In this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.
- Modified invariant subspace method
- Caputo-Fabrizio fractional derivative
- Atangana-Baleanu fractional derivative
- Exact solution
- Partial differential equations
Fractional calculus provides an important characteristic to describe the complicated physical phenomena with memory effects. For this reason, the fractional calculus is becoming increasingly used as a modeling tool in physics, engineering and control processing in various fields of sciences such as fluid dynamics, plasma physics, mathematical biology and chemical kinetics, diffusion, etc [1, 2, 3, 4]. Due to their properties, fractional derivatives and integrals make this kind of calculus a good candidate to describe such phenomena. Some fundamental definitions of fractional derivatives were given by Riemann-Liouville and Liouville-Caputo [5, 6, 7, 8, 9]. Recently, Caputo and Fabrizio defined a new fractional derivative without singular kernel  named Caputo-Fabrizio derivative with specific properties, the derivative of a constant is zero and the initial conditions used in the fractional differential equations having a physical interpretation. Later, Atangana and Baleanu proposed another fractional derivative with non-local and non-singular kernel named Atangana-Baleanu derivative . Besides, seeking exact solutions of fractional partial differential equations is not an easy task, and it's remain a relevant problem. Therefore, many powerful methods have been proposed for solving analytically the fractional partial differential equations. Such methods include; Homotopy Perturbation Method , Homotopy Perturbation coupled with Sumudu Transform , Adomian Mecomposition Method , Variational Iteration Method , Fractional Iteration Method , etc.
On the author hand, recent investigations show that the invariant subspace method, developed by V.A. Galaktionov and S.R. Svirshchevski , is an effective tool to construct exact solutions of some fractional partial differential equations with Caputo fractional derivative. R.Sahadevan and P.Prakash  used invariant subspace method to derive exact solutions of certain time fractional nonlinear partial differential equations, Hashemi  also adopted the same method to solve partial differential equations with conformable derivatives, Choudhary et al.  used this technique to explore solutions of some fractional differential equations, etc.
In the present paper, we present a modified version of the invariant subspace method which does not require any use of the Laplace transformation. We then make use of this novel technique to solve some fractional partial differential equations using fractional operators of Caputo-Fabrizio and also Atangana-Baleanu type. The exact solutions of these equations are obtained by solving the reduced systems constructed from the studied equations.
The laout of the paper is organized as follows: In section 2, we present some basic definitions of fractional derivatives and integrals. Section 3 describes the modified invariant subspace method. Construction of exact solutions to some partial differential equations with Caputo-Fabrizio and Atangan-Baleanu derivatives is presented in section 4. Finally, concluding remarks are given in section 5.
In this section, we present some important defnitions and mathematical concepts on fractional derivatives with nonsingular kernels and related tools.
The Mittag-Leffler function
This function arises naturally in the solution of fractional order integral equations or fractional order differential equations. It interpolate between a purely exponential law and power-law like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts.
On the other hand, Caputo and Fabrizio  developed a new fractional derivative as follows
From , we recall that if the function
The fractional integral operator associated to the CF fractional derivative is expressed as
It's clear that the Caputo-Fabrizio derivative has no singular kernel, since the kernel is based on exponential function.
Recently, Atangana and Baleanu proposed a new fractional derivative which has non-local and non-singular kernel based on the generalized Mittag-Leffler function. More recently, they claimed that there is two general definitions of their derivative in the Riemann-Liouville and Caputo sense. Moreover, this fractional derivative has a fractional integral as an anti-derivative of their operators.
The Atangana-Baleanu fractional derivative in Caputo sense (ABC) is given by
The AB fractional derivative of order
The AB fractional integral operator of order
The Atangana-Baleanu fractional integral of order
This section is devoted to descrive the invariant subspace method. Such method has been firstly used in  to construct particular exact solutions for partial differential equations of the form
Recently, Gazizov and Kasatkin  showed that the invariant subspace method can be applied also to equations with time fractional derivative.
In fact, consider the time fractional partial differential equation of the form
The modified invariant subspace method is based on the following basic definitions and results .
The important question concerning the modified invariant subspace method was how to obtain the corresponding invariant subspace of a given differential operator. The answer of this question is given by the following proposition, for more details we refer the reader to .
In this section, we apply the modified invariant subspace method to construct exact solutions of some partial differential equations with Caputo-Fabrizio derivative in time.
Consider the following time-fractional partial differential equation
Therefore, the exact solution of Eq.(12) can be written as
From the second equation of (16), we find that the function
Therefore, Eq.(12) has an exact solution of the form
Consider now, the following time-fractional partial differential equation
It is easy to check that
The second equation of (23) shows that the function
Finally, we obtain an exact solution of Eq.(19) as
Now we deal with the nonlinear time-fractional partial differential equation
Eq.(26) admits an invariant subspace defined through
As a deduction, an exact solution of Eq.(26) can take the form
Substituting Eq.(29) in Eq.(26) and equating coefficients of different powers of
Solving second equation of (30) gives
Let us consider the following equation
It is clear that the above equation Eq.(33) admits an invariant subspace
In an analogous way, the exact solution of Eq.(33) has the form
Similarly, we find that the function
Therefore, the first equation of (37) has the following solution:
This is leads eventually to an exact solution to the system Eq.(33) as:
In what follows, we discuss four examples of getting exact solutions to some partial differential equations with Atangana-Baleanu fractional derivative.
Consider the time-fractional partial differential equation:
Which admits an invariant subspace defined through
It then follows that the form of exact solution for Eq.(40) is
Substituting Eq.(42) in Eq.(40) and equating different powers of
From the second equation of (43), we find that the function
Consequently, the exact solution of Eq.(40) reads
Let us second consider the following time-fractional partial differential equation
It is easy to check that Eq.(46) admits an invariant subspace as
Hence, an exact solution of Eq(46) has the following form
From second equation of (50) it comes
Therefore, the solution of the first equation of (50) is
Finally, we obtain an exact solution of Eq.(46) as
Consider now the partial differential equation
Equation (26) admits an invariant subspace of the form
Then we can form an exact solution of Eq.(26) as
Solving second equation of (30), we get
We finally obtain an exact solution of Eq.(53) as
We finally consider the nonlinear time-fractional partial differential equation
It is easy to check that the above Eq.(60) admits an invariant subspace as
Therefore, the exact solution of Eq.(33) has the form
From (64), it can be infered that
The first equation of (64) has then the following solution
Accordingly, we get an exact solution of Eq.(60) as
The modifed invariant subspace method was used to seek exact solutions to a class of nonlinear equations with fractional derivatives having nonsingular kernels. Several examples illustrated the effectiveness of the invariant subspace theory for exploring solution of various structures. It is also worth mentionning that the present method does not need any use of laplace transform. Furthermore, some graphical reprensentations are given to show the profiles of the obtained solutions. We stress here that those solutions are very useful to test the efficiency of newly suggested numerical methods for solving partial differential equations with Caputo-Fabrizio or Atangana-Baleanu fractional derivatives.