We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

Recently,there have been several schemes devoted to the solution of fractional differential equations. These schemes can be broadly classified into two classes, numerical and analytical ([

In fact, as the theories of wavelets analyses improve day by day, the wavelet has become a powerful mathematical tool which widely used in signal processing, image compression and enhancement, pattern recognition, control systems, and other fields in the past two decades. But almost no papers or books have applied the theories of wavelets to solve the fractional differential equations. And our fundamental purpose of this paper is applying the fractional B-Splines wavelets to prove the existence and uniqueness of the solution of the nonhomogeneous linear fractional differential equations (also so-called linear multiterm fractional differential equations) with its initial conditions. Let us begin to discuss the solution of multiterms fractional ordinary differential equations with the following form:

The plan of this paper is as follows. In Section

The present paper is essentially based on the works of the Unser and Blu in [

We may recall the definition of the Left Riemann-Liouville differential operators of arbitrary order

Let us now introduce the case of the Caputo differential operators of arbitrary order

The Mittag-Leffler functions and its generalized forms have played a special role in solving the fractional differential equations. In this section, we just give the definition of the following series of representation of the Mittag-Leffler function

Splines have had a significant impact on the early development of the theory of the wavelet transform (see [

In this section, we will prove the existence and uniqueness properties of the solutions of the nonhomogeneous linear differential equations of arbitrary real order

Under the hypothesis of the existence for the solutions of (

Let

In order to prove this lemma, we divide the proof into two steps.

Firstly, we consider the case of

And then, let us consider the case of

To (

From the representation (

Considering the solution of the fractional differential equation

Let

From Lemma

Then substituting (

Let us suppose that the initial value problems (

The purpose of this section is to discuss the case of

Let

In Section

Similarity to the scheme of Theorem

Thus, taking the inverse Laplace Transform to (

To estimate the error of asymptotic solution of initial value problems (

For all

To (

Combining (

Thus, from the above discussion, it is evident to derive that the truncated error converges as

Noting that the process of the proof of the existence and uniqueness of the solution of the initial value problems (

Let

In this paper, we have proved the existence and uniqueness of the solution of the differential equations of arbitrary positive real order. And the representation of the solution of (