂t α (33) Solution series of time fractional order Fokker-Planck equation with (zeroth, first, second and third-order).
In the following v (s, t) is the approximate result of time fractional order Fokker-Planck equation.exact Solution of example2 with = 1

Example
Let us assume the following Fokker-Planck equation with time fractional order.Where A (s) and B (s) are drift and di¤usion coe¢ cients.@ v(s; t) @t = @ @s 4v s Exact Solution

Example
Let us assume the following Fokker-Planck equation with time fractional order.Where A (s) and B (s) are drift and diffusion coefficients.
Problem series of time fractional order Fokker-Planck equation with (zeroth, first, second and third-order).
p 0 : ∂ α v 0 (s, t) ∂t α (40) Solution series of time fractional order Fokker-Planck equation with (zeroth, first, second and third-order).v 0 (s, t) = s (44) In the following v (s, t) is the approximate result of time fractional order Fokker-Planck partial differential equation.

Conclusion
In this article, time-fractional order of partial di¤erential equations is given for Fokker-Planck system.The applications in hand are solved by OHAM.
Results indicate the best agreement between approximate solutions and exact solutions.This work illustrates that optimal homotopy asymptotic approach is promptly convergent technique.Therefore OHAM demonstrates its power and hidden strength for the solutions of fractional models in applications of non-linear systems.

Appendix
Table8:Third order approximate results for di¤erent values of at t = 0:25;of example-1

Conclusions
In this article, time-fractional order of partial differential equations is given for Fokker-Planck system.The applications in hand are solved by OHAM.Results indicate the best agreement between approximate solutions and exact solutions.This work illustrates that optimal homotopy asymptotic approach is promptly convergent technique.Therefore OHAM demonstrates its power and hidden strength for the solutions of fractional models in applications of non-linear systems.
) and 15 figures (Figures 1-15 ) in Section 4. Tables 2, 6 and 11 of examples 1, 2 and 3 respectively represents the values of auxiliary constants c 1 , c 2 and c 3 for different values of α.Tables 3-5 of example 1, Tables 7-10 of example 2 and Tables 12-15 of example 3 respectively represents approximate results, exact results and absolute error for different values of α, and also for different values of time t = 1, and error norms of examples 1, 2 and 3 represents the validity and accuracy of the method.Error norms are available in the following Table

Figure 5 :
Figure 5:2D Solutions of example1 for di¤erent values of

Figure 5 .
Figure 5. 2D Solutions of Example 1 for different values of α.

Figure 5 :
Figure 5:2D Solutions of example1 for di¤erent values of

Figure 10 .
Figure 10.2D Solutions of Example 2 for different values of α.

Figure 5 :
Figure 5:2D Solutions of example3 for di¤erent values of

Figure 15 .
Figure 15.2D Solutions of Example 3 for different values of α.

Table 1 .
Error norms of examples.

Table 2 .
Auxiliary Convergence-constants for dissimilar values of α of Example 1.

Table 3 .
Third order approximate results for different values of α at t = 0.25 of Example 1.

Table 4 .
Third order approximate results for different values of α at t = 0.75 of Example 1.

Table 5 .
Third order approximate results for different values of α at t = 1 of Example 1.

Table 6 .
Auxiliary Convergence-constants for dissimilar values of α of Example 2.

Table 7 .
Third order approximate results for different values of α at t = 0.25 of Example 2.

Table 8 .
Third order approximate results for different values of α at t = 0.50 of Example 2.

Table 9 .
Third order approximate results for different values of α at t = 0.75 of Example 2.

Table 10 .
Third order approximate results for different values of α at t = 1 of Example 2.

Table 11 .
Auxiliary Convergence-constants for dissimilar values of α of Example 3.

Table 12 .
Third order approximate results for different values of α at t = 0.25 of Example 3.

Table 13 .
Third order approximate results for different values of α at t = 0.50 of Example 3.

Table 14 .
Third order approximate results for different values of α at t = 0.75 of Example 3.

Table 15 .
Third order approximate results for different values of α at t = 1 of Example 3.