1 Introduction

In order to present the partial differential equation that is solved numerically, firstly,we give an introduction to the linear and nonlinear Fokker-Planck equations(FPEs)and provide a brief review and history of these equations in the following subsection.

1.1 The Governing Equations

The solution of the FPEs is important in various fields of natural science,including astrophysics problems,biological applications,chemical physics,polymer,circuit theory,dielectric relaxation,economics,electron relaxation in gases,nucleation,optical bistability,dynamics,quantum optics,reactive systems,solid-state physics,and numerous other applications.[1]The origin and history of FPEs go back to the time when Fokker-Planck described the Brownian motion of particles.[1−2]The theory of Brownian motion exists in many areas of physics and chemistry,and particularly in those that deal with the nature of metastable states and the rates at which these states decay.[3]Kramers equation is a special form of the FPEs utilized to describe the Brownian motion of a potential.[4]

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The general form of the FPEs,for the variable x and t,is

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where A(x)and B(x)are referred to as the drift and diffusion coefficients and in case the drift and diffusion coefficients depended on time we can show it as:

The above equation is considered as the equation of motion for the distribution function y(x,t),and is also called the forward Kolmogorov equation.

In this section,generally,the GL functions are introduced and suitable formulas for the first-and second-order derivative matrices of these functions are presented.

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The more general forms of FPEs are its nonlinear form of the equation.The nonlinear FPEs may be derived from the principles of linear nonequilibrium thermodynamics.[5]Nonlinear FPEs have important applications and advantages in miscellaneous fields of sciences:biophysics,neurosciences,Engineering,laser physics,nonlinear hydrodynamics,plasma physics,pattern formation,poly-mer physics,population dynamics,psychology,surface physics.[1,6]

In the nonlinear FPEs,the equation also depends on y where this dependency happens in the drift and di ff usion coefficients.The general form of this equation is

by which

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Although there can be analytical solutions for the FPEs,it is difficult to result in solutions when the number of variables are large and no separation of variables methods are demanded.

1.2 The Literature Review on the FPEs

In the early 1990s,Palleschi et al.[7−8]investigated FPEs. They discussed a fast and accurate algorithm for the numerical solution of Fokker-Planck-like equation.Vanaja[9]presented an iterative solution method for solving FPEs.Zorzano et al.[10]used the finite difference to investigate two-dimensional of this equation.Dehghan et al.[11]employed the He’s variational iteration method(VIM)to give a solution for this equation.Tatari et al.[12]applied the Adomian decomposition method for solving the FPEs.Using the cubic B-spline scaling functions,Lakestani et al.[2]obtained the numerical solution of FPEs.Kazem et al.[6]utilized RBF to solve the equation.

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Other insights for solving FPEs are numerical techniques.Among them,Wehner[13]applied path integrals to solve the nonlinear FPEs.Fourier transformations were employed by Brey et al.[14]Zhang et al.[15]applied distributed approximating functionals to solve the nonlinear FPEs.Further to these,for solving the one-dimensional nonlinear FPEs,the finite difference schemes[16]are also applied.

In recent years,dozens of scientists are attracted to Spectral and pseudo spectral methods.[17−18]Spectral methods are providing the solution of the problem with the aid of truncated series of smooth global functions;[19−20]They provide such an accurate approximation for a smooth solution with relatively few degrees of freedom.They are widely employed in the approximation of the solution of differential equations,variational problems,and function approximation.The reason existed beyond this accuracy is that the spectral coefficients tend to zero faster than any algebraic power of their index n.[21]As said in such papers,spectral methods can fall into 3 categories:Collocation,Galerkin,and Tau methods[22]Collocation method provides highly accurate solutions to nonlinear differential equations.[23−26]There are only two main steps to approximate a problem in collocation methods:First,as a common approach,appropriate nodes(Gauss/Gauss-Radau/Gauss-Lobatto)are chosen to represent a finite or discrete form of the differential equations.

Second,a system of algebraic equations from the discretization of the original equation is obtained.[27−29]The Tau spectral method is one of the most important methods used to approximate numerical solutions of various differential equations.This method approximates the solution as an expansion of certain orthogonal polynomials/functions and the coefficients,in the expansion,are considered so as to satisfy the differential equation as accurately as possible.[30]Spectral Tau method is,somehow,similar to Galerkin methods in the way that the differential equation is enforced.[21]In Galerkin Spectral method,a finite dimensional subspace of the Hilbert space(trial function space)are selected and trail and test functions are regarded the same.[31]

Moreover,some numerical methods like Finite difference method(FDM)and Finite element method(FEM)that are implemented locally and require a network of data.Such methods like Meshfree methods do not require to build a network of data.[32−33]Comparing to these mentioned numerical methods,spectral methods are globally performing and they are continuous and do not need network construction.

In addition to spectral methods,pseudospectral methods have been of high interest to authors presently.[34−37]

Actually,in standard pseudospectral methods,interpolation operators are used to reduce the cost of computation of the inner product,in some spectral methods.For this purpose,a set of distinct interpolation pointsis defined,where the corresponding Lagrange interpolants are achieved.In addition to this,in collocation points,the residual function is set to vanish on the same set of points.Generally speaking,these collocation points do not need to be the same as the interpolation points;however,to have the Kronecker property,they are considered to be the same:therefore,by this trick,they reduce computational cost remarkably.[38−39]

1.3 The Main Aim of This Paper

In this study,we develop an exponentially accurate generalized pseudospectral method for solving the linear and nonlinear FPEs:This method is a generalization of the classical Lagrange interpolation method.To reach this goal,in Sec.2 some preliminaries of Jacobi polynomials are brought.In this section,we introduce the GL Functions and develop the GLJGL collocation scheme.Section 3 describes the numerical method;it explains the methodology and estimation of the error.We carry out numerical experiments to validate the presented collocation scheme.Subsequently,the analysis will be implemented to linear and nonlinear FPEs.Finally,some concluding remarks are given in Sec.5.

then,the matrix form of Eqs.(28)and(29)will be

2 Preliminaries and Notations

2.1 Jacobi Polynomials

The Jacobi polynomials are the eigenfunctions of a singular Sturm-Liouville equation. There are several particular cases of them,such as Legendre,the four kinds of Chebyshev,and Gegenbauer polynomials.Jacobi polynomials are defined on[−1,1]and are of interest recently.[36,40−43]The recursive formula for Jacobi polynomials is as follows:[44]

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with the properties as:

(3) 试验初期，排水管壁面积的大小会影响土体梯度比Gr值下降速度。与小直径排水管壁试样相比，在大直径试样条件下，砾质黏性土下降速度变缓的时间比小直径试样早3 h，砂质黏性土早3 h，粉质黏性土早1 h。梯度比下降速度大小为：大直径排水管壁试样>小直径排水管壁试样。

and its weight function is wα,β(x)=(1 − x)α(1+x)β.

By the aid of these,we can write Eq.(25)as

Moreover,the Jacobi polynomials are orthogonal on[−1,1]:

where δm,nis the Kronecker delta function.

The set of Jacobi polynomials makes a completeorthogonal system for any g(x) ∈there is an expansion as follows.

where

2.2 Generalized Lagrange(GL)functions

In addition to the forward Kolmogorov equation,there is another form of the equation called backward Kolmogorov equation.

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Definition 1 Considering the generalized Lagrange(GL)functions formula can be shown as:[38−39]

where κj=u′j/∂uw(xj), ∂uw(x)=(1/u′)∂xw(x),and u(x)is a continuous and sufficiently differentiable function which will be chosen to fit in the problem’s characteristics.For simplicity u=u(x)and ui=u(xi)are considered.The GL functions have the Kronecker property:

Theorem 1 Considering the GL functions Luj(x)in Eq.(13),one can exhibit the first-order derivative matrices of GL functions as

where

Proof As the GL functions defined in Eq.(13),the first-order derivative formula for the case kj can be achieved as follows:

But,when k=j,with L’Hˆopital’s rule:

This completes the proof. ?

2.3 Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL)Collocation Method

In case of GLJGL collocation method,w(x)in Eq.(13)can be restated as:

where λ is a real constant and to simplify the notation,we write

with the following important properties:

Then,we have:

Recalling thatand using formulas in Eq.(15)–(20),we find the entry of the first-order derivative matrix of GL functions as:

Theorem 2 Let D(1)be the above matrix( first order derivative matrix of GL functions)and define matrix Q such thatthen,the second-order derivative matrix of GL functions can be formulated as:

Proof See Ref.[38]. ?

3 Numerical Method

In this section, firstly,the time discretization method is recalled.Secondly,GLJGL collocation method is implemented to solve the FPEs.In a matrix form,the method has been presented and the error of this method is estimated.

3.1 Discretization

For solving the FPEs,we first discretize the time domain;to do this,we apply the Crank-Nicolson method.The main reason for choosing this method is its good convergence order and its unconditional stability.[45]To apply this method, firstly,we approximate and simplify the first-order derivative of y(x,t),with respect to the time variable,and deriving a formula from finite difference approximations as follows:

As said in the previous subsection,in each time level,we approximate the solution of FPEs,and therefore,the time variable is omitted from the equation.In each time step,we approximate an equation like in Eq.(25).The unknown yi+1(x)is approximated as

Considering FPEs,one can read in which E0,k,E1,iand E2,iare the coefficient specified in the “Numerical Examples” section;in linear FPEs,E0,i=0.

Implementing Crank-Nicolson on FPEs

and can be simplified as

By applying this method,the problem can be discretized in small time levels.As shown,time variable is discretized using Crank-Nicolson method.In each time level,we are to approximate the FPEs.Solving in sufficiently large time levels,brings in a good approximation for FPEs.

3.2 Implementation of GLJGL Collocation Method for Solving FPEs

The domain Ω × [0,T)is decomposed as Ω × [0,T)=and∆t=T/s:The error of this approximation is of order O(∆t).From now on,for simplicity yi(x)=y(x,Ti).

where

As y(x,0)=y0(x)=f(x)we can calculate f(x)=LA0,and by collocating n+1 nodes we can result in:

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The boundary conditions,by considering Guass-Lobatto scheme and Eq.(26),are specified as:

therefore,by collocating n+1 points and de fi ning

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The first and last row of matrices H0,H1,H2,and first and last elements of vector R are defined as if they satisfy the boundary condition of FPEs.

Hence,we can achieve the numerical solution of y(x,t)at each time level.Notice that,at time level 0 the solution is computed from the initial condition;This is shown in Eq.(27).From the solution of the system in Eq.(30),at each time level,for the next time levels,we will achieve the unknown values.In other words,it means that by solving this system,in each step of i+1,the unknown coefficients Ai+1will be found.

This system of equations is solved by applying a proper method like Newton methods.To show the accuracy of this method,some examples in the next section,are illustrated.

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3.3 Error Estimation

Theorem 3 Let x0=a,xn=b andbe the roots shifting Jacobi polynomialfrom[−1,1]to[a,b].Then,there exists a unique set of quadrature weights if ned by Jie Shen[46](Jacobi Gauss-Lobatto quadratures),such that for all functions p(x)of degree 2n−1

where w(x)is the weight function and here this weight function is wα,β(u(x)).This is worth noticing that

{ti,are Jacobi Gauss-Lobatto quadratures nodes and weights.

Proof See Ref.[46]. ?

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In FPEs[a,b]=[0,1],u(x)=2x−1,then∫

based in the last theorem,when p(x)∈Pm,m>2n−1,the above relation between integral and summation is not exact;it produces an error term as

where ξ∈(a,b).Hence,

For two arbitrary functions g1(x)and g2(x)we define

then forwe have

In the same fashion,for

As er[q(x)]=0,as long as q(x)∈Pm,m≤2n−1.Obviously,if any of the above terms’degree is less and equal than 2n−1,the error of that term will be zero.In numerical examples,this error is shown and discussed.

With Eqs.(26)and(33)the following relations in xkwill be obtained:(j=i,i+1)

in which D[k,:]means that the k-th row of matrix D is taken.Now,by taking xkinto account.k=0,...,n

Comparing with the system in Eq.(30)we solved,V is the error term vector:V is defined as:

for k=1,...,n−1,and v0=0,vn=0.

Now,by multiplying Eq.(25)with(x)wα,β(x)and integration in both sides:

4 Numerical Examples

In this section,in order to illustrate the performance of the GLJGL collocation method,we give some computations based on preceding sections,to support our theoretical discussion.By the aid of the presented method,linear and nonlinear forms of FPEs are solved.To illustrate the good accuracy of these methods,we apply different error criteria:The root-mean-square(RMS),Ne,and L2errors.

where y(xj)and yn(xj)are exact and approximate value of FPEs on equidistant xj,j=1,...,r.

As FPEs are defined over[0,1],the shifting function u(x),considered in Subsecs.2.2 and 2.3,is u(x)=2x−1.

The CPU time for calculation of matrices D(1)and D(2),defined in Subsec.2.3,is brought in Table 1.

The CPU time is performed on a DELL laptop with the configuration:Intel(R)Core(TM)i7-2670QM CPU,2.20 GHz;and 6 GB RAM.

Example 1 Consider Refs.[2,6,11]Eq.(1)with:A(x)=−1,B(x)=1,f(x)=x,x∈[0,1].

The exact solution of this test problem is y(x,t)=x+t.In this example E0,k=0,E1,k=−A(xk)=1,and E2,k=B(xk)=1 for k=1,...,n−1.

As stated earlier,if the order of terms in Eq.(35)is less than 2n,the error terms vanish;so,the error vector for Ex.1,V in Eq.(35),can be simplified as

In Table 2,the numerical absolute errors of Example 1,and their comparison with B-Spline method are displayed.Table 3,by representing the values of RMS and Neerrors,reveals the difference between the presented method and both HRBF and Kansa’s approaches.[6]

In Fig.1,RMS,L2and Neerrors,for different values of n and∆t,have been illustrated.Figure 2 shows the plot of error for Ex.1.

Example 2 Consider Refs.[2,6,11]the backward Kolmogorov Eq.(4)with:A(x,t)=−(x+1),B(x,t)=x2et,f(x)=x+1,x∈[0,1].

The exact solution of this test problem is y(x,t)=(x+1)et.In this example E0,k=0,E1,k=−A(xk,t)=

Table 4 depicts the numerical absolute errors of Ex.2 and draws a distinction with the presented method and BSpline method.For showing the accuracy,the differences between the presented method and HRBF and Kansa’s approaches[6]are shown by calculating RMS and Nein Table 5.In Fig.3,the error measurements RMS,L2and Neare shown for different n and∆t.In this figure,CPU times have been depicted for different n and∆t.It explicitly says that when n increases or∆t decreases,the time of solving the system of Eq.(30)increases.As it shows,when∆t tends to a smaller value,it affects and decreases all RMS,Ne,L2and absolute errors.The plot of absolute error for Ex.2 is also shown in Fig.4.

Example 3 Consider Refs.[2,6,11]the nonlinear Eq.(5)with:A(x,t,y)=(7/2)y,B(x,t,y)=xy,f(x)=x,x∈[0,1].

The exact solution of this test problem is y(x,t)=x/(1+t).By this consideration,Eq.(5)can be rewritten as

By Eqs.(23)and(36)one can set:

The error vector,V in Eq.(35),for Ex.2 and 3 is

By the aid of Table 6.the numerical absolute errors for Ex.3 demonstrated and a comparison with the B-Spline method is made.For this example,also,RMS and Neare compared with the ones provided by HRBF[6]in Table 7.

Figure 5 shows the values of RMS,L2and Neerrors for different n and∆t.This Figure,illustrates the CPU times for solving the system of Eq.(30)for different n and∆t.It shows that when n increases or∆t decreases,the time of obtaining solution will increase.The fact is,as∆t becomes smaller,RMS,Ne,L2and absolute errors decrease.The plot of absolute error for Ex.3 is also shown in Fig.6.

Example 4 Consider Refs.[2,6,11]the nonlinear Eq.(5)with:A(x,t,y)=4(y/x)−x/3,B(x,t,y)=y,f(x)=x2,x∈[0,1].

The exact solution of this test problem is y(x,t)=x2et.This nonlinear FPEs can be restated as

It must be noted that:the way this relation is factorized is playing a central role in the exactness of solution.By Eqs.(23)and(37):

For Ex.4,the error vector specified in Eq.(35)is

for k=1,...,n−1 and v0=0,vn=0.

In Table 8,the numerical absolute errors for Ex.4 demonstrated and a comparison with the B-Spline method is given.The error measurements RMS and Neare calculated by the presented method and HRBF[6]method and the results depicted in Table 9.Figure 7 illustrates the values of RMS,L2and Neerrors for different n and∆t.This Figure,also,illustrates the CPU times for solving the system of Eq.(30)for different n and∆t.It implies that when n increases or∆t decreases,the time of obtaining solution increases.In fact,when∆t becomes smaller,RMS,Ne,L2and absolute errors will decrease.The plot of absolute error for Ex.4 is also shown in Fig.8.

5 Conclusion

The(linear and nonlinear)FPEs have many applications in science and engineering.So,in this work,a numerical method based on GLJGL collocation method is discussed and developed to investigate FPEs.Firstly,we introduced GL functions with the Kronecker property.The advantages of using GL functions can be:

(i)These functions are the generalization of the classical Lagrange polynomials and corresponding differentiation matrices of D(1)and D(2),as shown,can be reached by specific formulas;this helps create and introduce a derivative-free method.

(ii)With different consideration of u(x),different basis of GL functions are provided;therefore,different problems defined on various intervals can be solved.

(iii) The accuracy of the presented method by GL function has exponential convergence rate.

Moreover,the time derivative of the FPEs is discretized using Crank-Nicolson method.The main reason for using Crank-Nicolson method is its unconditional stability.[3,45]

By the aid of Crank-Nicolson technique,we solved the linear and nonlinear types of FPEs with GLJGL collocation method.We apply the pseudospectral method in a matrix based manner where the matrix based structure of the present method makes it easy to implement.Also,to show the accuracy and ability of the proposed method,several examples are solved.

Several examples are given and the results obtained using the method introduced in this article show that the new proposed numerical procedure is efficient

The results showed that the approximate solutions of the GLJGL collocation method can be acceptable and provides very accurate results even with using a small number of collocation points.To illustrate the suitable accuracy of the proposed method,we used three different error criteria,namely,RMS,L2and Ne.Additionally,the obtained results have been compared with B-Spline,HRBF and Kansa methods,showing the accuracy and reliability of the presented method.

This method can also be used as a powerful tool for investigation of other problems.

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