A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

Kamal Ait Touchent; Zakia Hammouch; Toufik Mekkaoui

doi:10.2478/amns.2020.2.00012

Source Title:: Applied Mathematics and Nonlinear Sciences
Volume/Issue:: Volume 5: Issue 2

A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

Kamal Ait Touchent¹, Zakia Hammouch¹, and Toufik Mekkaoui¹

¹ Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, BP 509 Boutalamine, Errachidia, Morocco

Kamal Ait Touchent

Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, BP 509 Boutalamine, 52000, Errachidia, Morocco
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, Zakia Hammouch

Corresponding author
Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, BP 509 Boutalamine, 52000, Errachidia, Morocco
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and Toufik Mekkaoui

Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, BP 509 Boutalamine, 52000, Errachidia, Morocco
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DOI:: https://doi.org/10.2478/amns.2020.2.00012
Published online:: 01 Jul 2020

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Abstract

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.

Keywords:: Modified invariant subspace method; Caputo-Fabrizio fractional derivative; Atangana-Baleanu fractional derivative; Exact solution; Partial differential equations; 26A33; 35R11; 47A15

1 Introduction

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 [10] 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 [11]. 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 [12], Homotopy Perturbation coupled with Sumudu Transform [13], Adomian Mecomposition Method [14], Variational Iteration Method [15], Fractional Iteration Method [16], etc.

On the author hand, recent investigations show that the invariant subspace method, developed by V.A. Galaktionov and S.R. Svirshchevski [17], is an effective tool to construct exact solutions of some fractional partial differential equations with Caputo fractional derivative. R.Sahadevan and P.Prakash [18] used invariant subspace method to derive exact solutions of certain time fractional nonlinear partial differential equations, Hashemi [19] also adopted the same method to solve partial differential equations with conformable derivatives, Choudhary et al. [20] 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.

2 Fractional Calculus tools

In this section, we present some important defnitions and mathematical concepts on fractional derivatives with nonsingular kernels and related tools.

Definition 1

The Mittag-Leffler function E_α [21], is defined as

E_{α} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + 1)},

(1)

where z is a complex variable, α ∈ ℂ and ℜ(α) > 0.

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 [10] developed a new fractional derivative as follows

Definition 2

Let u be a function in H¹(a,b), b > a et 0 < α < 1 then, the new Caputo-Fabrizio derivative of fractional order α is defined as [10]

^{CF} D_{t}^{α} u (x, t) = \frac{M (α)}{1 - α} \int_{0}^{t} \frac{\partial}{\partial τ} u (x, τ) exp [- α \frac{(t - τ)}{1 - α}] d τ,

(2)

where M(α) is a normalization function satisfying M(0) = M(1) = 1.

From [10], we recall that if the function u does not belong to H¹(a;b) then, CF derivative can be writted as

^{CF} D_{t}^{α} u (x, t) = \frac{α M (α)}{1 - α} \int_{0}^{t} (u (x, t) - u (x, τ)) exp [- α \frac{(t - τ)}{1 - α}] d τ .

(3)

The fractional integral operator associated to the CF fractional derivative is expressed as

^{CF} I_{t}^{α} u (x, t) = \frac{1 - α}{M (α)} u (x, t) + \frac{α}{M (α)} \int_{0}^{t} u (x, τ) d τ .

(4)

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

Definition 3

The AB fractional derivative of order α in Caputo sense is given by [11]

^{ABC} D_{t}^{α} u (x, t) = \frac{B (α)}{1 - α} \int_{0}^{t} \frac{\partial}{\partial τ} u (x, τ) E_{α} [- α \frac{{(t - τ)}^{α}}{1 - α}] d τ,

(5)

where B(α) is a normalization function and B(0) = B(1) = 1 and 0 < α < 1.

The AB fractional integral operator of order α is given by [11]

Definition 4

The Atangana-Baleanu fractional integral of order α is defined as [11]

^{AB} I_{t}^{α} u (x, t) = \frac{1 - α}{B (α)} u (x, t) + \frac{α}{B (α) Γ (α)} \int_{0}^{t} u (x, τ) (t - τ)^{α - 1} d τ .

(6)

3 Description of the Modified Method

This section is devoted to descrive the invariant subspace method. Such method has been firstly used in [17] to construct particular exact solutions for partial differential equations of the form

\frac{\partial u}{\partial t} = F (u, u_{1 x}, u_{2 x}, \dots, u_{k x}), k \in ℕ,

(7)

where u = u(x,t),

u_{ix} = \frac{\partial^{i} u}{\partial x^{i}}

is the i^th order derivative of u with respect to the space variable x and F is a nonlinear differential operator.

Recently, Gazizov and Kasatkin [22] 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

D_{t}^{α} u (x, t) = F [u],

(8)

where F[u] = F(u, u_1x, u_2x,...,u_kx) and

D_{t}^{α}

is the time fractional derivative.

The modified invariant subspace method is based on the following basic definitions and results [22].

Definition 5

Let f₁(x),..., f_n(x) be an n linearly independent functions and W_n is the n-dimensional linear space namely W_n = 〈f₁(x),..., f_n(x)〉. W_n is said to be invariant under the given operator F if F[u] ∈ W_n whenever u ∈ W_n.

Proposition 1

Let W_nbe an invariant subspace of F. A function

u (x, t) = \sum_{i = 1}^{n} f_{i} (x) u_{i} (t)

is a solution of equation (8) if and only if the expansion coeffcients u_i(t) satisfy the following system of fractional ordinary differential equations

{\begin{matrix} D_{t}^{α} u_{1} = F_{1} (u_{1}, \dots, u_{n}), \\ D_{t}^{α} u_{2} = F_{2} (u_{1}, \dots, u_{n}), \\ . \\ . \\ . \\ D_{t}^{α} u_{n} = F_{n} (u_{1}, \dots, u_{n}), \end{matrix}

where F₁,..., F_nare given by

F (c_{1} f_{1} (x) + \dots + c_{n} f_{n} (x)) = F_{1} (c_{1}, \dots, c_{n}) f_{1} (x) + \dots + F_{n} (c_{1}, \dots, c_{n}) f_{n} (x) .

(9)

Remark 1

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 [22].

Proposition 2

Let f₁(x),..., f_n(x) form the fundamental set of solutions of a linear nth-order ordinary differential equation

T [y] = y^{(n)} + a_{1} (x) y^{(n - 1)} + \dots + a_{n - 1} (x) y^{'} + a_{n} (x) y = 0,

(10)

and F[y] = F(x,y,y^′,...,y^k) a given differential operator of order k ≤ n − 1, then the subspace W_n = 〈f₁(x),..., f_n(x)〉 is invariant with respect to F if and only if

T [F [y]] = 0,

(11)

whenever y satisfies the equation (10).

4 Applications

4.1 Fractional partial differential equations with Caputo-Fabrizio derivative

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.

• Example 1:

Consider the following time-fractional partial differential equation

^{CF} D_{t}^{α} u (x, t) = u_{xx} (x, t) + t^{2} u_{x} (x, t),

(12)

where t > 0, x ∈ ℝ and 0 < α < 1.

Setting

F [u] : =^{CF} D_{t}^{α} u (x, t)

, it is obvious that Eq.(12) admits the following invariant subspace

W_{1} = L {1, x},

(13)

Since

F [c_{1} (t) + c_{2} (t) x] = c_{2} (t) t^{2} \in W_{1} .

(14)

Therefore, the exact solution of Eq.(12) can be written as

u (x, t) = c_{1} (t) + c_{2} (t) x,

(15)

where c₁(t) and c₂(t) satisfy the following system of FDEs

{\begin{matrix} ^{CF} D_{t}^{α} c_{1} (t) = c_{2} (t) t^{2}, \\ ^{CF} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(16)

From the second equation of (16), we find that the function c₂(t) is a constant and then we assume that c₂(t) = 1. Thus, the first equation of (16) has the following solution

c_{1} (t) = \frac{(1 - α) t^{2}}{M (α)} + \frac{1}{3} \frac{α t^{3}}{M (α)} .

(17)

Therefore, Eq.(12) has an exact solution of the form

u (x, t) = \frac{(1 - α) t^{2}}{M (α)} + \frac{1}{3} \frac{α t^{3}}{M (α)} + x .

(18)

• Example 2:

Consider now, the following time-fractional partial differential equation

^{CF} D_{t}^{α} u (x, t) = sin (t) u_{xx} (x, t),

(19)

where t > 0, x ∈ ℝ and 0 < α < 1.

It is easy to check that

W_{2} = L {1, x^{2}},

(20)

is an invariant subspace of Eq(19), seeing that

F [c_{1} (t) + c_{2} (t) x^{2}] = 2 c_{2} (t) sin (t) \in W_{2} .

(21)

consequently, an exact solution of Eq.(19) can be written as

u (x, t) = c_{1} (t) + c_{2} (t) x^{2},

(22)

where c₁(t) end c₂(t) are unknown functions to be determined.

Substituting Eq.(22) in Eq.(19) yields:

{\begin{matrix} ^{CF} D_{t}^{α} c_{1} (t) = 2 c_{2} (t) sin (t), \\ ^{CF} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(23)

The second equation of (23) shows that the function c₂(t) is a constant and we infer that

c_{2} (t) = \frac{1}{2}

. Accordingly, the first equation of (23) can be expressed as

c_{1} (t) = \frac{(1 - α) sin (t)}{M (α)} + \frac{α (1 - cos (t))}{M (α)} .

(24)

Finally, we obtain an exact solution of Eq.(19) as

u (x, t) = \frac{(1 - α) sin (t)}{M (α)} + \frac{α (1 - cos (t))}{M (α)} + \frac{1}{2} x^{2} .

(25)

• Example 3:

Now we deal with the nonlinear time-fractional partial differential equation

^{CF} D_{t}^{α} u (x, t) = {tu}_{x}^{2} (x, t) + u_{xx} (x, t),

(26)

where t > 0, x ∈ ℝ and 0 < α < 1.

Eq.(26) admits an invariant subspace defined through

W_{3} = L {1, x},

(27)

F [c_{1} (t) + c_{2} (t) x] = {tc}_{2}^{2} (t) \in W_{3} .

(28)

As a deduction, an exact solution of Eq.(26) can take the form

u (x, t) = c_{1} (t) + c_{2} (t) x,

(29)

Substituting Eq.(29) in Eq.(26) and equating coefficients of different powers of x, we get

{\begin{matrix} ^{CF} D_{t}^{α} c_{1} (t) = {tc}_{2}^{2} (t), \\ ^{CF} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(30)

Solving second equation of (30) gives c₂(t) = 1. Therefore, the solution of the first equation of (30) is given by:

c_{1} (t) = \frac{(1 - α) t}{M (α)} + \frac{1}{2} \frac{α t^{2}}{M (α)} .

(31)

it then follows that an exact solution of Eq.(26) is given by

u (x, t) = \frac{(1 - α) t}{M (α)} + \frac{1}{2} \frac{α t^{2}}{M (α)} + x .

(32)

• Example 4:

Let us consider the following equation

^{CF} D_{t}^{2 α} u (x, t) = \frac{1}{2} x^{2} u_{xx}^{2} (x, t) + {tu}_{x} (x, t),

(33)

where t > 0, x ∈ ℝ, α ∈ [0,1] and

α \neq \frac{1}{2}

It is clear that the above equation Eq.(33) admits an invariant subspace

W_{4} = L {1, x},

(34)

by cause of

F [c_{1} (t) + c_{2} (t) x] = c_{2} (t) t \in W_{4} .

(35)

In an analogous way, the exact solution of Eq.(33) has the form

u (x, t) = c_{1} (t) + c_{2} (t) x,

(36)

where c₁(t) and c₂(t) satisfy the following system of FDEs

{\begin{matrix} ^{CF} D_{t}^{2 α} c_{1} (t) = c_{2} (t) t, \\ ^{CF} D_{t}^{2 α} c_{2} (t) = 0 . \end{matrix}

(37)

Similarly, we find that the function c₂(t) is a constant and then we assume that c₂(t) = 1.

Therefore, the first equation of (37) has the following solution:

c_{1} (t) = \frac{(1 - 2 α) t}{M (α)} + \frac{α t^{2}}{M (α)} .

(38)

This is leads eventually to an exact solution to the system Eq.(33) as:

u (x, t) = \frac{(1 - 2 α) t}{M (α)} + \frac{α t^{2}}{M (α)} + x .

(39)

4.2 Fractional partial differential equations with Atangana-Baleanu derivative

In what follows, we discuss four examples of getting exact solutions to some partial differential equations with Atangana-Baleanu fractional derivative.

• Example 1:

Consider the time-fractional partial differential equation:

^{ABC} D_{t}^{α} u (x, t) = u_{xx} (x, t) + t^{2} u_{x} (x, t),

(40)

where t > 0, x ∈ ℝ and 0 < α < 1.

Which admits an invariant subspace defined through

W_{1} = L {1, x},

(41)

by virtue of

F [c_{1} (t) + c_{2} (t) x] = c_{2} (t) t^{2} \in W_{1} .

It then follows that the form of exact solution for Eq.(40) is

u (x, t) = c_{1} (t) + c_{2} (t) x,

(42)

Substituting Eq.(42) in Eq.(40) and equating different powers of x to zero yields

{\begin{matrix} ^{ABC} D_{t}^{α} c_{1} (t) = c_{2} (t) t^{2}, \\ ^{ABC} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(43)

From the second equation of (43), we find that the function c₂(t) is a constant, then we assume that c₂(t) = 1. We then conclude that the solution of the first equation of (43) is expressed as

c_{1} (t) = \frac{(1 - α) t^{2}}{B (α)} + \frac{2 α t^{α + 2}}{Γ (α) B (α) (α^{2} + 2 α + 2)} .

(44)

Consequently, the exact solution of Eq.(40) reads

u (x, t) = \frac{(1 - α) t^{2}}{B (α)} + \frac{2 α t^{α + 2}}{Γ (α) B (α) (α^{2} + 2 α + 2)} + x .

(45)

• Example 2:

Let us second consider the following time-fractional partial differential equation

^{ABC} D_{t}^{α} u (x, t) = sin (t) u_{xx} (x, t),

(46)

where t > 0, x ∈ ℝ,0 < α < 1.

It is easy to check that Eq.(46) admits an invariant subspace as

W_{2} = L {1, x^{2}},

(47)

Since

F [c_{1} (t) + c_{2} (t) x^{2}] = 2 c_{2} (t) sin (t) \in W_{2} .

(48)

Hence, an exact solution of Eq(46) has the following form

u (x, t) = c_{1} (t) + c_{2} (t) x^{2},

(49)

In a similar way, substitution of Eq.(49) in Eq.(46) gives

{\begin{matrix} ^{ABC} D_{t}^{α} c_{1} (t) = 2 c_{2} (t) sin (t), \\ ^{ABC} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(50)

From second equation of (50) it comes c₂(t) is a constant then we assume that $c_{2} (t) = \frac{1}{2}$ .

Therefore, the solution of the first equation of (50) is

c_{1} (t) = \frac{(1 - α) sin (t)}{B (α)} + \frac{- \sqrt{t} LommelS 1 (\frac{3}{2} + α, \frac{1}{2}, t) + t^{1 + α}}{Γ (α) B (α) (1 + α)} .

(51)

Finally, we obtain an exact solution of Eq.(46) as

u (x, t) = \frac{(1 - α) sin (t)}{B (α)} + \frac{- \sqrt{t} LommelS 1 (\frac{3}{2} + α, \frac{1}{2}, t) + t^{1 + α}}{Γ (α) B (α) (1 + α)} + \frac{1}{2} x^{2} .

(52)

• Example 3:

Consider now the partial differential equation

^{ABC} D_{t}^{α} u (x, t) = t u_{x}^{2} (x, t) + u_{xx} (x, t),

(53)

where t > 0, x ∈ ℝ and 0 < α < 1.

Equation (26) admits an invariant subspace of the form

W_{3} = L {1, x},

(54)

as far as

F [c_{1} (t) + c_{2} (t) x] = {tc}_{2}^{2} (t) \in W_{3} .

(55)

Then we can form an exact solution of Eq.(26) as

u (x, t) = c_{1} (t) + c_{2} (t) x,

(56)

where c₁(t) and c₂(t) satisfy the following system of FDEs

{\begin{matrix} ^{ABC} D_{t}^{α} c_{1} (t) = {tc}_{2}^{2} (t), \\ ^{ABC} D_{t}^{α} c_{2} (t) = 0 . \end{matrix}

(57)

Solving second equation of (30), we get c₂(t) = 1. Therefore, the solution of the first equation of (57) is constructed as

c_{1} (t) = \frac{(1 - α) t}{B (α)} + \frac{t^{1 + α}}{Γ (α) B (α) (1 + α)} .

(58)

We finally obtain an exact solution of Eq.(53) as

u (x, t) = \frac{(1 - α) t}{B (α)} + \frac{t^{1 + α}}{Γ (α) B (α) (1 + α)} + x .

(59)

• Example 4:

We finally consider the nonlinear time-fractional partial differential equation

^{ABC} D_{t}^{2 α} u (x, t) = \frac{1}{2} x^{2} u_{xx}^{2} (x, t) + {tu}_{x} (x, t),

(60)

where t > 0, x ∈ ℝ, α ∈ [0,1] and

α \neq \frac{1}{2}

It is easy to check that the above Eq.(60) admits an invariant subspace as

W_{4} = L {1, x},

(61)

Since

F [c_{1} (t) + c_{2} (t) x] = c_{2} (t) t \in W_{4} .

(62)

Therefore, the exact solution of Eq.(33) has the form

u (x, t) = c_{1} (t) + c_{2} (t) x .

(63)

The functions c₁(t) and c₂(t) satisfy the following system of FDEs

{\begin{matrix} ^{CF} D_{t}^{2 α} c_{1} (t) = c_{2} (t) t, \\ ^{CF} D_{t}^{2 α} c_{2} (t) = 0 . \end{matrix}

(64)

From (64), it can be infered that c₂(t) is a constant, let us assume that c₂(t) = 1.

The first equation of (64) has then the following solution

c_{1} (t) = \frac{(1 - 2 α) t}{B (α)} + \frac{1}{2} \frac{t^{2 α + 1}}{Γ (α) B (α) (2 α + 1)} .

(65)

Accordingly, we get an exact solution of Eq.(60) as

u (x, t) = \frac{(1 - 2 α) t}{B (α)} + \frac{1}{2} \frac{t^{2 α + 1}}{Γ (α) B (α) (2 α + 1)} + x .

(66)

5 Conclusion

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.

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Article information

Received:: 2019-02-16

Accepted:: 2019-08-28

Published Online:: 2020-07-01

Citation Information:: Applied Mathematics and Nonlinear Sciences, Volume 5, Issue 2, Pages 35–48, eISSN 2444-8656, DOI: https://doi.org/10.2478/amns.2020.2.00012.