A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.


Introduction
Fractional calculus is a significantly important and useful branch of mathematics having a broad range of applications at almost any branch of science.Techniques of fractional calculus have been employed at the modeling of many different phenomena in engineering, physics, and mathematics.Problems in fractional calculus are not only important but also quite challenging which usually involves hard mathematical solution techniques (see, e.g., [1]).Unfortunately, a general solution theory for almost each problem in this area has yet to be established.Each application venue has developed its own approaches and implementations.As a consequence, a single standard method for the problems in fractional calculus has not emerged.Therefore, finding reliable and efficient solution techniques along with fast implementation methods is a significantly important and active research area.
In the literature, a number of methods have been developed for the numerical or analytical solutions for FPDEs.We can list some of these methods as follows: Adomian decomposition method [2], the collocation method [3], the fractional differential transform method [4], homotopy analysis method [5], homotopy perturbation method [6], and some other methods [7,8] listed on the references of these papers.In this paper, we present a new method for the analytical solutions of FPDEs.The solutions are expressed as a finite sum of a vector type functional.By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient.The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.Now let us briefly review some significant concepts in fractional calculus.The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notions of integer-order differentiation and -fold integration.We have well-known definitions of a fractional derivative of order  > 0 such as Riemann-Liouville, Grunwald-Letnikow, Caputo, and generalized functions approach [9].The most commonly used definitions of fractional derivative belongs to Riemann-Liouville and Caputo.For the purpose of this paper, Caputo's definition of fractional differentiation will be used, taking the advantage of Caputo's approach that the initial conditions for fractional differential equations with Caputo's derivatives take on the traditional form as for integer-order differential equations.We give some basic definitions and properties of the fractional calculus theory which were used through paper.

Mathematical Problems in Engineering
Definition 1.A real function (),  > 0, is said to be in the space   ,  ∈  if there exists a real number ( > ), such that () =    1 (), where  1 () ∈ [0, ∞), and it is said to be in the space    if   ∈   ,  ∈ .
Definition 2. The Riemann-Liouville fractional integral operator of order  ≥ 0 of a function  ∈   ,  ≥ −1 is defined as It has the following properties.
For  ∈   ,  ≥ −1, ,  ≥ 0, and  > 1, (1) The Riemann-Liouville fractional derivative is mostly used by mathematicians but this approach is not suitable for the physical problems of the real world since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet.Caputo introduced an alternative definition, which has the advantage of defining integer order initial conditions for fractional order differential equations.
Definition 3. The fractional derivative of () in the Caputo sense is defined as The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem.
Definition 5.For  to be the smallest integer that exceeds , the Caputo time-fractional derivative operator of order  > 0 is defined as and the space-fractional derivative operator of order  > 0 is defined as Another concept which plays a very significant role in the fractional calculus is the Gamma function.Next we briefly overview the definition and some important properties of Gamma function.Definition 6.For 0 <  < ∞, the integral (sometimes known as Euler integral) defined as is said to be Gamma function.From (6), it is clear that for  > 0, Again for  > 0, the function defined as is known as the Gamma function.Detailed information about the significant features of the Gamma function might be obtained from any well-written differential equation book.
Organization of the paper is in the following way.Firstly, we overview basic concepts of fractional derivative.Because we employ Caputo sense derivative, we describe it in detail.Secondly, we introduce a new method for analytical solutions of FPDEs.In the third section, we illustrate three computational examples as the application of the present method and complete the paper with a discussion section.

A Novel Method for Analytical Solutions of FPDEs
Let us consider the FPDE given as where   is a real number for every  and  is a natural number.Initial conditions for this equation are given by Writing we can express (9) as Now let us assume that the solution of ( 12) is given by where  is an  dimensional vector function and  is also a vector type function having the same dimension with .
If the solution ( 13) is plugged into (12) and the higher order derivatives are ignored, a linear equation expressed in terms of  is obtained.As a consequence,  is plugged in (13) and solution is obtained.If the same process is repeated for the higher order derivatives, series solution of (9) (or (13)) from an arbitrary degree is obtained.Now in order to express the solution in a new power series form, let us explain the application of this method to the power series.When ( 13) is plugged into (9), we can write where  1 ,  2 , . . .,   are constant real numbers,  1 ,  2 , . . .,   are the bases of , and  is the dimension of the vector .It is clear that using (13) it is possible to define that If  and its derivatives are written at (14), one obtains where   is an element of (,   , ) appearing at (9) and  is an index which is only 1 less than the degree of the highest derivative given in the initial problem.By solving (16), one obtains   , for each  = 1, 2, . . ., .By writing these   values at (15), for each , the polynomial   having the degree of  is obtained.This gives us power series solution of (9) from an arbitrary order.Finally let us suppose that step size of  is ℎ.If this value is plugged in  and   ,  and   are obtained at the point of  =  0 + ℎ.Repeating the same process, one obtains solution of (9).In the next section, we illustrate the application of this new and novel method to the analytical solutions of some FPDEs.

Computational Applications
Example 7. In the first example, we consider a special case of Bagley-Torvik equation with the initial conditions Now bearing in mind the aforementioned solution procedure, let us assume that the solution of ( 17) is given by For  = 2, from (13) we can write that Taking the derivatives of , we get which implies that  = 0.In a similar way, one obtains Again by taking the derivatives of , one obtains that  = 0.
In the next step, we have which implies that  = 0. Repeating this procedure, one obtains the solution of (17) as Example 8.In this example, we consider the initial value problem studied by Diethelm and given as  0.5  () =  () +  2 + 2 Γ (2.5)  1.5  (25)   with initial conditions Let us suppose that the solution of (25) is given by  = .When we write this solution in the original equation (25), we get  = 0.In the same way, if we consider the solution we obtain  = 1.Again by considering the solution we obtain  = 0. Repeating the procedure in this way, we obtain the solution of the initial value problem (25) as which is also exact solution of the problem.
In the last example, we consider a more general example to illustrate the application of the novel method.