# Fractional-View Analysis of Space-Time Fractional Fokker-Planck Equations within Caputo Operator

Authors: Saleh Alshammari; Naveed Iqbal; Mohammad Yar
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this article, we investigate the fractional-order Fokker-Planck equations with the help of the Yang transform decomposition method (YTDM). The YTDM combines Yang transform, Adomian decomposition method, and Adomian polynomials into one method. In the Caputo sense, fractional derivatives of space and time are studied. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Compared to other methods of finding approximate and exact solutions for nonlinear partial differential equations, this technique is more efficient and time-consuming.


---

## Body

## 1. Introduction

Fractional calculus, which can be thought of as a generalization of integer-order differentiation and integration, has received much attention in recent decades. Many definitions have been proposed for fractional derivatives, including Riesz, Grunwald-Letnikov, Caputo, Riemann-Liouville, and conformable fractional definitions [1–4]. Noninteger order integral and differential operators contain all historical conditions of the function in a weighted form known as the memory effect. In any case, fractional differential equations (FDEs), specifically fractional partial differential equations, are used to analyze a broad range of physical systems (FPDEs). FPDEs have gained attention due to their widespread application in electrical circuits, electrochemistry, quantum physics, and theoretical biology [5–8]. Furthermore, the nonlocal property of FPDEs is the most important feature for using them in such and other applications, whereas the differential operator having order integer is local. In this light, the next state of a fractional system is determined by both its current and historical states. This ensures that the mathematical model components in physical processes and dynamic systems are highly consistent. However, it is not easy to solve those FDEs, particularly for numerical calculations [9–11]. To handle partial differential equations (PDEs), having order fraction is of physical importance, and effective, trustworthy, and appropriate numerical methods are required [12–14]. Several major strategies have been utilized in this regard, including the fractional operational matrix method (FOMM) [15], Elzaki transform decomposition method (ETDM) [16, 17], homotopy analysis method (HAM) [18], homotopy perturbation method (HPM) [19, 20], iterative Laplace transform method [21], and variational iteration method (FVIM) [22].The Fokker-Planck equation is a well-known statistical physics equation that Fokker and Planck first proposed to describe a particle’s Brownian motion and the change in probability of a random function in time and space [23]. An uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation can also be used to obtain the chemical Fokker-Planck equation. This equation proves to be more accurate than the chemical master equation’s linear-noise approximation. The Fokker-Planck equation appears in many natural science phenomena, such as probability flux, polymer dynamics, electron relaxation, solid-state systems, quantum optics, and other practical and theoretical models [24].We have studied Fokker-Planck equations of fractional-order having general form as(1)φIγμ,I=Lφμμ,I+φμμμ,I+Nφμμμ,I,μ,I>0,γ∈0,1,with the initial condition
(2)φμ,0=ζμ.In biological molecules, chemical physics, energy consumption, and engineering, the fractional Fokker-Planck equation (F-FPE) has been successfully applied. Indeed, fractional diffusion, a special kind of F-FPE, has also been used in numerous scenarios such as frequency-dependent damping behaviour of materials, viscoelasticity, and diffusion processes [17]. Unfortunately, finding an accurate solution for FDEs, in general, is difficult. To approximate these solutions, various numerical and analytical techniques are used. Some of the advanced numerical and approximate methods used for F-FPEs include the Laplace transform method [18], the multistep reduced differential transform method [25], the predictor-corrector approach [26], the Adomian decomposition method (ADM) [27], and the variational iteration method (VIM) [28].In this research, we used the Yang transform decomposition method (YTDM) to solve time-fractional F-FPEs. The Yang transform was proposed by Xiao-Jun Yang and can be utilized to solve a variety of differential equations with constant coefficients. The Adomian decomposition method [29] is a well-known methodology to solve linear and nonlinear differential and partial differential equations and integrodifferential and FDEs that yield accurate solutions in a concurrent series form. The results of the suggested strategy are convincing and offering specific solutions to the problems at work. The fractional problem results obtained through the given approach are also used to analyze the problems fractionally. It has been confirmed that the proposed technique can be implemented to solve various fractional PDEs and related systems.


## 2. Preliminaries

We covered several fundamental definitions of fractional calculus as well as Yang transform theory features in this part.Definition 1.
The fractional Caputo derivative is defined as(3)Dφγφμ,I=1Γk−γ∫0II−ϑk−γ−1φkμ,ϑdϑ,k−1<γ≤k,k∈N.Definition 2.
Xiao-Jun Yang introduced the Yang Laplace transform in 2018.φI or Mu determines the Yang transform for a function φI and is provided as
(4)YφI=Mu=∫0∞e−I/uφIdI,I>0,u∈−I1,I2.
The inverse Yang transform is given as(5)Y−1Mu=φI.Definition 3.
Fornth derivatives, the Yang transform is given as
(6)YφnI=Muun−∑k=0n−1φk0un−k−1,∀n=1,2,3,⋯.Definition 4.
For derivative having fractional order, the Yang transform is(7)YφγI=Muuγ−∑k=0n−1φk0uγ−k+1,0<γ≤n.


## 3. Idea of YTDM

The general methodology for solving fractional partial differential equations is given as(8)DIγφμ,I=P1μ,I+Q1μ,I,0<γ≤1,with initial sources
(9)φμ,0=φμ,∂∂Iφμ,0=ζμ,where Caputo fractional derivative having order γ is represented by DIγ=∂γ/∂Iγ; P1 and Q1 are linear and nonlinear functions, respectively.On employing Yang transform, we get(10)YDIγφμ,I=YP1μ,I+Q1μ,I.By using Yang differentiation property, we have(11)1uγMu−uφ0−u2φ′0=YP1μ,I+Q1μ,I.From above equation(12)Mφ=uφ0+u2φ′0+uγYP1μ,I+Q1μ,I.On applying inverse Yang transform, we have(13)φμ,I=φ0+φ′0+Y−1uγYP1μ,I+Q1μ,I.The solution in terms of infinite sequenceφμ,I by means of YTDM is
(14)φμ,I=∑m=0∞φmμ,I.Now, the nonlinear terms by means of Adomian polynomials are decomposed as(15)Q1μ,I=∑m=0∞Am.The Adomian polynomials all forms of nonlinearity are given as(16)Am=1m!∂m∂ℓmQ1∑k=0∞ℓkμk,∑k=0∞ℓkIkℓ=0.By substituting Equation (35) and Equation (38) into (34), we have
(17)∑m=0∞φmμ,I=φ0+φ′0+Y−1uγYP1∑m=0∞μm,∑m=0∞Im+∑m=0∞Am.The below terms are derived.(18)φ0μ,I=φ0+Iφ′0,φ1μ,I=Y−1uγY+P1μ0,I0+A0,thus for m≥1, the general term is given as
(19)φm+1μ,I=Y−1uγY+P1μm,Im+Am.Theorem 5.
Here, we will study the convergence analysis as same manner in [30] of the YTDM applied to the fractional order partial differential equation. Let us consider the Hilbert space H which may define by H=L2α,βX0,T the set of applications:
(20)u:α,βX0,T⟶with∫α,βX0,Tu2x,sdsdθ<+∞.
Now, we consider the fractional partial differential equation in the above assumptions and let us denote(21)Yu=∂γu∂Iγ,then the fractional partial differential equation becomes in an operator form
(22)Yu=−φ∂νx,I∂x−w∂3νx,I∂x3.
The YTDM is convergence if the following two hypotheses are satisfied:
H1:Yu−Yv,u−v≥ku−v2;k>0,∀u,vεH
H2: whatever may beM>0, there exist a constant CM>0 such that for u,vεH with u≤M and v≤M we have Yu−Yv,u−v≤CMu−vw for every wεH


## 4. Applications

Here, in this part, we implemented YTDM for solving various time-fractional Fokker-Planck equation.Example 1.
Consider F-FPEs of the form(23)∂γ∂Iγφμ,I+∂∂μμ6φμ,I−∂2∂μ2μ212φμ,I=0,μ,I>0,γ∈0,1,with the initial condition
(24)φμ,0=μ2.
On employing Yang transform, we get(25)Y∂γφ∂Iγ=Y−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I.
By using Yang differentiation property, we have(26)1uγMu−uφ0=Y−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I,Mu=uφ0+uγY−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I.
On applying inverse Yang transform, we have(27)φμ,I=φ0+Y−1uγY−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I,φμ,I=μ2+Y−1uγY−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I.
The solution in terms of infinite sequenceφμ,I by means of YTDM is
(28)φμ,I=∑m=0∞φmμ,I,∑m=0∞φmμ,I=μ2+Y−1uγY−∂∂μμ6φμ,I+∂2∂μ2μ212φμ,I.
By comparing Equation (28) both sides, we get
(29)φ0μ,I=μ2.
Onm=0,
(30)φ1μ,I=μ2Iγ2Γγ+1.
Onm=1,
(31)φ2μ,I=μ2I2γ8Γ2γ+1.
Onm=2,
(32)φ3μ,I=μ2I3γ24Γ3γ+1.
The YTDM solution remaining componentsφm for m≥3 are calculated easily. Thus, we define the series form solution as
(33)φμ,I=∑m=0∞φmμ,I=φ0μ,I+φ1μ,I+φ2μ,I+φ3μ,I+⋯,φμ,I=μ2+μ2Iγ2Γγ+1+μ2I2γ8Γ2γ+1+μ2I3γ24Γ3γ+1+⋯.
The YTDM solution atγ=1 is
(34)φμ,I=μ2expI/2.In Figure1, the first graph shows the exact and second the analytical solution graph, which shows the close contact with each other. In Figure 1, the third and fourth graphs are the three- and two-dimensional graphs concerning different fractional order of problem 1. The figures show that the suggested technique agrees with the actual solution for the given problem. As fractional order approaches integer order, fractional-order solution surfaces converge to the integer-order surface, as depicted by graphs. It means that we may physically model any surface based on the physical events observed in nature.Example 2.
Consider F-FPEs of the form(35)∂∂Iγφμ,I+∂∂μμφμ,I−∂2∂μ2μ22φμ,I=0,μ,I>0,γ∈0,1,with the initial condition
(36)φμ,0=μ.
On employing Yang transform, we get(37)Y∂γφ∂Iγ=Y−∂∂μμφμ,I+∂2∂μ2μ22φμ,I.
By using Yang differentiation property, we have(38)1uγMu−uφ0=Y−∂∂μμφμ,I+∂2∂μ2μ22φμ,I,(39)Mu=uφ0+uγY−∂∂μμφμ,I+∂2∂μ2μ22φμ,I.
On applying inverse Yang transform, we have(40)φμ,I=φ0+Y−1uγY−∂∂μμφμ,I+∂2∂μ2μ22φμ,I,φμ,I=μ+Y−1uγY−∂∂μμφμ,I+∂2∂μ2μ22φμ,I.
The solution in terms of infinite sequenceφμ,I by means of YTDM is
(41)φμ,I=∑m=0∞φmμ,I,∑m=0∞φmμ,I=μ+Y−1uγY−∂∂μμφμ,I+∂2∂μ2μ22φμ,I.
By comparing Equation (41) both sides, we get
(42)φ0μ,I=μ.
Onm=0,
(43)φ1μ,I=μIγΓγ+1.
Onm=1,
(44)φ2μ,I=μI2γΓ2γ+1.
Onm=2,
(45)φ3μ,I=μI3γΓ3γ+1.
The YTDM solution remaining componentsφm with m≥3 are calculated easily. Thus, we define the series form solution as
(46)φμ,I=∑m=0∞φmμ,I=φ0μ,I+φ1μ,I+φ2μ,I+φ3μ,I+⋯,φμ,I=μ+μIγΓγ+1+μI2γΓ2γ+1+μI3γΓ3γ+1+⋯.
The YTDM solution atγ=1 is
(47)φμ,I=μexpI.Figure 1
Nature of the exact solution, analytical solution, and solution at various fractional orders of problem 1.In Figure2, the first graph shows the exact and second the analytical solution graph, which shows the close contact with each other. In Figure 2, the third and fourth graphs are the three- and two-dimensional graphs concerning different fractional order of problem 2. The figures show that the suggested technique agrees with the actual solution for the given problem. As fractional order approaches integer order, fractional-order solution surfaces converge to the integer-order surface, as depicted by graphs. It means that we may physically model any surface based on the physical events observed in nature.Example 3.
Consider F-FPEs of the form(48)∂∂Iγφμ,I+∂∂μ4μφ2μ,I−∂∂μμ3φμ,I−∂2∂μ2φ2μ,I=0,μ,I>0,γ∈0,1,with the initial condition
(49)φμ,0=μ2.
On employing Yang transform, we get(50)Y∂γφ∂Iγ=Y∂∂μμ3φμ,I+∂2∂μ2φ2μ,I−∂∂μ4μφ2μ,I.
By using Yang differentiation property, we have(51)1uγMu−uφ0=Y∂∂μμ3φμ,I+∂2∂μ2φ2μ,I−∂∂μ4μφ2μ,I,Mu=uφ0+uγY∂∂μμ3φμ,I+∂2∂μ2φ2μ,I−∂∂μ4μφ2μ,I.
On applying inverse Yang transform, we have(52)φμ,I=φ0+Y−1uγY∂∂μμ3φμ,I+∂2∂μ2φ2μ,I−∂∂μ4μφ2μ,I,φμ,I=μ2+Y−1uγY∂∂μμ3φμ,I+∂2∂μ2φ2μ,I−∂∂μ4μφ2μ,I.
Now, by assuming that the infinite series form the functionφμ,I which is unknown, it has the solution as
(53)φμ,I=∑m=0∞φmμ,I.
Thus, the nonlinear terms are defined by the Adomian polynomialφ2=∑m=0∞Am. Using specific concepts, Equation (52) can be rewritten in the form
(54)∑m=0∞φmμ,I=φμ,0+Y−1uγY∂∂μμ3φμ,I+∑m=0∞Am,∑m=0∞φmμ,I=μ2+Y−1uγY∂∂μμ3φμ,I+∑m=0∞Am.
Now, by Adomian polynomialQ1, the nonlinear terms are decomposed according to Equation (38),
(55)A0=φ02,A1=2φ0φ1,A2=2φ0φ2+φ22.
By comparing Equation (54) both sides, we get
(56)φ0μ,I=μ2.
Onm=0,
(57)φ1μ,I=μ2IγΓγ+1.
Onm=1,
(58)φ2μ,I=μ2I2γΓ2γ+1.
Onm=2,
(59)φ3μ,I=μ2I3γΓ3γ+1.
The YTDM solution remaining componentsφm with m≥3 are calculated easily. Thus, we define the series form solution as
(60)φμ,I=∑m=0∞φmμ,I=φ0μ,I+φ1μ,I+φ2μ,I+φ3μ,I+⋯,φμ,I=μ2+μ2IγΓγ+1+μ2I2γΓ2γ+1+μ2I3γΓ3γ+1+⋯.
The YTDM solution atγ=1 is
(61)φμ,I=μ2expI.Figure 2
Nature of the exact solution, analytical solution, and solution at various fractional orders of problem 2.In Figure3, the first graph shows the exact and second the analytical solution graph, which shows the close contact with each other. In Figure 3, the third and fourth graphs are the three- and two-dimensional graphs concerning different fractional order of problem 3. The figures show that the suggested technique agrees with the actual solution for the given problem. As fractional order approaches integer order, fractional-order solution surfaces converge to the integer-order surface, as depicted by graphs. It means that we may physically model any surface based on the physical events observed in nature.Example 4.
Consider F-FPEs of the form(62)∂∂Iγφμ,I−∂∂μφμ,I−∂2∂μ2φμ,I=0,I>0,γ∈0,1,with the initial condition
(63)φμ,0=μ.
On employing Yang transform, we get(64)Y∂γφ∂Iγ=Y∂∂μφμ,I+∂2∂μ2φμ,I.
By using Yang differentiation property, we have(65)1uγMu−uφ0=Y∂∂μφμ,I+∂2∂μ2φμ,I,Mu=uφ0+uγY∂∂μφμ,I+∂2∂μ2φμ,I.
On applying inverse Yang transform, we have(66)φμ,I=φ0+Y−1uγY∂∂μφμ,I+∂2∂μ2φμ,I,φμ,I=μ+Y−1uγY∂∂μφμ,I+∂2∂μ2φμ,I.
The solution in terms of infinite sequenceφμ,I by means of YTDM is
(67)φμ,I=∑m=0∞φmμ,I,∑m=0∞φmμ,I=μ+Y−1uγY∂∂μφμ,I+∂2∂μ2φμ,I.
By comparing Equation (67) both sides, we get
(68)φ0μ,I=μ.
Onm=0,
(69)φ1μ,I=IγΓγ+1.
Onm=1,
(70)φ2μ,I=0.
Onm=2,
(71)φ3μ,I=0.
The YTDM solution remaining componentsφm with m≥3 are calculated easily. Thus, we define the series form solution as
(72)φμ,I=∑m=0∞φmμ,I=φ0μ,I+φ1μ,I+φ2μ,I+φ3μ,I+⋯,φμ,I=μ+IγΓγ+1+0+0+⋯.
The YTDM solution atγ=1 is
(73)φμ,I=μ+I.Figure 3
Nature of the exact solution, analytical solution, and solution at various fractional orders of problem 3.In Figure4, the first graph shows the exact and second the analytical solution graph, which shows the close contact with each other. In Figure 4, the third and fourth graphs are the three- and two-dimensional graphs concerning different fractional order of problem 4. The figures show that the suggested technique agrees with the actual solution for the given problem. As fractional order approaches integer order, fractional-order solution surfaces converge to the integer-order surface, as depicted by graphs. It means that we may physically model any surface based on the physical events observed in nature.Example 5.
Consider F-FPEs of the form(74)∂γ∂Iγφμ,I−1−μ∂∂μφμ,I−eIμ2∂2∂μ2φμ,I=0,I>0,γ∈0,1,with the initial condition
(75)φμ,0=1+μ.
On employing Yang transform, we get(76)Y∂γφ∂Iγ=Y1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I.
By using Yang differentiation property, we have(77)1uγMu−uφ0=Y1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I,Mu=uφ0+uγY1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I.
On applying inverse Yang transform, we have(78)φμ,I=φ0+Y−1uγY1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I,φμ,I=1+μ+Y−1uγY1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I.
The solution in terms of infinite sequenceφμ,I by means of YTDM is
(79)φμ,I=∑m=0∞φmμ,I,(80)∑m=0∞φmμ,I=1+μ+Y−1uγY1−μ∂∂μφμ,I+eIμ2∂2∂μ2φμ,I.
By comparing Equation (80) both sides, we get
(81)φ0μ,I=1+μ.
Onm=0,
(82)φ1μ,I=1+μIγΓγ+1.
Onm=1,
(83)φ2μ,I=1+μI2γΓ2γ+1.
Onm=2,
(84)φ3μ,I=1+μI3γΓ3γ+1.
The YTDM solution remaining componentsφm with m≥3 are calculated easily. Thus, we define the series form solution as
(85)φμ,I=∑m=0∞φmμ,I=φ0μ,I+φ1μ,I+φ2μ,I+φ3μ,I+⋯,φμ,I=1+μ+1+μIγΓγ+1+1+μI2γΓ2γ+1+1+μI3γΓ3γ+1+⋯.
The YTDM solution atγ=1 is
(86)φμ,I=expI1+μ.Figure 4
Nature of the exact solution, analytical solution, and solution at various fractional orders of problem 4.In Figure5, the first graph shows the exact and second the analytical solution graph, which shows the close contact with each other. In Figure 5, the third and fourth graphs are the three- and two-dimensional graphs concerning different fractional order of problem 5. The figures show that the suggested technique agrees with the actual solution for the given problem. As fractional order approaches integer order, fractional-order solution surfaces converge to the integer-order surface, as depicted by graphs. It means that we may physically model any surface based on the physical events observed in nature.Figure 5
Nature of the exact solution, analytical solution, and solution at various fractional orders of problem 5.


## 5. Conclusion

The Adomian decomposition approach was expanded in this paper to find explicit and numerical solutions to the F-FPEs. The proposed method is an effective and powerful strategy for solving the proposed equations. The plotted graphs confirm the strong relationship between the exact and analytical results. The approaches provide series form solutions with a higher convergence rate to exact results. While providing quantitatively accurate results, the Adomian decomposition method requires less computational work than existing approaches.



---

*Source: 4471757-2022-07-04.xml*