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

Kamal Ait Touchent 1 , Zakia Hammouch 1  and Toufik Mekkaoui 1
  • 1 Faculty of Sciences and Techniques, Moulay Ismail University of Meknes, BP 509 Boutalamine 52000, 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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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.

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 Eα [21], is defined as

Eα(z)=k=0zkΓ(αk+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,
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.

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.

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,
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.

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

ut=F(u,u1x,u2x,,ukx),k,
where u = u(x,t), uix=iuxi 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],
where F[u] = F(u, u1x, u2x,...,ukx) and Dtα 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)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),
where F1,..., Fnare given by
F(c1f1(x)++cnfn(x))=F1(c1,,cn)f1(x)++Fn(c1,,cn)fn(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 equation

T[y]=y(n)+a1(x)y(n1)++an1(x)y+an(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 if
T[F[y]]=0,
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

CFDtαu(x,t)=uxx(x,t)+t2ux(x,t),
where t > 0, x ∈ ℝ and 0 < α < 1.

Setting F[u]:=CFDtαu(x,t) , it is obvious that Eq.(12) admits the following invariant subspace

W1=L{1,x},

Since

F[c1(t)+c2(t)x]=c2(t)t2W1.

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

u(x,t)=c1(t)+c2(t)x,
where c1(t) and c2(t) satisfy the following system of FDEs
{CFDtαc1(t)=c2(t)t2,CFDtαc2(t)=0.

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).

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

u(x,t)=(1α)t2M(α)+13αt3M(α)+x.

Fig. 1
Fig. 1

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 2:

Consider now, the following time-fractional partial differential equation

CFDtαu(x,t)=sin(t)uxx(x,t),
where t > 0, x ∈ ℝ and 0 < α < 1.

It is easy to check that

W2=L{1,x2},
is an invariant subspace of Eq(19), seeing that
F[c1(t)+c2(t)x2]=2c2(t)sin(t)W2.
consequently, an exact solution of Eq.(19) can be written as
u(x,t)=c1(t)+c2(t)x2,
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.

The second equation of (23) shows that the function c2(t) is a constant and we infer that c2(t)=12 . Accordingly, the first equation of (23) can be expressed as

c1(t)=(1α)sin(t)M(α)+α(1cos(t))M(α).

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

u(x,t)=(1α)sin(t)M(α)+α(1cos(t))M(α)+12x2.

Fig. 2
Fig. 2

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 3:

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

CFDtαu(x,t)=tux2(x,t)+uxx(x,t),
where t > 0, x ∈ ℝ and 0 < α < 1.

Eq.(26) admits an invariant subspace defined through

W3=L{1,x},
as
F[c1(t)+c2(t)x]=tc22(t)W3.

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

u(x,t)=c1(t)+c2(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.

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).
it then follows that an exact solution of Eq.(26) is given by
u(x,t)=(1α)tM(α)+12αt2M(α)+x.

Fig. 3
Fig. 3

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 4:

Let us consider the following equation

CFDt2αu(x,t)=12x2uxx2(x,t)+tux(x,t),
where t > 0, x ∈ ℝ, α ∈ [0,1] and α12 .

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

W4=L{1,x},
by cause of
F[c1(t)+c2(t)x]=c2(t)tW4.

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

u(x,t)=c1(t)+c2(t)x,
where c1(t) and c2(t) satisfy the following system of FDEs
{CFDt2αc1(t)=c2(t)t,CFDt2αc2(t)=0.

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).

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

u(x,t)=(12α)tM(α)+αt2M(α)+x.

Fig. 4
Fig. 4

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

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:

ABCDtαu(x,t)=uxx(x,t)+t2ux(x,t),
where t > 0, x ∈ ℝ and 0 < α < 1.

Which admits an invariant subspace defined through

W1=L{1,x},
by virtue of
F[c1(t)+c2(t)x]=c2(t)t2W1.

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

u(x,t)=c1(t)+c2(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.

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).

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

u(x,t)=(1α)t2B(α)+2αtα+2Γ(α)B(α)(α2+2α+2)+x.

Fig. 5
Fig. 5

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 2:

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

ABCDtαu(x,t)=sin(t)uxx(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},

Since

F[c1(t)+c2(t)x2]=2c2(t)sin(t)W2.

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

u(x,t)=c1(t)+c2(t)x2,

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

{ABCDtαc1(t)=2c2(t)sin(t),ABCDtαc2(t)=0.

From second equation of (50) it comes c2(t) is a constant then we assume that c2(t)=12 .

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).

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.

Fig. 6
Fig. 6

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 3:

Consider now the partial differential equation

ABCDtαu(x,t)=tux2(x,t)+uxx(x,t),
where t > 0, x ∈ ℝ and 0 < α < 1.

Equation (26) admits an invariant subspace of the form

W3=L{1,x},
as far as
F[c1(t)+c2(t)x]=tc22(t)W3.

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

u(x,t)=c1(t)+c2(t)x,
where c1(t) and c2(t) satisfy the following system of FDEs
{ABCDtαc1(t)=tc22(t),ABCDtαc2(t)=0.

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).

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

u(x,t)=(1α)tB(α)+t1+αΓ(α)B(α)(1+α)+x.

Fig. 7
Fig. 7

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

• Example 4:

We finally consider the nonlinear time-fractional partial differential equation

ABCDt2αu(x,t)=12x2uxx2(x,t)+tux(x,t),
where t > 0, x ∈ ℝ, α ∈ [0,1] and α12 .

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

W4=L{1,x},

Since

F[c1(t)+c2(t)x]=c2(t)tW4.

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

u(x,t)=c1(t)+c2(t)x.

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

{CFDt2αc1(t)=c2(t)t,CFDt2αc2(t)=0.

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).

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

u(x,t)=(12α)tB(α)+12t2α+1Γ(α)B(α)(2α+1)+x.

Fig. 8
Fig. 8

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

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00012

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.

References

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    J. Singh. Chaos: An Interdisciplinary Journal of Nonlinear Science. 013137, (2019).

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    J. Singh, and D. Kumar, and D. Baleanu. An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Applied Mathematics and Computation. 12–24, (2018).

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    J. Singh, and D. Kumar, and D. Baleanu. On the analysis of fractional diabetes model with exponential law. Advances in Difference Equations. 231, (2018).

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    Z. Hammouch, T. Mekkaoui: Approximate analytical and numerical solutions to fractional KPP-like equations. Gen. maths Notes. N.2, Vol.14, pp.1–9 (2013).

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    Z. Hammouch, T. Mekkaoui: A Laplace-Variational Iteration Method for Solving the Homogeneous Smoluchowski Coagulation Equation. Applied Mathematical Sciences; 6.18, pp.879–886 (2012).

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    T. Mekkaoui, Z. Hammouch: Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method. Annals of the University of Craiova-Mathematics and Computer Science Series. 39.2, pp.251–256 (2012).

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    V. Galaktionov, S. Svirshchevskii: Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman and Hall/CRC applied mathematics and nonlinear science series, 2007.

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    R. Sahadevan Ramajayam, B. Thangarasu Bakkyaraj: Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fractional Calculus and Applied Analysis. 18.1, 146–162 (2015).

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    R. Gazizov and A. Kasatkin: Construction of exact solutions for fractional order differential equations by the invariant subspace method, Computers and Mathematics with Applications, vol. 66, no. 5, 576–584, 2013.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    J. Singh, D. Kumar, Z. Hammouch, A. Atangana: A fractional epidemiological model for computer viruses pertaining to a new fractional derivative. Applied Mathematics and Computation. 504–515, (2018).

  • [2]

    J. Singh. Chaos: An Interdisciplinary Journal of Nonlinear Science. 013137, (2019).

  • [3]

    J. Singh, and D. Kumar, and D. Baleanu. An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Applied Mathematics and Computation. 12–24, (2018).

  • [4]

    J. Singh, and D. Kumar, and D. Baleanu. On the analysis of fractional diabetes model with exponential law. Advances in Difference Equations. 231, (2018).

  • [5]

    I. Podlubny: Fractional Differential Equations. Academic Press, San Diego, CA (1999).

  • [6]

    G. Samko, A.A. Kilbas and S. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).

  • [7]

    A.A Kilbas, M.H. Srivastava and J.J. Trujillo: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204 (2006).

  • [8]

    K. Miller and B. Ross: An introduction to the fractional calculus and fractional differential Equations, John Wiley, Sons Inc., New York, 1993.

  • [9]

    D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012).

  • [10]

    M. Caputo and M. Fabrizio: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015).

  • [11]

    J.D. Djida, A. Atangana and I. Area: Numerical computation of a fractional derivative with non-local and non-singular kernel. MMNP. Vol. 7, pp. 32–41 (2012).

  • [12]

    H. Jafari, A. Golbabai, S. Seifi and K. Sayvand: Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order, Comput. Math. Appl. 59, pp. 1337–1344 (2010).

    • Crossref
    • Export Citation
  • [13]

    K. Ait Touchent et al: Implementation and convergence analysis of homotopy perturbation coupled with sumudu transform to construct solutions of local-fractional pdes. Fractal and Fractional. 2.3, pp.22 (2018).

    • Crossref
    • Export Citation
  • [14]

    Z. Hammouch, T. Mekkaoui: Approximate analytical and numerical solutions to fractional KPP-like equations. Gen. maths Notes. N.2, Vol.14, pp.1–9 (2013).

  • [15]

    Z. Hammouch, T. Mekkaoui: A Laplace-Variational Iteration Method for Solving the Homogeneous Smoluchowski Coagulation Equation. Applied Mathematical Sciences; 6.18, pp.879–886 (2012).

  • [16]

    T. Mekkaoui, Z. Hammouch: Approximate analytical solutions to the Bagley-Torvik equation by the fractional iteration method. Annals of the University of Craiova-Mathematics and Computer Science Series. 39.2, pp.251–256 (2012).

  • [17]

    V. Galaktionov, S. Svirshchevskii: Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman and Hall/CRC applied mathematics and nonlinear science series, 2007.

  • [18]

    R. Sahadevan Ramajayam, B. Thangarasu Bakkyaraj: Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fractional Calculus and Applied Analysis. 18.1, 146–162 (2015).

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