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
[1] | Ye Yue, Ghulam Farid, Ayșe Kübra Demirel, Waqas Nazeer, Yinghui Zhao . Hadamard and Fejér-Hadamard inequalities for generalized k-fractional integrals involving further extension of Mittag-Leffler function. AIMS Mathematics, 2022, 7(1): 681-703. doi: 10.3934/math.2022043 |
[2] | Khaled Matarneh, Suha B. Al-Shaikh, Mohammad Faisal Khan, Ahmad A. Abubaker, Javed Ali . Close-to-convexity and partial sums for normalized Le Roy-type q-Mittag-Leffler functions. AIMS Mathematics, 2025, 10(6): 14288-14313. doi: 10.3934/math.2025644 |
[3] | Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407 |
[4] | Ghulam Farid, Maja Andrić, Maryam Saddiqa, Josip Pečarić, Chahn Yong Jung . Refinement and corrigendum of bounds of fractional integral operators containing Mittag-Leffler functions. AIMS Mathematics, 2020, 5(6): 7332-7349. doi: 10.3934/math.2020469 |
[5] | Maryam Saddiqa, Ghulam Farid, Saleem Ullah, Chahn Yong Jung, Soo Hak Shim . On Bounds of fractional integral operators containing Mittag-Leffler functions for generalized exponentially convex functions. AIMS Mathematics, 2021, 6(6): 6454-6468. doi: 10.3934/math.2021379 |
[6] | Bushra Kanwal, Saqib Hussain, Thabet Abdeljawad . On certain inclusion relations of functions with bounded rotations associated with Mittag-Leffler functions. AIMS Mathematics, 2022, 7(5): 7866-7887. doi: 10.3934/math.2022440 |
[7] | Sabir Hussain, Rida Khaliq, Sobia Rafeeq, Azhar Ali, Jongsuk Ro . Some fractional integral inequalities involving extended Mittag-Leffler function with applications. AIMS Mathematics, 2024, 9(12): 35599-35625. doi: 10.3934/math.20241689 |
[8] | Gauhar Rahman, Iyad Suwan, Kottakkaran Sooppy Nisar, Thabet Abdeljawad, Muhammad Samraiz, Asad Ali . A basic study of a fractional integral operator with extended Mittag-Leffler kernel. AIMS Mathematics, 2021, 6(11): 12757-12770. doi: 10.3934/math.2021736 |
[9] | Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386 |
[10] | Miguel Vivas-Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Artion Kashuri, Muhammad Aslam Noor . Generalized (p,q)-analogues of Dragomir-Agarwal's inequalities involving Raina's function and applications. AIMS Mathematics, 2022, 7(6): 11464-11486. doi: 10.3934/math.2022639 |
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,z∈C,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,z∈C,[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,z∈C,[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,z∈C,[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
kkjk∏m=1(ρ+m−1k)j if k∈N. |
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 f∗F, is defined as
(f∗F)(z)=zp+∞∑j=p+1ajdjzj=(F∗f)(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) (z∈O)⇔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[j−1]q…[2]q[1]q, j=1,2,3,…1, j=0 |
where [j]q=1−qj1−q=1+∑j−1m=1 qm and [0]q=0.
Definition 1.2 ([18]). The q-generalized Pochhammer symbol [ρ]j,q, ρ∈C, is given as
[ρ]j,q=[ρ]q[ρ+1]q[ρ+2]q…[ρ+j−1]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 q→1−, 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(j−p)Γq(μ+σ(j−p))zj(j−p)!. | (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(j−p)Γq(μ+σ(j−p))zj(j−p)!. | (1.4) |
Definition 1.3. For f∈S(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:
1p−τ(z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−τ)≺1+Mz1+Nz, (z∈O) | (1.6) |
or equivalently
z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)≺p+(pN+(M−N)(p−τ))z1+Nz, (z∈O) |
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+(M−N)(p−τ)]|<1, | (1.7) |
where −1≤M<N≤1, 0≤τ<p, and p∈N.
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)=S∗p(M,N;τ,p) was provided by Aouf [2].
(2) Q0,0,10,1(M,N;0,p)=S∗p(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)(j−p)+(M−N)(p−τ))χj|aj|≤(M−N)(p−τ), | (2.1) |
where 1≤M<N≤1, 0≤τ<p, and p∈N.
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))′−[(M−N)(p−τ)+pN]Aμ,ρ;kσ;p,qf(z)|=|∞∑j=p+1(j−p)χjajzj|−|(M−N)(p−τ)zj−∞∑j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj|≤−(M−N)(p−τ)+∞∑j=p+1[(1+N)(j−p)+((M−N)(p−τ))]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+(M−N)(p−τ)]|=|∑∞j=p+1(j−p)χjajzj(M−N)(p−τ)zj−∑∞j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj|<1. |
Since R(z)≤|z|, we get
R{∑∞j=p+1(j−p)χjajzj(M−N)(p−τ)zj−∑∞j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj}<1. |
Taking z→1−, we have
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj|aj|≤(M−N)(p−τ). |
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 f1∗f2∈Qμ,ρ;kσ;q(M,N;τ,p), where
τ1=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)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)(j−p)+(M−N)(p−τ1))χj(M−N)(p−τ1)aj,1aj,2≤1. | (2.3) |
Since f1,f2∈Qμ,ρ;kσ;q(M,N;τ,p), by the condition (2.1) and the Cauchy-Schwarz inequality, we get
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)√aj,1aj,2≤1. | (2.4) |
From (2.3) and (2.4), we observe that
√aj,1aj,2≤[((1+N)(j−p)+(M−N)(p−τ))χj](p−τ1)[((1+N)(j−p)+(M−N)(p−τ1))χj](p−τ). |
From (2.4), it is necessary to prove
(M−N)(p−τ)((1+N)(j−p)+(M−N)(p−τ))χj≤[((1+N)(j−p)+(M−N)(p−τ))χj](p−τ1)[((1+N)(j−p)+(M−N)(p−τ1))χj](p−τ). | (2.5) |
Furthermore, from the inequality (2.5) it follows that
τ1≤p−(j−p)(1+N)(M−N)(p−τ)2χj[((1+N)(j−p)+(M−N)(p−τ1))χj]2−(M−N)2(p−τ)2χj. |
Now, set
E(j)=p−(j−p)(1+N)(M−N)(p−τ)2χj[((1+N)(j−p)+(M−N)(p−τ1))χj]2−(M−N)2(p−τ)2χj. |
We observe that the function E(j) is increasing for j∈N. Putting j=1, we have
τ1=E(1)=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)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)zj∈Qμ,ρ;kσ;q(M,N;τ,p), |
where
η=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)2χ1. |
Proof. By Theorem 2.1, we have
∞∑j=p+1 [((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2a2j,s≤∞∑j=p+1 [((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)aj,s]2≤1, (s=1,2). |
From the above inequality, we obtain
∞∑j=p+1 12[((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2(a2j,1+a2j,2)≤1. |
Therefore, the largest η can be obtained such that
((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)≤12[((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2. |
That is,
η≤p−2(j−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(j−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
Now, set
E(j)=p−2(j−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(j−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
We observe that the function E(j) is increasing for j∈N. Putting j=1, we have
η=E(1)=p−2(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
This completes the proof.
Theorem 2.4. Let f1,f2∈Qμ,ρ;kσ;q(M,N;τ,p). Then for γ∈[0,1], the function F(z)=(1−γ)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)=(1−γ)f1+γf2=zp+∞∑j=p+1ηjzj, |
where ηj=(1−γ)aj,1+γaj,2.
By (2.1), we observe that
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj[(1−γ)aj,1+γaj,2]=(1−γ)∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χjaj,1+γ∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χjaj,2≤(1−γ)(M−N)(p−τ)+γ(M−N)(p−τ). |
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=1ℵsfs, where ∑ms=1ℵs=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)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)aj,s≤1. |
Since
P(z)=m∑s=1ℵsfs=m∑s=1ℵs(zp+∞∑j=p+1aj,szj)=zp+∞∑j=p+1(m∑s=1ℵsaj,s)zj, |
∞∑j=p+1((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)m∑s=1ℵsaj,s≤1. |
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) )δ−1f(v)dv=z+∞∑j=2 (2j+1)δajzj(δ>0;f∈S). |
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 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent starlike of order τ in |z|≤r1, where
r1=infj≥p+1{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)(j−2p+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|≤p−τ, | (3.3) |
|z(Jmu,pf(z))′Jmu,pf(z)−p|=|∑∞j=p+1(j−p)(p+uj+u)δajzj∑∞j=p+1(p+uj+u)δajzj|≤∑∞j=p+1(j−p)(p+uj+u)δaj|z|j∑∞j=p+1(p+uj+u)δaj|z|j. |
By (3.2), we have
∞∑j=p+1(j−2p+τ)(p+u)δaj|z|j(p−τ)(j+u)δ≤1. |
By (2.1) in Theorem 2.1, it is clear that
(j−2p+τ)(p+u)δ(p−τ)(j+u)δ|z|j≤((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ). |
Therefore,
|z|≤{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)(j−2p+τ)(p+u)δ}1j. |
This completes the proof.
Theorem 3.2. If f(z)∈Qμ,ρ;kσ;q(M,N;τ,p) and 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent convex of order τ in |z|≤r2, where
r2=infj≥p+1{((1+N)(j−p)+(M−N)(p−τ))χjp(j+u)δ(M−N)[j(j−2p+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|≤p−τ, |
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 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent close-to-convex of order τ in |z|≤r3, where
r3=infj≥1{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)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.
[1] |
J. G. Liu, X. J. Yang, Symmetry group analysis of several coupled fractional partial differential equations, Chaos Soliton. Fract., 173 (2023), 113603. https://doi.org/10.1016/j.chaos.2023.113603 doi: 10.1016/j.chaos.2023.113603
![]() |
[2] |
J. G. Liu, Y. F. Zhang, J. J. Wang, Investigation of the time fractional generalized (2+1)-dimensional Zakharov-Kuznetsov equation with single-power law nonlinearity, Fractals, 31 (2023), 2350033. https://doi.org/10.1142/S0218348X23500330 doi: 10.1142/S0218348X23500330
![]() |
[3] |
B. Shiri, H. Kong, G. C. Wu, C. Luo, Adaptive learning neural network method for solving time-fractional diffusion equations, Neural Comput., 34 (2022), 971–990. https://doi.org/10.1162/neco_a_01482 doi: 10.1162/neco_a_01482
![]() |
[4] |
M. K. Sadabad, A. J. Akbarfam, B. Shiri, A numerical study of eigenvalues and eigenfunctions of fractional Sturm-Liouville problems via Laplace transform, Indian J. Pure Appl. Math., 51 (2020), 857–868. https://doi.org/10.1007/s13226-020-0436-2 doi: 10.1007/s13226-020-0436-2
![]() |
[5] |
Z. Li, S. Zhao, Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation, AIMS Math., 9 (2024), 22590–22601. https://doi.org/10.3934/math.20241100 doi: 10.3934/math.20241100
![]() |
[6] |
L. C. Yu, Zero-r law on the analyticity and the uniform continuity of fractional resolvent families, Integr. Equ. Oper. Theory, 96 (2024), 34. https://doi.org/10.1007/s00020-024-02785-4 doi: 10.1007/s00020-024-02785-4
![]() |
[7] |
Y. Z. Guo, L. Z. Guo, S. A. Billings, D. Coca, Z. Q. Lang, Volterra series approximation of a class of nonlinear dynamical systems using the Adomian decomposition method, Nonlinear Dyn., 74 (2013), 359–371. https://doi.org/10.1007/s11071-013-0975-8 doi: 10.1007/s11071-013-0975-8
![]() |
[8] |
S. H. Chang, A variational iteration method involving Adomian polynomials for a strongly nonlinear boundary value problem, East Asian J. Appl. Math, 9 (2019), 153–164. https://doi.org/10.4208/eajam.041116.291118 doi: 10.4208/eajam.041116.291118
![]() |
[9] |
N. Herisanu, V. Marinca, An iteration procedure with application to Van der Pol oscillator, Int. J. Nonlin. Sci. Num. Simul., 10 (2009), 353–361. https://doi.org/10.1515/IJNSNS.2009.10.3.353 doi: 10.1515/IJNSNS.2009.10.3.353
![]() |
[10] |
Y. Khan, A novel Laplace decomposition method for non-linear stretching sheet problem in the presence of MHD and slip condition. Int. J. Num. Meth. Heat Fl. Flow, 24 (2013), 73–85. https://doi.org/10.1108/HFF-02-2012-0048 doi: 10.1108/HFF-02-2012-0048
![]() |
[11] | M. Kurulay, A. Secer, M. A. Akinlar, A new approximate analytical solution of Kuramoto-Sivashinsky equation using homotopy analysis method, Appl. Math. Inform. Sci., 7 (2013), 267–271. |
[12] |
A. Shah, S. Hussain, An analytical approach to the new solution of family of Kuramoto Sivashinsky equation by q-Homotopy analysis technique, Int. J. Differ. Equat., 2024 (2024), 6652990. https://doi.org/10.1155/2024/6652990 doi: 10.1155/2024/6652990
![]() |
[13] |
A. H. Khater, R. S. Temsah, Numerical solutions of the generalized Kuramoto-Sivashinsky equation by Chebyshev spectral collocation methods, Comput. Math. Appl., 56 (2008), 1465–1472. https://doi.org/10.1016/j.camwa.2008.03.013 doi: 10.1016/j.camwa.2008.03.013
![]() |
[14] |
M. Lakestani, M. Dehghan, Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions, Appl. Math. Model., 36 (2012), 605–617. https://doi.org/10.1016/j.apm.2011.07.028 doi: 10.1016/j.apm.2011.07.028
![]() |
[15] |
Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356–369. https://doi.org/10.1143/PTP.55.356 doi: 10.1143/PTP.55.356
![]() |
[16] | G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Annu. Rev. Fluid Mech., 15 (1982), 179–199. |
[17] |
G. D. Akrivis, Finite difference discretization of the Kuramoto-Sivashinsky equation, Numer. Math., 63 (1992), 1–11. https://doi.org/10.1007/BF01385844 doi: 10.1007/BF01385844
![]() |
[18] |
M. Benlahsen, G. Bognar, Z. Csati, M. Guedda, K. Hriczo, Dynamical properties of a nonlinear Kuramoto-Sivashinsky growth equation, Alex. Eng. J., 6 (2021), 3419–3427. https://doi.org/10.1016/j.aej.2021.02.003 doi: 10.1016/j.aej.2021.02.003
![]() |
[19] |
S. R. Jena, G. S. Gebremedhin, Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation, Arab J. Basic Appl. Sci., 28 (2021), 283–291. https://doi.org/10.1080/25765299.2021.1949846 doi: 10.1080/25765299.2021.1949846
![]() |
[20] |
W. S. Yin, F. Xu, W. P. Zhang, Y. X. Gao, Asymptotic expansion of the solutions to time-space fractional Kuramoto-Sivashinsky equations, Adv. Math. Phy., 2016 (2016), 4632163. https://doi.org/10.1155/2016/4632163 doi: 10.1155/2016/4632163
![]() |
[21] |
P. Veeresha, D. G. Prakasha, Solution for fractional Kuramoto-Sivashinsky equation using novel computational technique, Int. J. Appl. Comput. Math., 7 (2021), 33. https://doi.org/10.1007/s40819-021-00956-0 doi: 10.1007/s40819-021-00956-0
![]() |
[22] |
J. W. Wang, X. X. Jiang, X. H. Yang, H. X. Zhang, A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto-Sivashinsky equation, Comp. Appl. Math., 43 (2024), 381. https://doi.org/10.1007/s40314-024-02883-4 doi: 10.1007/s40314-024-02883-4
![]() |
[23] |
R. Choudhary, S. Singh, P. Das, D. Kumar, A higher order stable numerical approximation for time-fractional non-linear Kuramoto-Sivashinsky equation based on quintic B-spline, Math. Method. Appl. Sci., 47 (2024), 11953–11975. https://doi.org/10.1002/mma.9778 doi: 10.1002/mma.9778
![]() |
[24] |
R. Shah, H. Khan, D. Baleanu, P. Kumam, M. Arif, A semi-analytical method to solve family of Kuramoto-Sivashinsky equations, J. Taibah Univ. Sci., 14 (2020), 402–411. https://doi.org/10.1080/16583655.2020.1741920 doi: 10.1080/16583655.2020.1741920
![]() |
[25] | M. Ali, M. Alquran, I. Jaradat, N. A. Afouna, D. Baleanu, Dynamics of integer-fractional time-derivative for the new two-mode Kuramoto-Sivashinsky model, Rom. Rep. Phys, 72 (2020), 103. |
[26] |
A. Burqan, R. Saadeh, A. Qazza, S. Momani, ARA-residual power series method for solving partial fractional differential equations, Alex. Eng. J., 62 (2023), 47–62. https://doi.org/10.1016/j.aej.2022.07.022 doi: 10.1016/j.aej.2022.07.022
![]() |
[27] |
A. Qazza, A. Burqan, R. Saadeh, Application of ARA-Residual power series method in solving systems of fractional differential equations, Math. Probl. Eng., 2022 (2022), 6939045. https://doi.org/10.1155/2022/6939045 doi: 10.1155/2022/6939045
![]() |
[28] |
J. K. Zhang, X. D. Tian, Laplace-residual power series method for solving fractional generalized long wave equations, Ocean Eng., 310 (2024), 118693. https://doi.org/10.1016/j.oceaneng.2024.118693 doi: 10.1016/j.oceaneng.2024.118693
![]() |
[29] |
A. Shafee, Y. Alkhezi, R. Shah, Efficient solution of fractional system partial differential equations using Laplace residual power series method, Fractal Fract., 7 (2023), 429. https://doi.org/10.3390/fractalfract7060429 doi: 10.3390/fractalfract7060429
![]() |
[30] |
M. A. N. Oqielat, T. Eriqat, O. Ogilat, A. El-Ajou, S. E. Alhazmi, S. Al-Omari, Laplace-residual power series method for solving time-fractional reaction-diffusion model, Fractal Fract., 7 (2023), 309. https://doi.org/10.3390/fractalfract7040309 doi: 10.3390/fractalfract7040309
![]() |
[31] |
R. Pant, G. Arora, H. Emadifar, Elzaki residual power series method to solve fractional diffusion equation, Plos One, 19 (2024), e0298064. https://doi.org/10.1371/journal.pone.0298064 doi: 10.1371/journal.pone.0298064
![]() |
[32] | G. A. Anastassiou, On right fractional calculus, Chaos Solition. Fract., 42 (2009), 365–376. https://doi.org/10.1016/j.chaos.2008.12.013 |
[33] |
S. Kumar, A. Yildirim. Y. Khan, L. Wei, A fractional model of the diffusion equation and its analytical solution using Laplace transform, Sci. Iran., 19 (2012), 1117–1123. https://doi.org/10.1016/j.scient.2012.06.016 doi: 10.1016/j.scient.2012.06.016
![]() |
[34] |
K. Shah, H. Khalil, R. A. Khan, Analytical solutions of fractional order diffusion equations by natural transform method, Iran. J. Sci. Technol. Trans. Sci., 42 (2018), 1479–1490. https://doi.org/10.1007/s40995-016-0136-2 doi: 10.1007/s40995-016-0136-2
![]() |
[35] |
A. S. Alshehry, M. Imran, A. Khan, R. Shan, W. Weera, Fractional view analysis of Kuramoto-Sivashinsky equations with non-singular kernel operators, Symmetry, 14 (2022), 1463. https://doi.org/10.3390/sym14071463 doi: 10.3390/sym14071463
![]() |
1. | H. M. Srivastava, Sarem H. Hadi, Maslina Darus, Some subclasses of p-valent γ-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator, 2023, 117, 1578-7303, 10.1007/s13398-022-01378-3 | |
2. | Alina Alb Lupaş, Applications of the q-Sălăgean Differential Operator Involving Multivalent Functions, 2022, 11, 2075-1680, 512, 10.3390/axioms11100512 | |
3. | Ali Mohammed Ramadhan, Najah Ali Jiben Al-Ziadi, New Class of Multivalent Functions with Negative Coefficients, 2022, 2581-8147, 271, 10.34198/ejms.10222.271288 | |
4. | Sarem H. Hadi, Maslina Darus, Alina Alb Lupaş, A Class of Janowski-Type (p,q)-Convex Harmonic Functions Involving a Generalized q-Mittag–Leffler Function, 2023, 12, 2075-1680, 190, 10.3390/axioms12020190 | |
5. | Abdullah Alatawi, Maslina Darus, Badriah Alamri, Applications of Gegenbauer Polynomials for Subfamilies of Bi-Univalent Functions Involving a Borel Distribution-Type Mittag-Leffler Function, 2023, 15, 2073-8994, 785, 10.3390/sym15040785 | |
6. | Abdulmtalb Hussen, An application of the Mittag-Leffler-type Borel distribution and Gegenbauer polynomials on a certain subclass of bi-univalent functions, 2024, 10, 24058440, e31469, 10.1016/j.heliyon.2024.e31469 | |
7. | Sarem H. Hadi, Maslina Darus, Firas Ghanim, Alina Alb Lupaş, Sandwich-Type Theorems for a Family of Non-Bazilevič Functions Involving a q-Analog Integral Operator, 2023, 11, 2227-7390, 2479, 10.3390/math11112479 | |
8. | Sarem H. Hadi, Maslina Darus, Rabha W. Ibrahim, Third-order Hankel determinants for q -analogue analytic functions defined by a modified q -Bernardi integral operator , 2024, 47, 1607-3606, 2109, 10.2989/16073606.2024.2352873 | |
9. | Haewon Byeon, Manivannan Balamurugan, T. Stalin, Vediyappan Govindan, Junaid Ahmad, Walid Emam, Some properties of subclass of multivalent functions associated with a generalized differential operator, 2024, 14, 2045-2322, 10.1038/s41598-024-58781-6 | |
10. | Timilehin Gideon Shaba, Serkan Araci, Babatunde Olufemi Adebesin, 2023, Investigating q-Exponential Functions in the Context of Bi-Univalent Functions: Insights into the Fekctc-Szcgö Problem and Second Hankel Determinant, 979-8-3503-5883-4, 1, 10.1109/ICMEAS58693.2023.10429891 | |
11. | Sarem H. Hadi, Maslina Darus, 2024, 3023, 0094-243X, 070002, 10.1063/5.0172085 | |
12. | Sarem H. Hadi, Maslina Darus, Badriah Alamri, Şahsene Altınkaya, Abdullah Alatawi, On classes of ζ -uniformly q -analogue of analytic functions with some subordination results , 2024, 32, 2769-0911, 10.1080/27690911.2024.2312803 | |
13. | Sarem H. Hadi, Khalid A. Challab, Ali Hasan Ali, Abdullah A. Alatawi, A ϱ-Weyl fractional operator of the extended S-type function in a complex domain, 2024, 13, 22150161, 103061, 10.1016/j.mex.2024.103061 | |
14. | Ehsan Mejeed Hameed, Elaf Ali Hussein, Rafid Habib Buti, 2025, 3264, 0094-243X, 050109, 10.1063/5.0258939 | |
15. | Girish D. Shelake, Sarika K. Nilapgol, Priyanka D. Jirage, 2025, 3283, 0094-243X, 040016, 10.1063/5.0265526 |