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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.

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., [

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 [

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

A real function

The Riemann-Liouville fractional integral operator of order

For

The fractional derivative of

If

For

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.

For

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.

Let us consider the FPDE given as

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 (

In the next section, we illustrate the application of this new and novel method to the analytical solutions of some FPDEs.

In the first example, we consider a special case of Bagley-Torvik equation

Now bearing in mind the aforementioned solution procedure, let us assume that the solution of (

For

In this example, we consider the initial value problem studied by Diethelm and given as

Let us suppose that the solution of (

In the last example, we consider a more general example to illustrate the application of the novel method.

Let us consider the equation

Following the steps of aforementioned solution algorithm, one can obtain the solution of (

A new technique for the analytical solutions of FPDEs has been successfully developed in this paper. 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 partial differential equations as well as a more general fractional differential equation. Experimental results prove that the present method is a useful and highly efficient technique.

The authors declare no conflict of interests.