By using a complex transform, we impose a system of fractional order in the sense of Riemann-Liouville fractional operators. The analytic solution for this system is discussed. Here, we introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover, applications are illustrated.

Fractional models have been studied by many researchers to sufficiently describe the operation of variety of computational, physical, and biological processes and systems. Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations, and fractional partial differential equations of physical phenomena. Most of these fractional differential equations have analytic solutions, approximation, and numerical techniques [

The idea of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was planted over 300 years ago. Abel in 1823 investigated the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem. Later Liouville applied fractional calculus to problems in potential theory. Since that time the fractional calculus has drawn the attention of many researchers in all areas of sciences (see [

One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms. In this paper, we will deal with scalar linear time-space fractional differential equations. The time is taken in sense of the Riemann-Liouville fractional operators. Also, This type of differential equation arises in many interesting applications. For example, the Fokker-Planck partial differential equation, bond pricing equations, and the Black-Scholes equations are in this class of differential equations (partial and fractional).

In [

This section concerns with some preliminaries and notations regarding the fractional calculus.

The fractional (arbitrary) order integral of the function

The fractional (arbitrary) order derivative of the function

From Definitions

The Caputo fractional derivative of order

Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator

In this paper, we consider the following fractional differential equation:

The above equation involves well-known time fractional diffusion equations.

In this section, we will transform the fractional differential equation (

Let us put

Consider the fractional differential system (

The homotopy perturbation technique implies that the initial value problem ((

We proceed to prove the analytical convergence of our solution.

Suppose the sequence

Let

Recently the homotopy methods are used to obtain approximate analytic solutions of the time-fractional nonlinear equation and time-space-fractional nonlinear equation (see [

In this section, we will consider the pump wave equations along the fiber (Schrödinger equations). These types of equations are the fundamental equations for describing non-relativistic quantum mechanical behavior taking the form

Under the transform

We suggested two types of complex transforms for systems of fractional differential equations. We concluded that the complex fractional differential equations can be transformed into coupled and uncoupled system of homogeneous and nonhomogeneous types. Moreover, we employed the homotopy perturbation scheme for solving the nonlinear complex fractional differential systems. The convergence of the method is discussed in a domain that contains the initial solution. The Schrödinger equation is illustrated as an application. This type of equation is used in the quantum mechanics, which describes how the quantum state of a physical system changes with time. In the standard quantum mechanics, the wave function is the most complete explanation that can be specified to a physical system. Solutions of the Schrdinger’s equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems (see Figure

((a)–(d)) The solution