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Research article Special Issues

Mathematical analysis of time-fractional nonlinear Kuramoto-Sivashinsky equation

  • Received: 03 February 2025 Revised: 09 March 2025 Accepted: 25 March 2025 Published: 22 April 2025
  • MSC : 34G20, 35A20, 35A22, 35R11

  • This research explored fractional Kuramoto-Sivashinsky equation analytical solutions through application of the residual power series transform method (RPSTM). The Caputo derivative approach served as the basis to analyze fractional systems since it offered a strong foundation for modeling intricate nonlinear processes. Researchers extended the Kuramoto-Sivashinsky equation through fractional domain application to capture anomalous behavior and memory effects since it showed practical uses in turbulence and plasma dynamics and flame propagation. The RPSTM proposed solution united residual power series method capabilities with integral transforms features to develop an accurate and efficient method for fractional nonlinear partial differential equation solutions. The method enabled accurate approximate solution acquisition while the procedure underwent complete convergence examination. This paper demonstrated the effectiveness and reliability of the developed method through numerical simulation results. The RPSTM proved itself as an effective analytical method for fractional differential problems which reveals vital information about the fractional Kuramoto-Sivashinsky equation behavior. The research added value to existing fractional calculus studies focused on nonlinear science applications.

    Citation: Qasem M. Tawhari. Mathematical analysis of time-fractional nonlinear Kuramoto-Sivashinsky equation[J]. AIMS Mathematics, 2025, 10(4): 9237-9255. doi: 10.3934/math.2025424

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  • This research explored fractional Kuramoto-Sivashinsky equation analytical solutions through application of the residual power series transform method (RPSTM). The Caputo derivative approach served as the basis to analyze fractional systems since it offered a strong foundation for modeling intricate nonlinear processes. Researchers extended the Kuramoto-Sivashinsky equation through fractional domain application to capture anomalous behavior and memory effects since it showed practical uses in turbulence and plasma dynamics and flame propagation. The RPSTM proposed solution united residual power series method capabilities with integral transforms features to develop an accurate and efficient method for fractional nonlinear partial differential equation solutions. The method enabled accurate approximate solution acquisition while the procedure underwent complete convergence examination. This paper demonstrated the effectiveness and reliability of the developed method through numerical simulation results. The RPSTM proved itself as an effective analytical method for fractional differential problems which reveals vital information about the fractional Kuramoto-Sivashinsky equation behavior. The research added value to existing fractional calculus studies focused on nonlinear science applications.



    In 1903, Mittag-Leffler [22] provided the function Eσ(z) defined by

    Eσ(z)=j=0 zjΓ(σj+1),(σ,zC,R(σ)>0),

    where Γ is the gamma function and R means the real part.

    Wiman [34] introduced the following generalized Mittag-Leffler function

    Eσ,μ(z)=j=0 zjΓ(σj+μ),(σ,μ,zC,[R(σ),R(μ)]>0).

    Prabhakar [25] introduced the following function Eρσ,μ(z) in the form

    Eρσ,μ(z)=j=0 (ρ)jΓ(μ+σj).zjj!,   (σ,μ,ρ,zC,[R(σ),R(μ),R(ρ)]>0).

    Later, Shukla and Prajapati [27] (see also [32]) defined another generalized Mittag-Leffler function

    Eρ,kσ,μ(z)=j=0 (ρ)kjΓ(μ+σj)zjj!,(σ,μ,ρ,zC,[R(σ),R(μ),R(ρ)]>0)

    where k(0,1)N and (ρ)kj=Γ(ρ+kj)Γ(ρ) is the generalized Pochhammer symbol defined as

    kkjkm=1(ρ+m1k)j if kN.

    Bansal and Prajapat [5] and Srivastava and Bansal [31] investigated geometric properties of the Mittag-Leffler function Eσ,μ(z), including starlikeness, convexity, and close-to-convexity (see [1,4,6,8,12,13,17,28,29]). In reality, the generalized Mittag-Leffler function Eσ,μ(z) and its extensions are still widely used in geometric function theory and in a variety of applications (see, for details, [2,3,7,16,24]).

    Let S(p) be the class of functions of the form

    f(z)=zp+j=p+1ajzj, (1.1)

    where f is holomorphic and multivalent in the open unit disk O={z:|z|<1}.

    Let f and F be two functions in S(p). Then the convolution (or Hadamard product), denoted by fF, is defined as

    (fF)(z)=zp+j=p+1ajdjzj=(Ff)(z),

    where f(z) is in (1.1) and F(z)=zp+j=p+1djzj.

    Let f(z) and h(z) be two analytic functions defined in O. The function f(z) is called subordinate to h(z), or h(z) is superordinate to f(z), denoted by f(z)h(z) and h(z)f(z), respectively, if there is a Schwarz function φ with φ(z)=0,|φ(z)|<1 and f(z)=h(φ(z)). If the function h is univalent in O, then the following equivalence is true if

    f(z)h(z)  (zO)f(0)=h(0) and f(O)h(O).

    Definition 1.1. ([18]). Let 0<q<1. Then [j]q! denotes the q-factorial, which is defined as follows:

    [j]q!={[j]q[j1]q[2]q[1]q,    j=1,2,3,1,    j=0

    where [j]q=1qj1q=1+j1m=1 qm and [0]q=0.

    Definition 1.2 ([18]). The q-generalized Pochhammer symbol [ρ]j,q, ρC, is given as

    [ρ]j,q=[ρ]q[ρ+1]q[ρ+2]q[ρ+j1]q,

    and the q-Gamma function is defined as

    Γq(ρ+1)=[ρ]qΓq(ρ) and Γq(1)=1.

    It follows that Γq(j+1)=[j]q!.

    Lately, many results have been given for some related special functions such as the Wright function [3] and multivalent functions (see [10,23,26]).

    Here, we propose a q-extension of specific extensions of the Mittag-Leffler function, motivated by the success of Mittag-Leffler function applications in physics, biology, engineering, and applied sciences. We generalize the Mittag-Leffler function given by Shukla and Prajapati [27] and obtain a new generalized q-Mittag-Leffler function.

    Now, we present a new generalized q-Mittag-Leffler function as follows

    Eρσ,μ(q;z)=z+j=2 (ρ)kjΓq(μ+σj)zjj!. (1.2)

    It is obvious that, when q1, the resulting function is the generalized Mittag-Leffler function, which is given by Shukla and Prajapati [27].

    Corresponding to the function Eρσ,μ(q;z) in (1.2), we establish the following generalized q-Mittag-Leffler function Eρσ,μ(p,q;z) in multivalent functions S(p), as given below

    Eρσ,μ(p,q;z)=zp+j=p+1 (ρ)k(jp)Γq(μ+σ(jp))zj(jp)!. (1.3)

    Again, using the new function (1.3), we define the following function:

    Gρσ,μ(p,q;z):=zpΓq(μ)Eρσ,μ(p,q;z)=zp+j=p+1 Γq(μ)(ρ)k(jp)Γq(μ+σ(jp))zj(jp)!. (1.4)

    Definition 1.3. For fS(p), we define the new linear operator Aμ,ρ;kσ;p,qf(z):S(p)S(p) by

    Aμ,ρ;kσ;p,qf(z)=Gρσ,μ(p,q;z)f(z)=zp+j=p+1 χjajzj, (1.5)

    where χj=Γq(μ)(ρ)kjΓq(μ+σj)j!.

    We now define a subclass Qμ,ρ;kσ;q(M,N;τ,p) of the family S(p) using the multivalent linear operator in (1.5) and the subordination concept.

    Definition 1.4. Let Aμ,ρ;kσ;p,qf(z) be an operator in (1.5). A function f(z)S(p) is said to be in the class Qμ,ρ;kσ;q(M,N;τ,p) if satisfies the following subordination condition:

    1pτ(z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)τ)1+Mz1+Nz,  (zO) (1.6)

    or equivalently

    z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)p+(pN+(MN)(pτ))z1+Nz,  (zO)

    and

    |z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)pNz(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)[pN+(MN)(pτ)]|<1, (1.7)

    where 1M<N1, 0τ<p, and pN.

    Remark 1.1. Some well-known special classes of the class Qμ,ρ;kσ;q(M,N;τ,p) can be obtained by choosing the values of the parameters ς,μ,ρ;τ,k,p,q, M, and N.

    (1) Q0,0,10,1(M,N;τ,p)=Sp(M,N;τ,p) was provided by Aouf [2].

    (2) Q0,0,10,1(M,N;0,p)=Sp(M,N;p) was provided by Goel and Sohi [16].

    In this work, we introduce a new subclass of multivalent functions Qμ,ρ;kσ;q(M,N;τ,p) defined by the new linear operator Aμ,ρ;kσ;p,qf(z). And we study some geometric properties for the class Qμ,ρ;kσ;q(M,N;τ,p) such as the coefficient estimates, convexity and convex linear combination. Finally, the radius theorems associated with the generalized Srivastava-Attiya integral operator will be investigated.

    The first theorem in this section presents the necessary and sufficient condition for the function f(z) in (1.1) belong to the class Qμ,ρ;kσ;q(M,N;τ,p).

    Theorem 2.1. A function f(z) is in the class Qμ,ρ;kσ;q(M,N;τ,p) if and only if

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj|aj|(MN)(pτ), (2.1)

    where 1M<N1, 0τ<p, and pN.

    Proof. Assume that the condition (2.1) is true. Then by (1.7), we have

    |z(Aμ,ρ;kσ;p,qf(z))pAμ,ρ;kσ;p,qf(z)||Nz(Aμ,ρ;kσ;p,qf(z))[(MN)(pτ)+pN]Aμ,ρ;kσ;p,qf(z)|=|j=p+1(jp)χjajzj||(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj|(MN)(pτ)+j=p+1[(1+N)(jp)+((MN)(pτ))]χj|aj|0.

    By maximum modulus theorem [11], we get f(z)Qμ,ρ;kσ;q(M,N;τ,p).

    Conversely, suppose that f(z)Qμ,ρ;kσ;q(M,N;τ,p). Then

    |z(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)pNz(Aμ,ρ;kσ;p,qf(z))Aμ,ρ;kσ;p,qf(z)[pN+(MN)(pτ)]|=|j=p+1(jp)χjajzj(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj|<1.

    Since R(z)|z|, we get

    R{j=p+1(jp)χjajzj(MN)(pτ)zjj=p+1[Nj((MN)(pτ)+pN)]χjajzj}<1.

    Taking z1, we have

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj|aj|(MN)(pτ).

    This completes the proof.

    Theorem 2.2. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p). Then f1f2Qμ,ρ;kσ;q(M,N;τ,p), where

    τ1=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1, (2.2)

    where χ1=Γq(μ)(ρ)kΓq(μ+ς).

    Proof. We will show that τ1 is the largest satisfying

    j=p+1 ((1+N)(jp)+(MN)(pτ1))χj(MN)(pτ1)aj,1aj,21. (2.3)

    Since f1,f2Qμ,ρ;kσ;q(M,N;τ,p), by the condition (2.1) and the Cauchy-Schwarz inequality, we get

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,1aj,21. (2.4)

    From (2.3) and (2.4), we observe that

    aj,1aj,2[((1+N)(jp)+(MN)(pτ))χj](pτ1)[((1+N)(jp)+(MN)(pτ1))χj](pτ).

    From (2.4), it is necessary to prove

    (MN)(pτ)((1+N)(jp)+(MN)(pτ))χj[((1+N)(jp)+(MN)(pτ))χj](pτ1)[((1+N)(jp)+(MN)(pτ1))χj](pτ). (2.5)

    Furthermore, from the inequality (2.5) it follows that

    τ1p(jp)(1+N)(MN)(pτ)2χj[((1+N)(jp)+(MN)(pτ1))χj]2(MN)2(pτ)2χj.

    Now, set

    E(j)=p(jp)(1+N)(MN)(pτ)2χj[((1+N)(jp)+(MN)(pτ1))χj]2(MN)2(pτ)2χj.

    We observe that the function E(j) is increasing for jN. Putting j=1, we have

    τ1=E(1)=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1.

    This completes the proof.

    Theorem 2.3. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p) of forms given in (1.1) with aj,1 and aj,2, respectively. Then

    w(z)=zp+j=p+1(a2j,1+a2j,2)zjQμ,ρ;kσ;q(M,N;τ,p),

    where

    η=p(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]2(MN)2(pτ)2χ1.

    Proof. By Theorem 2.1, we have

    j=p+1 [((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2a2j,sj=p+1 [((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,s]21, (s=1,2).

    From the above inequality, we obtain

    j=p+1 12[((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2(a2j,1+a2j,2)1.

    Therefore, the largest η can be obtained such that

    ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)12[((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)]2.

    That is,

    ηp2(jp)(1+N)(MN)(pτ)2χ1[((1+N)(jp)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    Now, set

    E(j)=p2(jp)(1+N)(MN)(pτ)2χ1[((1+N)(jp)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    We observe that the function E(j) is increasing for jN. Putting j=1, we have

    η=E(1)=p2(1p)(1+N)(MN)(pτ)2χ1[((1+N)(1p)+(MN)(pτ1))χ1]22(MN)2(pτ)2χ1.

    This completes the proof.

    Theorem 2.4. Let f1,f2Qμ,ρ;kσ;q(M,N;τ,p). Then for γ[0,1], the function F(z)=(1γ)f1+γf2 belongs to the class Qμ,ρ;kσ;q(M,N;τ,p).

    Proof. Since the functions f1 and f2 belong to the class Qμ,ρ;kσ;q(M,N;τ,p),

    F(z)=(1γ)f1+γf2=zp+j=p+1ηjzj,

    where ηj=(1γ)aj,1+γaj,2.

    By (2.1), we observe that

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj[(1γ)aj,1+γaj,2]=(1γ)j=p+1 ((1+N)(jp)+(MN)(pτ))χjaj,1+γj=p+1 ((1+N)(jp)+(MN)(pτ))χjaj,2(1γ)(MN)(pτ)+γ(MN)(pτ).

    Hence F(z)Qμ,ρ;kσ;q(M,N;τ,p).

    Theorem 2.5. Let fs(z)=zp+j=p+1aj,szj be in the class Qμ,ρ;kσ;q(M,N;τ,p) for s=1,2,,m. Then the function P(z)=ms=1sfs, where ms=1s=1, is also in the class Qμ,ρ;kσ;q(M,N;τ,p).

    Proof. By Theorem 2.1, we have

    j=p+1 ((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)aj,s1.

    Since

    P(z)=ms=1sfs=ms=1s(zp+j=p+1aj,szj)=zp+j=p+1(ms=1saj,s)zj,
    j=p+1((1+N)(jp)+(MN)(pτ))χj(MN)(pτ)ms=1saj,s1.

    Thus P(z)Qμ,ρ;kσ;q(M,N;τ,p).

    In this section, we investigate radii of multivalent starlikeness, multivalent convexity, and multivalent close-to-convex for the function f(z) in the class Qμ,ρ;kσ;q(M,N;τ,p) with the generalized integral operator of Srivastava-Attiya.

    Jung et al. [19] introduced an integral operator with one parameter as follows:

    Iδ(f)(z):=2δzΓ(δ)z0 (log(zv) )δ1f(v)dv=z+j=2 (2j+1)δajzj(δ>0;fS).

    In 2007, Srivastava and Attiya [30] investigated a new integral operator, which is called Srivastava-Attiya operator, given by

    Ju,mf(z)=z+j=1(1+uj+u)δajzj.

    Many studies are concerned with the study of the operator of Srivastava-Attiya (see [9,14,15,20]).

    Mishra and Gochhayat [21] (also [33]) provided a fractional differintegral operator Jmu,pf(z):S(p)S(p) which is called a generalized of Srivastava-Attiya integral operator, defined by

    Jmu,pf(z)=zp+j=p+1(p+uj+u)δajzj. (3.1)

    Theorem 3.1. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent starlike of order τ in |z|r1, where

    r1=infjp+1{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)(j2p+τ)(p+u)δ}. (3.2)

    Proof. According to the definition of a starlike function in [28], we have

    |z(Jmu,pf(z))Jmu,pf(z)p|pτ, (3.3)
    |z(Jmu,pf(z))Jmu,pf(z)p|=|j=p+1(jp)(p+uj+u)δajzjj=p+1(p+uj+u)δajzj|j=p+1(jp)(p+uj+u)δaj|z|jj=p+1(p+uj+u)δaj|z|j.

    By (3.2), we have

    j=p+1(j2p+τ)(p+u)δaj|z|j(pτ)(j+u)δ1.

    By (2.1) in Theorem 2.1, it is clear that

    (j2p+τ)(p+u)δ(pτ)(j+u)δ|z|j((1+N)(jp)+(MN)(pτ))χj(MN)(pτ).

    Therefore,

    |z|{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)(j2p+τ)(p+u)δ}1j.

    This completes the proof.

    Theorem 3.2. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent convex of order τ in |z|r2, where

    r2=infjp+1{((1+N)(jp)+(MN)(pτ))χjp(j+u)δ(MN)[j(j2p+τ)](p+u)δ}. (3.4)

    Proof. To verify (3.4), it is necessary to prove

    |(1+z(Jmu,pf(z))(Jmu,pf(z)))p|pτ,

    but the result is obtained by repeating the steps in Theorem 3.1.

    Corollary 3.1. If f(z)Qμ,ρ;kσ;q(M,N;τ,p) and 0τ<p, then Jmu,pf(z) in (3.1) is multivalent close-to-convex of order τ in |z|r3, where

    r3=infj1{((1+N)(jp)+(MN)(pτ))χj(j+u)δ(MN)j(p+u)δ}. (3.5)

    In this work, we established and investigated a new generalized Mittag-Leffler function, which is a generalization of q-Mittag-Leffler function defined by Shukla and Prajapati [27]. Also, we studied some of the geometric properties of a certain subclass of multivalent functions. In addition, we introduced radius theorem using a generalized Srivastava-Attiya integral operator. Since the Mittag-Leffler function is of importance, it is related to a wide range of problems in mathematical physics, engineering, and the applied sciences. The results obtained in this article may have many other applications in special functions.

    The authors express many thanks to the Editor-in-Chief, handling editor, and the reviewers for their outstanding comments that improve our paper.

    The authors declare that they have no competing interests concerning the publication of this article.



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