Exponential differential operators for singular integral equations and space fractional Fokker-Planck equation

In this article, it is shown that the combined use of exponential operators and integral transform provides a powerful tool to solve certain type of fractional PDEs. It is shown that exponential operators are powerful and effective method for solving certain differential equations and fractional Fokker-Planck equations with non-constant coefficients. Constructive examples are also provided.


Introduction
We present a general method of operational nature to obtain solutions for several types of partial differential equations.The integral transform technique is one of the most useful tools of applied mathematics, employed in many branches of science, mathematical physics and engineering.The most interesting and useful applications of the Laplace transformation are solving linear differential equations with discontinuous or impulsive forcing functions which are common place in mechanical systems and circuit analysis problems.(1.1)

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A. Aghili and J.Aghili If L{f (t)} = F (s), then L −1 {F (s)} is given by [3] f (1. 3) The left Riemann-Liouville fractional integral of order α > 0 is defined as [7] (1.4) It follows that D RL,α a φ(x) exists for all Φ(t) belongs to C[a, b] ,and a < t < b .Note: A very useful fact about the R-L operators is that they satisfy semi group properties of fractional integrals.The special case of fractional derivative when α = 0.5 is called semi -derivative. (1.5) Lemma 1.1:The following identities hold true.
The following exponential identities hold true.

exp(±α
where F (t) is primitive of (q(t)) −1 and Q(t) is inverse of F (t). Proof.See [4].Like Fourier transform, the Laplace transforms is used in a variety of applications.
The most common usage of the Laplace transforms is in the solution of initial value problems.The Laplace transform is useful tool in applied mathematics, for instance for solving singular integral equations, partial differential equations, and in automatic control, where it defines a transfer function.
Problem 1.Let us consider the following non-linear impulsive differential equation ( D 2 t − a 2 )y(t) = t k δ(t − λ).Solution.The above differential equation can be written as below from which we deduce that Using Lemma 1.2 leads to the following thus, by using elementary properties of Dirac-delta function, we obtain Note:In the above relation I 0 stands for the modified Bessel's function of the first kind of order zero.
Problem 2. Let us solve the following fractional Volterra integral equation of convolution type.The Laplace transform provides a useful technique for the solution of such integral equations.
Solution.Let us make a change of variable ξ − β = η and taking the Laplace transform of the given integral equation, we obtain , at this point, taking inverse Laplace transform term wise, after simplifing we have In this section, we will also develop a more general procedure to treat singular integral equation whose solution requires exponential differential operators.Singular integral equations arise in many problems of mathematical physics.The mathematical formulation of physical phenomena often involves singular integral equations.Applications in many important fields like elastic contacts problems, the theory of porous filtering, fracture mechanics contain integral and integrodifferential equation with singular kernel.
Corollary 1.1 Let us consider the following singular integral equation the above integral equation has the following formal solution Where J ν (. : .)stands for the Bessel -Wright function of order ν.
Note:the special function of the form defined by the series representation is known as Bessel-Maitland function or the Bessel-Wright function.It has a wide application in the problem of physics, chemistry, applied sciences.
Proof.Let us rewrite the left hand side of the above equation(1.7)as below in relation (1.8), we used the following exponential identity (1.9) At this point, we may rewrite relation (1.9) in terms of Gamma function as follows (1.10) Exponential Differential Operators For Singular Integral Equations 227 From the above operational relationship and Taylor expansion of the exponential function results in Corollary 1.2.Let us consider the following singular integral equation the above integral equation has the following formal solution where I ν (2 xµ), stands for the modified Bessel function of the first kind of order ν .Proof.Let us rewrite the left hand side of the above equation as below and treating the derivative operator as a constant, the evaluation of the integral yields after writing Taylor expansion of exponential function, we arrive at (1.15) Note: From operational relation (xD x )x n = nx n we get the following identity and g(x) has Taylor series expansion.Lemma 1.3.The following exponential operator identity holds true dξ. (1.17) Proof.Let us introduce the following integral By making the change of variable k + ξ = ζ in the above integral we get after simplifying, we get or, Let us choose p = d dt , we obtain dξ.
(1.21) Special case .For n = 2 , we have from which we deduce Lemma1.4.The following second order exponential operator relations hold true.
(1.27) by using lemma 1.1, we get finally 2. In the above integral relation, we set r by using Lemma 1.1, we get finally .
Proof.Let us rewrite the right hand side of the above equation as below and treating the derivative operator as a constant, the evaluation of the integral yields x exp(βx 2 ), (1.32) at this point, using relation (1.23) leads to ))du, (1.33) from which and after some easy calculations, we get .

Evaluation of certain integrals
The main purpose of this section is to introduce the use of the exponential differential operator technique for evaluation of certain types of integrals.Lemma 2.1.Considering the integral as a function of parameters ν ,µ, show thatI(x, ν) satisfies the following relationship (2.2) Proof.By making a change of variable t = ky ,and letting x = k 2µ r , we get The above integral can be written in the following operational form after evaluation and simplifying the right hand side integral, this last result leads to (2.5) By using Taylor expansion of the Bessel's function of order ν, we obtain finally, (2.7) Lemma 2.2.Let us Consider the following integral as a function of parameters k ,µ , show that I 0 satisfies the following relationship (2.9) Proof.By making a change of variable t = ky ,and letting x = k 2µ r , we get (2.10) The above integral can be written in the following operational form after evaluation and simplifying the right hand side integral, this last result leads to Exponential Differential Operators For Singular Integral Equations 231 (2.12) By using Taylor expansion of the Error -function, one has finally we get (2.14)

Main Results
Solution to generalized space fractional Fokker-Planck equation with non-constant coefficients, which is used to study the beam life time due to quantum fluctuation in the storage ring.Fokker-Planck equation arises frequently in the theory of stochastic processes.The physical interpretation of the variables in this equation is that, the probability that a random variable has the value x at time t.For example, u(x,t) might be the probability distribution of the position of a harmonically bound particle in Brownian motion, or probability distribution of the deflection x of an electrical noise traces at time t.Problem 3. Let us consider the following generalized Fokker-Planck equation Solution: In order to obtain a solution for equation (3.1) in view of [2], [3] first by solving the first order PDE with respect to t, and applying the initial condition (3.2), we get the following relationship In order to find the result of the action of exponential operator, we make use of part three of Lemma 1.1, by choosing dr, where, g(r) = e −r λ (at ν+1 cos λπ) sin(at ν+1 r λ sin λπ)( then after some manipulation, and using the following relationship [5] exp we get the formal solution as below dτ )dr.At this point, in order to simplify the above relationship, we consider the following -well -known relationship for Laguerre polynomials of two variable as below.
Remark.For the general case,0 < λ < 1,λ = 0.5 , and f (x) is any differentiable function of all orders (we assume that,f (x) has Taylor series expansion ), we may solve the above fractional partial differential equation by making use of the above procedure.The procedure as described above should be generally applicable to the most fractional partial differential equations with non -constant coefficients.

Conclusion
Operational methods provide fast and universal mathematical tool for obtaining the solution of PDEs or even FPDEs.Combination of integral transforms, operational methods and special functions give more powerful analytical instrument for solving a wide range of engineering and physical problems.The paper is devoted to study exponential operators and their applications in solving certain boundary value problems such as the space fractional Fokker-Planck equation.We show that the present technique could be used to solve different kind of fractional partial differential equations.The present method can be readily applied to certain singular integral equations such as generalized Lamb -Bateman equation.The results of these developments will be published in future papers.

Aknowledgments
The authors wish to express their sincere thanks to the referees for valuable comments and suggestions that lead to a vast improvement in the paper.