Fractional partial differential equations provide an excellent model for the description of memory and hereditary properties of various processes and materials. This is the main advantage of fractional partial differential equations in comparison with classical integer - order models, in which such effects are in fact neglected. It is well - known that the mixed boundary value problems occur in the theory of elasticity in connection with punching and crack problems. The main objective of present study is to justify, in a clear fasion, the interesting possibility that fractional methods represent for modelling the dynamics certain phenomena which ordinary models cannot. The solution of the mixed boundary value problems requires considerable mathematical skills. Most mixed boundary value problems are solved using integral transform method or separation of variables [7,13,14]. Transform method are usually led to the problem of solving dual or triple Fourier or Bessel integral equations. For a discussion of such equation see [4,5]. The main goal of this study is to give an updated treatment of this subject. An alternative method of solving mixed boundary value problem involves Green’s function. Conformal mapping is a mathematical technique to solve certain types of mixed boundary value problems [4]. It should be emphasized that the focus of this paper is only on integral transform method for solving fractional partial differential equations. However, some papers have recently presented numerical techniques for this class of problems. In [9], the authors used a q- homotopy analysis transform method to find the solution for fractional Drinfeld - Sokolov - Wilson equation, where fractional derivative defined with Atangana - Baleanu (AB) operator. In [16], P. Veeresha used a numerical technique called q - homotopy analysis transform method to solve a non - linear Fisher’s equation of fractional order. Finally, we list a number of research articles where the background and many applications of numerical methods of solution could be found (see [8,9,12,15,16,17]) and focus mostly on the solution of non - linear equations.
In the last three decades or so, fractional derivatives and notably fractional calculus have played a very important role in the various fields such as chemistry, biology, engineering, economics and signal processing. At this point, it should be pointed out that several definitions have been proposed of a fractional derivative, among those the Riemann - Liouville and Caputo fractional derivatives are the most popular. The differential equations defined in terms of Riemann - Liouville derivatives require fractional initial conditions, whereas the differential equations defined in terms of Caputo derivatives require regular boundary conditions. For this reason, the Caputo fractional derivatives are popular among engineers and scientists.
The following integral identities hold true [11]
Note
Note: With
Many problems of physical interest lead to Laplace transform whose inverses are not readily expressed in terms of tabulated functions. Therefore, it is highly desirable to have methods for inversion. In this section an algorithm to invert the Laplace transform is presented [1,2,3]
In the next Lemmas, we need the following integral representation for the modified Bessel’s function of the second kind
By using an appropriate integral representation for the modified Bessel’s functions of the second kind of order zero,
In view of the definition1.1 taking the inverse Laplace transform of the given
By setting relation (8) in (7), we obtain
In view of the relation (1.3), we obtain the following integral relation
Thus, we get the following integral representation for the modified Bessel’s function of the second kind of order zero,
In order for a transformation to be useful in solving boundary value problems, it must have an inverse. The inverse Hankel transform of a function
We have the following relation
Let us start with the following Laplace transform relation
The following integral identities hold true
Parseval’s relation for Hankel transform
If
Like the Laplace transform, the Hankel transforms used in a variety of applications. Perhaps the most common usage of the Hankel transform is in the solution of boundary value problems. However, there are other situations for which the properties of the Hankel transform are also very useful, such as in the evaluation of certain integrals.
By using Parseval’s relation show that
Let us take
At this point, using Parseval’s relation leads to
After simplifying, we arrive at
The following integral relations hold true.
Let us start with the following elementary integral identity
By substituting this integral on the left hand side of the first integral and interchanging the order of integration, we obtain
At this point, let us evaluate the inner integral by making a change of variable
Making a change of variable
But, the value of the integral in the braces is
At this stage, the above relation can be written as Hankel transform of a function as below
Thus, by taking the inverse Hankel transform, we obtain
Let us choose
Let us consider the following relations,
Combination of (38) and (39) leads to the following integral relation
Let us assume that
The right hand side of the above relation can be written as follows
In the above double integral, we set
Let us assume that
See [6].
The following integral identity holds true
Let us take
On the other hand, by using part three of the Lemma 1.5, we arrive at
In the past three decades, considerable research efforts have been expended to study anomalous diffusion using the time fractional equation. Anomalous diffusion transport appears to be a universal experimental phenomenon. A number of works have been published dealing with anomalous transport in fractals and disordered media, glass - forming liquids and colloidal structures. Let us consider the following two-dimensional heat conduction problem that arises during the manufacture of p-n junctions. To the best of the author’s knowledge this kind of fractional mixed boundary value problem is not considered in the literature.
Let us consider the following time fractional diffusion problem with mixed boundary conditions
In order to solve the above mixed boundary value problem, we reformulating it in cylindrical coordinates, to obtain
The above mixed boundary value problem can be solved via the Laplace transform.
Let us define
Then the transformed equation becomes
At this stage, let us choose
Then we have the following relations
Now, we express the Fourier series solution to the above equation as follows
Note that the Equation (77) satisfies the boundary conditions (74),(75) and each Fourier coefficient
Equation (78) is known as non - homogeneous modified Bessel equation of order (
One of the principal uses of the Hankel transform is in the solution of boundary value problems involving cylindrical coordinates. At this stage, we apply the Hankel transform of order (
Inverting this result by means of the Hankel inversion formula, we have
Finally, we get the solution to transformed Equation (67) as follows
Note: It is easy to check that
In this work, the author presents analytical techniques to solve time fractional diffusion problem with mixed bounadry conditions. We consider a generalization of the fractional heat conduction problem in two dimensions that arises during analysis of the impurity atom distribution near the diffusion mask for a planar p - n junction. The article is intended for scientists and researchers of different disciplines of engineering and science dealing with the solutions of fractional mixed boundary value problems. The results reveal that the integral transforms method is very convenient and effective. It is hoped that this study will lead to further investigations in the field and more elegant solutions would be found.