1. bookVolume 5 (2020): Issue 2 (July 2020)
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A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

Published Online: 01 Jul 2020
Page range: 35 - 48
Received: 16 Feb 2019
Accepted: 28 Aug 2019
Journal Details
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Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
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

MSC 2010

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.

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 Eα [21], is defined as Eα(z)=k=0zkΓ(αk+1),{E_\alpha}(z) = \sum\limits_{k = 0}^\infty {{{z^k}} \over {\Gamma (\alpha k + 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] CFDtαu(x,t)=M(α)1α0tτu(x,τ)exp[α(tτ)1α]dτ,^{CF}D_t^\alpha u(x,t) = {{M(\alpha )} \over {1 - \alpha}}\int_0^t {\partial \over {\partial \tau}}u(x,\tau )\exp \left[ {- \alpha {{(t - \tau )} \over {1 - \alpha}}} \right]d\tau, 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 CFDtαu(x,t)=αM(α)1α0t(u(x,t)u(x,τ))exp[α(tτ)1α]dτ.^{CF}D_t^\alpha u(x,t) = {{\alpha M(\alpha )} \over {1 - \alpha}}\int_0^t (u(x,t) - u(x,\tau ))\exp \left[ {- \alpha {{(t - \tau )} \over {1 - \alpha}}} \right]d\tau.

The fractional integral operator associated to the CF fractional derivative is expressed as CFItαu(x,t)=1αM(α)u(x,t)+αM(α)0tu(x,τ)dτ.^{CF}I_t^\alpha u(x,t) = {{1 - \alpha} \over {M(\alpha )}}u(x,t) + {\alpha \over {M(\alpha )}}\int_0^t u(x,\tau )d\tau.

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] ABCDtαu(x,t)=B(α)1α0tτu(x,τ)Eα[α(tτ)α1α]dτ,^{ABC}D_t^\alpha u(x,t) = {{B(\alpha )} \over {1 - \alpha}}\int_0^t {\partial \over {\partial \tau}}u(x,\tau ){E_\alpha}\left[ {- \alpha {{{{(t - \tau )}^\alpha}} \over {1 - \alpha}}} \right]d\tau, 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] ABItαu(x,t)=1αB(α)u(x,t)+αB(α)Γ(α)0tu(x,τ)(tτ)α1dτ.^{AB}I_t^\alpha u(x,t) = {{1 - \alpha} \over {B(\alpha )}}u(x,t) + {\alpha \over {B(\alpha )\Gamma (\alpha )}}\int_0^t u(x,\tau )(t - \tau {)^{\alpha - 1}}d\tau.

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 ut=F(u,u1x,u2x,,ukx),k,{{\partial u} \over {\partial t}} = F(u,{u_{1x}},{u_{2x}},...,{u_{kx}}),\quad \quad k \in \mathbb{N}, where u = u(x,t), uix=iuxi{u_{ix}} = {{{\partial ^i}u} \over {\partial {x^i}}} is the ith 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 Dtαu(x,t)=F[u],D_t^\alpha u(x,t) = F[u], where F[u] = F(u, u1x, u2x,...,ukx) and DtαD_t^\alpha 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),..., fn(x) be an n linearly independent functions and Wn is the n-dimensional linear space namely Wn = 〈f1(x),..., fn(x)〉. Wn is said to be invariant under the given operator F if F[u] ∈ Wn whenever uWn.

Proposition 1

Let Wnbe an invariant subspace of F. A functionu(x,t)=i=1nfi(x)ui(t)u(x,t) = \sum\nolimits_{i = 1}^n {f_i}(x){u_i}(t)is a solution of equation (8) if and only if the expansion coeffcients ui(t) satisfy the following system of fractional ordinary differential equations{Dtαu1=F1(u1,,un),Dtαu2=F2(u1,,un),...Dtαun=Fn(u1,,un),\left\{{\matrix{{D_t^\alpha {u_1} = {F_1}({u_1},...,{u_n}),} \hfill \cr {D_t^\alpha {u_2} = {F_2}({u_1},...,{u_n}),} \hfill \cr {\quad \quad.} \hfill & {} \hfill & {} \hfill \cr {\quad \quad.} \hfill \cr {\quad \quad.} \hfill \cr {D_t^\alpha {u_n} = {F_n}({u_1},...,{u_n}),} \hfill \cr}} \right.where F1,..., Fnare given byF(c1f1(x)++cnfn(x))=F1(c1,,cn)f1(x)++Fn(c1,,cn)fn(x).F({c_1}{f_1}(x) +... + {c_n}{f_n}(x)) = {F_1}({c_1},...,{c_n}){f_1}(x) +... + {F_n}({c_1},...,{c_n}){f_n}(x).

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),..., fn(x) form the fundamental set of solutions of a linear nth-order ordinary differential equationT[y]=y(n)+a1(x)y(n1)++an1(x)y+an(x)y=0,T[y] = {y^{(n)}} + {a_1}(x){y^{(n - 1)}} +... + {a_{n - 1}}(x)y' + {a_n}(x)y = 0,and F[y] = F(x,y,y,...,yk) a given differential operator of order kn − 1, then the subspace Wn = 〈f1(x),..., fn(x)〉 is invariant with respect to F if and only ifT[F[y]]=0,T[F[y]] = 0,whenever y satisfies the equation (10).

Applications
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 CFDtαu(x,t)=uxx(x,t)+t2ux(x,t),^{CF}D_t^\alpha u(x,t) = {u_{xx}}(x,t) + {t^2}{u_x}(x,t), where t > 0, x ∈ ℝ and 0 < α < 1.

Setting F[u]:=CFDtαu(x,t)F[u]: = {{\kern 1pt} ^{CF}}D_t^\alpha u(x,t) , it is obvious that Eq.(12) admits the following invariant subspace W1=L{1,x},{W_1} = \mathfrak{L}\{1,x\},

Since F[c1(t)+c2(t)x]=c2(t)t2W1.F[{c_1}(t) + {c_2}(t)x] = {c_2}(t){t^2} \in {W_1}.

Therefore, the exact solution of Eq.(12) can be written as u(x,t)=c1(t)+c2(t)x,u(x,t) = {c_1}(t) + {c_2}(t)x, where c1(t) and c2(t) satisfy the following system of FDEs {CFDtαc1(t)=c2(t)t2,CFDtαc2(t)=0.\left\{{\matrix{{^{CF}D_t^\alpha {c_1}(t) = {c_2}(t){t^2},} \hfill\cr {^{CF}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

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 c1(t)=(1α)t2M(α)+13αt3M(α).{c_1}(t) = {{\left( {1 - \alpha} \right){t^2}} \over {M(\alpha )}} + {1 \over 3}{\kern 1pt} {{\alpha {\kern 1pt} {t^3}} \over {M(\alpha )}}.

Therefore, Eq.(12) has an exact solution of the form u(x,t)=(1α)t2M(α)+13αt3M(α)+x.u(x,t) = {{\left( {1 - \alpha} \right){t^2}} \over {M(\alpha )}} + {1 \over 3}{\kern 1pt} {{\alpha {\kern 1pt} {t^3}} \over {M(\alpha )}} + x.

Fig. 1

Profile of the solution (18) for α = 0.9.

• Example 2:

Consider now, the following time-fractional partial differential equation CFDtαu(x,t)=sin(t)uxx(x,t),^{CF}D_t^\alpha u(x,t) = \sin (t){u_{xx}}(x,t), where t > 0, x ∈ ℝ and 0 < α < 1.

It is easy to check that W2=L{1,x2},{W_2} = \mathfrak{L}\{1,{x^2}\}, is an invariant subspace of Eq(19), seeing that F[c1(t)+c2(t)x2]=2c2(t)sin(t)W2.F[{c_1}(t) + {c_2}(t){x^2}] = 2{c_2}(t)\sin (t) \in {W_2}. consequently, an exact solution of Eq.(19) can be written as u(x,t)=c1(t)+c2(t)x2,u(x,t) = {c_1}(t) + {c_2}(t){x^2}, where c1(t) end c2(t) are unknown functions to be determined.

Substituting Eq.(22) in Eq.(19) yields: {CFDtαc1(t)=2c2(t)sin(t),CFDtαc2(t)=0.\left\{{\matrix{{^{CF}D_t^\alpha {c_1}(t) = 2{c_2}(t)\sin (t),} \hfill\cr {^{CF}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

The second equation of (23) shows that the function c2(t) is a constant and we infer that c2(t)=12{c_2}(t) = {1 \over 2} . Accordingly, the first equation of (23) can be expressed as c1(t)=(1α)sin(t)M(α)+α(1cos(t))M(α).{c_1}(t) = {{\left( {1 - \alpha} \right)\sin \left( t \right)} \over {M\left( \alpha \right)}} + {{\alpha {\kern 1pt} \left( {1 - \cos \left( t \right)} \right)} \over {M\left( \alpha \right)}}.

Finally, we obtain an exact solution of Eq.(19) as u(x,t)=(1α)sin(t)M(α)+α(1cos(t))M(α)+12x2.u(x,t) = {{\left( {1 - \alpha} \right)\sin \left( t \right)} \over {M\left( \alpha \right)}} + {{\alpha {\kern 1pt} \left( {1 - \cos \left( t \right)} \right)} \over {M\left( \alpha \right)}} + {1 \over 2}{x^2}.

Fig. 2

Profile of the solution (25) for α = 0.9.

• Example 3:

Now we deal with the nonlinear time-fractional partial differential equation CFDtαu(x,t)=tux2(x,t)+uxx(x,t),^{CF}D_t^\alpha u(x,t) = tu_x^2(x,t) + {u_{xx}}(x,t), where t > 0, x ∈ ℝ and 0 < α < 1.

Eq.(26) admits an invariant subspace defined through W3=L{1,x},{W_3} = \mathfrak{L}\{1,x\}, as F[c1(t)+c2(t)x]=tc22(t)W3.F[{c_1}(t) + {c_2}(t)x] = tc_2^2(t) \in {W_3}.

As a deduction, an exact solution of Eq.(26) can take the form u(x,t)=c1(t)+c2(t)x,u(x,t) = {c_1}(t) + {c_2}(t)x,

Substituting Eq.(29) in Eq.(26) and equating coefficients of different powers of x, we get {CFDtαc1(t)=tc22(t),CFDtαc2(t)=0.\left\{{\matrix{{^{CF}D_t^\alpha {c_1}(t) = tc_2^2(t),} \hfill \cr {^{CF}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

Solving second equation of (30) gives c2(t) = 1. Therefore, the solution of the first equation of (30) is given by: c1(t)=(1α)tM(α)+12αt2M(α).{c_1}(t) = {{\left( {1 - \alpha} \right)t} \over {M\left( \alpha \right)}} + {1 \over 2}{\kern 1pt} {{\alpha {\kern 1pt} {t^2}} \over {M\left( \alpha \right)}}. it then follows that an exact solution of Eq.(26) is given by u(x,t)=(1α)tM(α)+12αt2M(α)+x.u(x,t) = {{\left( {1 - \alpha} \right)t} \over {M\left( \alpha \right)}} + {1 \over 2}{\kern 1pt} {{\alpha {\kern 1pt} {t^2}} \over {M\left( \alpha \right)}} + x.

Fig. 3

Profile of the solution (32) for α = 0.9.

• Example 4:

Let us consider the following equation CFDt2αu(x,t)=12x2uxx2(x,t)+tux(x,t),^{CF}D_t^{2\alpha}u(x,t) = {1 \over 2}{x^2}u_{xx}^2(x,t) + t{u_x}(x,t), where t > 0, x ∈ ℝ, α ∈ [0,1] and α12\alpha \ne {1 \over 2} .

It is clear that the above equation Eq.(33) admits an invariant subspace W4=L{1,x},{W_4} = \mathfrak{L}\{1,x\}, by cause of F[c1(t)+c2(t)x]=c2(t)tW4.F[{c_1}(t) + {c_2}(t)x] = {c_2}(t)t \in {W_4}.

In an analogous way, the exact solution of Eq.(33) has the form u(x,t)=c1(t)+c2(t)x,u(x,t) = {c_1}(t) + {c_2}(t)x, where c1(t) and c2(t) satisfy the following system of FDEs {CFDt2αc1(t)=c2(t)t,CFDt2αc2(t)=0.\left\{{\matrix{{^{CF}D_t^{2\alpha}{c_1}(t) = {c_2}(t)t,} \hfill\cr {^{CF}D_t^{2\alpha}{c_2}(t) = 0.} \hfill\cr}} \right.

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: c1(t)=(12α)tM(α)+αt2M(α).{c_1}(t) = {{\left( {1 - 2{\kern 1pt} \alpha} \right)t} \over {M\left( \alpha \right)}} + {{\alpha {\kern 1pt} {t^2}} \over {M\left( \alpha \right)}}.

This is leads eventually to an exact solution to the system Eq.(33) as: u(x,t)=(12α)tM(α)+αt2M(α)+x.u(x,t) = {{\left( {1 - 2{\kern 1pt} \alpha} \right)t} \over {M\left( \alpha \right)}} + {{\alpha {\kern 1pt} {t^2}} \over {M\left( \alpha \right)}} + x.

Fig. 4

Profile of the solution (39) for α = 0.9.

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: ABCDtαu(x,t)=uxx(x,t)+t2ux(x,t),^{ABC}D_t^\alpha u(x,t) = {u_{xx}}(x,t) + {t^2}{u_x}(x,t), where t > 0, x ∈ ℝ and 0 < α < 1.

Which admits an invariant subspace defined through W1=L{1,x},{W_1} = \mathfrak{L}\{1,x\}, by virtue of F[c1(t)+c2(t)x]=c2(t)t2W1.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)=c1(t)+c2(t)x,u(x,t) = {c_1}(t) + {c_2}(t)x,

Substituting Eq.(42) in Eq.(40) and equating different powers of x to zero yields {ABCDtαc1(t)=c2(t)t2,ABCDtαc2(t)=0.\left\{{\matrix{{^{ABC}D_t^\alpha {c_1}(t) = {c_2}(t){t^2},} \hfill\cr {^{ABC}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

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 c1(t)=(1α)t2B(α)+2αtα+2Γ(α)B(α)(α2+2α+2).{c_1}(t) = {{\left( {1 - \alpha} \right){t^2}} \over {B\left( \alpha \right)}} + {{2\alpha {t^{\alpha + 2}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {{\alpha ^2} + 2{\kern 1pt} \alpha + 2} \right)}}.

Consequently, the exact solution of Eq.(40) reads u(x,t)=(1α)t2B(α)+2αtα+2Γ(α)B(α)(α2+2α+2)+x.u(x,t) = {{\left( {1 - \alpha} \right){t^2}} \over {B\left( \alpha \right)}} + {{2\alpha {t^{\alpha + 2}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {{\alpha ^2} + 2{\kern 1pt} \alpha + 2} \right)}} + x.

Fig. 5

Profile of the solution (45) for α = 0.9.

• Example 2:

Let us second consider the following time-fractional partial differential equation ABCDtαu(x,t)=sin(t)uxx(x,t),^{ABC}D_t^\alpha u(x,t) = \sin (t){u_{xx}}(x,t), where t > 0, x ∈ ℝ,0 < α < 1.

It is easy to check that Eq.(46) admits an invariant subspace as W2=L{1,x2},{W_2} = \mathfrak{L}\{1,{x^2}\},

Since F[c1(t)+c2(t)x2]=2c2(t)sin(t)W2.F[{c_1}(t) + {c_2}(t){x^2}] = 2{c_2}(t)\sin (t) \in {W_2}.

Hence, an exact solution of Eq(46) has the following form u(x,t)=c1(t)+c2(t)x2,u(x,t) = {c_1}(t) + {c_2}(t){x^2},

In a similar way, substitution of Eq.(49) in Eq.(46) gives {ABCDtαc1(t)=2c2(t)sin(t),ABCDtαc2(t)=0.\left\{{\matrix{{^{ABC}D_t^\alpha {c_1}(t) = 2{c_2}(t)\sin (t),} \hfill\cr {^{ABC}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

From second equation of (50) it comes c2(t) is a constant then we assume that c2(t)=12{c_2}(t) = {1 \over 2} .

Therefore, the solution of the first equation of (50) is c1(t)=(1α)sin(t)B(α)+tLommelS1(32+α,12,t)+t1+αΓ(α)B(α)(1+α).{c_1}(t) = {{\left( {1 - \alpha} \right)\sin \left( t \right)} \over {B\left( \alpha \right)}} + {{- \sqrt t LommelS{\it 1}\,\left( {{3 \over 2} + \alpha,{1 \over 2},t} \right) + {t^{1 + \alpha}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {1 + \alpha} \right)}}.

Finally, we obtain an exact solution of Eq.(46) as u(x,t)=(1α)sin(t)B(α)+tLommelS1(32+α,12,t)+t1+αΓ(α)B(α)(1+α)+12x2.u(x,t) = {{\left( {1 - \alpha} \right)\sin \left( t \right)} \over {B\left( \alpha \right)}} + {{- \sqrt t LommelS{\it 1}\,\left( {{3 \over 2} + \alpha,{1 \over 2},t} \right) + {t^{1 + \alpha}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {1 + \alpha} \right)}} + {1 \over 2}{\kern 1pt} {x^2}.

Fig. 6

Profile of the solution (52) for α = 0.9.

• Example 3:

Consider now the partial differential equation ABCDtαu(x,t)=tux2(x,t)+uxx(x,t),^{ABC}D_t^\alpha u(x,t) = tu_x^2(x,t) + {u_{xx}}(x,t), where t > 0, x ∈ ℝ and 0 < α < 1.

Equation (26) admits an invariant subspace of the form W3=L{1,x},{W_3} = \mathfrak{L}\{1,x\}, as far as F[c1(t)+c2(t)x]=tc22(t)W3.F[{c_1}(t) + {c_2}(t)x] = tc_2^2(t) \in {W_3}.

Then we can form an exact solution of Eq.(26) as u(x,t)=c1(t)+c2(t)x,u(x,t) = {c_1}(t) + {c_2}(t)x, where c1(t) and c2(t) satisfy the following system of FDEs {ABCDtαc1(t)=tc22(t),ABCDtαc2(t)=0.\left\{{\matrix{{^{ABC}D_t^\alpha {c_1}(t) = tc_2^2(t),} \hfill\cr {^{ABC}D_t^\alpha {c_2}(t) = 0.} \hfill\cr}} \right.

Solving second equation of (30), we get c2(t) = 1. Therefore, the solution of the first equation of (57) is constructed as c1(t)=(1α)tB(α)+t1+αΓ(α)B(α)(1+α).{c_1}(t) = {{\left( {1 - \alpha} \right)t} \over {B\left( \alpha \right)}} + {{{t^{1 + \alpha}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {1 + \alpha} \right)}}.

We finally obtain an exact solution of Eq.(53) as u(x,t)=(1α)tB(α)+t1+αΓ(α)B(α)(1+α)+x.u(x,t) = {{\left( {1 - \alpha} \right)t} \over {B\left( \alpha \right)}} + {{{t^{1 + \alpha}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {1 + \alpha} \right)}} + x.

Fig. 7

Profile of the solution (59) for α = 0.9.

• Example 4:

We finally consider the nonlinear time-fractional partial differential equation ABCDt2αu(x,t)=12x2uxx2(x,t)+tux(x,t),^{ABC}D_t^{2\alpha}u(x,t) = {1 \over 2}{x^2}u_{xx}^2(x,t) + t{u_x}(x,t), where t > 0, x ∈ ℝ, α ∈ [0,1] and α12\alpha \ne {1 \over 2} .

It is easy to check that the above Eq.(60) admits an invariant subspace as W4=L{1,x},{W_4} = \mathfrak{L}\{1,x\},

Since F[c1(t)+c2(t)x]=c2(t)tW4.F[{c_1}(t) + {c_2}(t)x] = {c_2}(t)t \in {W_4}.

Therefore, the exact solution of Eq.(33) has the form u(x,t)=c1(t)+c2(t)x.u(x,t) = {c_1}(t) + {c_2}(t)x.

The functions c1(t) and c2(t) satisfy the following system of FDEs {CFDt2αc1(t)=c2(t)t,CFDt2αc2(t)=0.\left\{{\matrix{{^{CF}D_t^{2\alpha}{c_1}(t) = {c_2}(t)t,} \hfill\cr {^{CF}D_t^{2\alpha}{c_2}(t) = 0.} \hfill \cr}} \right.

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 c1(t)=(12α)tB(α)+12t2α+1Γ(α)B(α)(2α+1).{c_1}(t) = {{\left( {1 - 2{\kern 1pt} \alpha} \right)t} \over {B\left( \alpha \right)}} + {1 \over 2}{\kern 1pt} {{{t^{2{\kern 1pt} \alpha + 1}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {2{\kern 1pt} \alpha + 1} \right)}}.

Accordingly, we get an exact solution of Eq.(60) as u(x,t)=(12α)tB(α)+12t2α+1Γ(α)B(α)(2α+1)+x.u(x,t) = {{\left( {1 - 2{\kern 1pt} \alpha} \right)t} \over {B\left( \alpha \right)}} + {1 \over 2}{\kern 1pt} {{{t^{2{\kern 1pt} \alpha + 1}}} \over {\Gamma \left( \alpha \right)B\left( \alpha \right)\left( {2{\kern 1pt} \alpha + 1} \right)}} + x.

Fig. 8

Profile of the solution (66) for α = 0.9.

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.

Fig. 1

Profile of the solution (18) for α = 0.9.
Profile of the solution (18) for α = 0.9.

Fig. 2

Profile of the solution (25) for α = 0.9.
Profile of the solution (25) for α = 0.9.

Fig. 3

Profile of the solution (32) for α = 0.9.
Profile of the solution (32) for α = 0.9.

Fig. 4

Profile of the solution (39) for α = 0.9.
Profile of the solution (39) for α = 0.9.

Fig. 5

Profile of the solution (45) for α = 0.9.
Profile of the solution (45) for α = 0.9.

Fig. 6

Profile of the solution (52) for α = 0.9.
Profile of the solution (52) for α = 0.9.

Fig. 7

Profile of the solution (59) for α = 0.9.
Profile of the solution (59) for α = 0.9.

Fig. 8

Profile of the solution (66) for α = 0.9.
Profile of the solution (66) for α = 0.9.

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