The time-fractional coupled Schrödinger-KdV equation is an interesting mathematical model because of its wide and significant application in mathematics and applied sciences. A fractional coupled Schrödinger-KdV equation in the sense of Caputo derivative is investigated in this article. Namely, we provide a comparative study of the considered model using the Adomian decomposition method and the homotopy perturbation method with Shehu transform. Approximate solutions obtained using the Adomian decomposition and homotopy perturbation methods were numerically evaluated and presented in graphs and tables. Then, these solutions were compared to the exact solutions, demonstrating the simplicity, effectiveness, and good accuracy of the applied method. To demonstrate the accuracy and efficiency of the suggested techniques, numerical problem are provided.
Citation: Rasool Shah, Abd-Allah Hyder, Naveed Iqbal, Thongchai Botmart. Fractional view evaluation system of Schrödinger-KdV equation by a comparative analysis[J]. AIMS Mathematics, 2022, 7(11): 19846-19864. doi: 10.3934/math.20221087
[1] | Nehad Ali Shah, Iftikhar Ahmed, Kanayo K. Asogwa, Azhar Ali Zafar, Wajaree Weera, Ali Akgül . Numerical study of a nonlinear fractional chaotic Chua's circuit. AIMS Mathematics, 2023, 8(1): 1636-1655. doi: 10.3934/math.2023083 |
[2] | Ihtisham Ul Haq, Shabir Ahmad, Sayed Saifullah, Kamsing Nonlaopon, Ali Akgül . Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels. AIMS Mathematics, 2022, 7(10): 18809-18823. doi: 10.3934/math.20221035 |
[3] | Najat Almutairi, Sayed Saber . Chaos control and numerical solution of time-varying fractional Newton-Leipnik system using fractional Atangana-Baleanu derivatives. AIMS Mathematics, 2023, 8(11): 25863-25887. doi: 10.3934/math.20231319 |
[4] | Dongjie Gao, Peiguo Zhang, Longqin Wang, Zhenlong Dai, Yonglei Fang . A novel high-order symmetric and energy-preserving continuous-stage Runge-Kutta-Nyström Fourier pseudo-spectral scheme for solving the two-dimensional nonlinear wave equation. AIMS Mathematics, 2025, 10(3): 6764-6787. doi: 10.3934/math.2025310 |
[5] | Mohsen DLALA, Abdelhamid ZAIDI, Farida ALHARBI . Event-triggered impulsive control for exponential stabilization of fractional-order differential system. AIMS Mathematics, 2025, 10(7): 16551-16569. doi: 10.3934/math.2025741 |
[6] | Anastacia Dlamini, Emile F. Doungmo Goufo, Melusi Khumalo . On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system. AIMS Mathematics, 2021, 6(11): 12395-12421. doi: 10.3934/math.2021717 |
[7] | Abdulaziz Khalid Alsharidi, Saima Rashid, S. K. Elagan . Short-memory discrete fractional difference equation wind turbine model and its inferential control of a chaotic permanent magnet synchronous transformer in time-scale analysis. AIMS Mathematics, 2023, 8(8): 19097-19120. doi: 10.3934/math.2023975 |
[8] | Qingkai Xu, Chunrui Zhang, Xingjian Wang . Codimension one and codimension two bifurcations analysis of discrete mussel-algae model. AIMS Mathematics, 2025, 10(5): 11514-11555. doi: 10.3934/math.2025524 |
[9] | Nisar Gul, Saima Noor, Abdulkafi Mohammed Saeed, Musaad S. Aldhabani, Roman Ullah . Analytical solution of the systems of nonlinear fractional partial differential equations using conformable Laplace transform iterative method. AIMS Mathematics, 2025, 10(2): 1945-1966. doi: 10.3934/math.2025091 |
[10] | Xiaojun Zhou, Yue Dai . A spectral collocation method for the coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2022, 7(4): 5670-5689. doi: 10.3934/math.2022314 |
The time-fractional coupled Schrödinger-KdV equation is an interesting mathematical model because of its wide and significant application in mathematics and applied sciences. A fractional coupled Schrödinger-KdV equation in the sense of Caputo derivative is investigated in this article. Namely, we provide a comparative study of the considered model using the Adomian decomposition method and the homotopy perturbation method with Shehu transform. Approximate solutions obtained using the Adomian decomposition and homotopy perturbation methods were numerically evaluated and presented in graphs and tables. Then, these solutions were compared to the exact solutions, demonstrating the simplicity, effectiveness, and good accuracy of the applied method. To demonstrate the accuracy and efficiency of the suggested techniques, numerical problem are provided.
Probability theory and other related fields describe stochastic processes as groups of random variables. In verifiable terms, the random variables were correlated or listed by a large number of numbers, usually considered as time foci, providing a translation of stochastic processes into numerical estimation of a system that changes over time, such as bacteria growing and decaying, electrical circuits fluctuating due to thermal noise, or gas molecules forming. A stochastic process can be a scientific model for systems that appear to change at random. Many applications of these techniques exist in various fields, such as biology, ecology, chemistry, and physics, as well as engineering and technology fields, such as picture preparing, data theory, computer science, signal processing, telecommunications, and cryptography.
Moore's book prompted a systematic investigation into interval arithmetic in different countries after its appearance. Although he dealt with a whole bunch of general ideas for intervals, he mainly focused on bounding solutions to various boundary value and initial value problems, see Ref. [1]. In numerical analysis, compact intervals have played an increasingly important role in verifying or enclosing solutions to a variety of mathematical problems or demonstrating that such problems cannot be solved within a given domain over the past three decades. In order to accomplish this, intervals need to be regarded as extensions of real and complex numbers, interval functions need to be introduced and interval arithmetic applied, and appropriate fixed point theorems have to be applied. A thorough and sophisticated computer implementation of these arithmetic concepts combined with partly new concepts such as controlled rounding, variable precision, and epsilon inflation resulted in the theory being useful in practice, and the computer was able to automatically verify and enclose solutions in many fields. For some recent developments in variety of other practical fields, see Refs. [2,3,4]
Mathematics is always captivating and fascinating to study when it comes to convex functions. Mathematicians work hard to find and explore a large array of results with fruitful applications. Mathematicians as well as people from physics, economics, data mining, and signal processing have been studying convex functions for years. In the optimization of stochastic processes, especially in numerical approximations when there are probabilistic quantities, convexity is of utmost importance. The convex stochastic processes were defined by Nikodem in 1980, along with some properties known from classical convex functions, see Ref. [5]. The Jensen convex mapping plays a very important role in the inequalities theory. Several well known inequalities are derived from this and this was further utilized by Skowronski in stochastic sense in 1992, see Ref. [6]. Using these notions, Kotrys showed how integral operators can be used to calculate the lower and upper bounds for the Hermite-Hadamard inequality for convex stochastic processes, as well as properties of convex and Jensen-convex processes, see Ref. [7]. Many studies on various types of convexity for stochastic processes have been published in the literature in recent years. Okur et al. [8] developed (H.H) inequality for stochastic p-convexity. Erhan constructs these inequalities using the stochastic s-convex concept, see Ref. [9]. Budak et al. [10] extends the results of the following authors [11] by using the concept of h-convexity to present inequalities in a more generalised form. Check out some recent developments in the stochastic process using a variety of other classes, see Refs. [12,13,14,15,16,17,18,19,20,21]. Tunc created the Ostrowski type inequality for the h-convex function, see Ref. [22]. Later authors extended the results for the Ostrowski inequality and converted them to a stochastic process, see Ref. [23]. We know that Zhao et al. [24] recently connected the term inequalities to interval calculus, and they present the inequalities in this form. Gaining inspiration from this by the end of 2022 Afzal et al. [25] introduces the concept of stochastic processes in relation to interval analysis, as well as the (H.H) and Jensen versions of stochastic processes developed. Eliecer and his co-author developed (m,h1,h2)-G-convex stochastic process and represent the following inequalities, see Ref. [26]. Cortez utilized the concept of fractional operator and developed inequalities using (m,h1,h2)-convex stochastic process, see Ref. [27]. There have been some recent developments related to convex stochastic processes and convexity with the use of IVFS, see Refs. [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52].
The stochastic process has been used in various fields related to mathematics and statistics in recent years, and their connection with interval analysis contributes greatly to the accuracy of desired results. Authors have used the stochastic process as a means of analyzing convex analysis and its related integral and variational inequalities over the past years, but the present results are considered novel because the integration of integral inequalities with interval analysis is very new with stochastic processes. They were constructed because we know inequalities play an essential role in ensuring the regularity, stability, and uniqueness of many interval stochastic differential equations, which was not possible with ordinary stochastic inequalities. Using interval analysis, we were able to explore a whole new dimension of inequalities. Some recent advances in interval stochastic processes can be found in the following disciplines, see Refs. [53,54,55].
Strong literature and specific articles, [10,22,23,24,25,27,51] served as our inspiration as we introduced the concept of the (h1,h2)-convex stochastic process and developed Hermite-Hadamard, Ostrowski and Jensen type inclusions. In addition, we provide some numerically non-trivial examples to demonstrate the validity of the main results. The article is organised as follows: After reviewing the necessary and pertinent information regarding interval-valued analysis in Section 2, we give some introduction related to Stochastic process under Section 3. In Section 4, we discuss our main results. Section 5 explores a brief conclusion.
This paper employs concepts that are not defined here but are used, see Ref. [25]. There is an interval J that is closed and bounded, and it is defined by the relation given below:
J=[J_,¯J]={r∈R:J_≤r≤¯J}. |
These J_,¯J∈R are the endpoints of interval J. When J_=¯J, then the interval J is called to be degenerated. When J_>0 or ¯J<0, we say that it is positive or negative respectively. Therefore, we are talking about the collection of all intervals in R by RI and positive intervals by RI+. The widely used Hausdorff separation of the intervals J and K is defined as:
D(J,K)=D([J_,¯J],[K_,¯K])=max{|J_−K_|,|¯J−¯K|}. |
It's obvious that (RI,D) is complete metric space. The definitions of the fundamental interval arithmetic operations are provided below for J and K:
J−K=[J_−¯K,¯J−K_],J+K=[J_+K_,¯J+¯K],J⋅K=[minO,maxO]where O={J_ K_,J_ ¯K, ¯JK_,¯J ¯K},J/K=[minJ,maxJ]where J={J_/K_,J_/¯K,¯J/K_,¯J/¯K} and 0∉K. |
The interval J can be multiplied scalarly by
ϱ[J_,¯J]={[ϱJ_,ϱ¯J],ϱ>0;{0},ϱ=0;[ϱ¯J,ϱJ_],ϱ<0. |
By clarifying how operations are defined on, its algebraic characteristics that make it quasilinear can be explained on RI. They fit into the following categories.
● (Associative w.r.t addition) (J+K)+ϱ=J+(K+ϱ) ∀ J,K,ϱ∈RI
● (Commutative w.r.t addition) J+ϱ=ϱ+J ∀ J,ϱ∈RI,
● (Additive element) J+0=0+J ∀ J∈RI,
● (Law of Cancellation) ϱ+J=ϱ+ϱ⇒J=ϱ ∀ J,ϱ,ϱ∈RI,
● (Associative w.r.t multiplication) (J⋅ϱ)⋅ϱ=J⋅(ϱ⋅ϱ) ∀ J,ϱ,ϱ∈RI,
● (Commutative w.r.t multiplication) J⋅ϱ=ϱ⋅J ∀ J,ϱ∈RI,
● (Unity element) J⋅1=1⋅J ∀ J∈RI.
A set's inclusion ⊆ is another property that is given by
J⊆K⟺K_≤J_and¯J≤¯K. |
When we combine inclusion and arithmetic operations, we get the following relationship, known as the isotone of intervals for inclusion. Consider ⊙ it will represent the basic arithmetic operations. If J,K,ϱ and ι are intervals, then
J⊆Kandϱ⊆ι. |
This implies that the following relationship is true.
J⊙ϱ⊆K⊙ι. |
This proposition is concerned with the preservation of inclusion in scalar multiplication.
Proposition 2.1. Let J and K be intervals and ϱ∈R, then ϱJ⊆ϱK.
The ideas presented below serve as the foundation for this section's discussion of the concept of integral for IVFS: A function F is known as IVF at Ja∈[J1,J2], if it gives each a nonempty interval Ja∈[J1,J2]
F(Ja)=[F_(Ja),¯F(Ja)]. |
A partition of any arbitrary subset P of [J1,J2] can be represented as:
P:J1=Ja<Jb<⋯<Jm=J2. |
The mesh of partition of P is represented by
Mesh(P)=max{Ji−Ji−1:i=1,2,…,m}. |
The pack of all partitions of [J1,J2] can be represented by P([J1,J2]). Let P(Δ,[J1,J2]) be the pack of all P∈P([J1,J2]) with satisfying this mesh(P)<Δ for any arbitrary point in interval, then sum is denoted by:
S(F,P,Δ)=n∑i=1F(Jii)[Ji−Ji−1], |
where F:[J1,J2]→RI. We say that S(F,P,Δ) is a sum of F with reference to P∈P(Δ,[J1,J2]).
Definition 2.1. (See [25]) A function F:[J1,J2]→RI is known as Riemann integrable for IVF or it can be represented by (IR) on [J1,J2], if ∃ γ∈RI such that, for every J2>0 ∃ Δ>0 such that
d(S(F,P,Δ),γ)<J2 |
for each Riemann sum S of F with reference to P∈P(Δ,[J1,J2]) and unrelated to the choice of Jii∈[Ji−1,Ji], ∀ 1≤i≤m. In this scenario, γ is known as (IR)-integral of F on [J1,J2] and is represented by
γ=(IR)∫J2J1F(Ja)dJa. |
The pack of all (IR)-integral functions of F on [J1,J2] can be represented by IR([J1,J2]).
Theorem 2.1. (See [25]) Let F:[J1,J2]→RI be an IVF defined as F(Ja)=[F_(Ja),¯F(Ja)]. F∈IR([J1,J2]) iff F_(Ja),¯F(Ja)∈R([J1,J2]) and
(IR)∫J2J1F(Ja)dJa=[(R)∫J2J1F_(Ja)dJa,(R)∫J2J1¯F(Ja)dJa], |
where R([J1,J2]) represent the bunch of all R-integrable functions. If F(Ja)⊆G(Ja) for all Ja∈[J1,J2], then this holds
(IR)∫J2J1F(Ja)dJa⊆(IR)∫J2J1G(Ja)dJa. |
Definition 3.1. A function F:Δ→R defined on probability space (Δ,A,P) is called to be random variable if it is A-measurable. A function F:I×Δ→R where I⊆R is known as stochastic process if, ∀ J1∈I the function F(J1,⋅) is a random variable.
Properties of stochastic process
A stochastic process F:I×Δ→R is
● Continuous over interval I, if ∀ Jo∈I, one has
P−limf→JoF(J1,.)=F(Jo,.) |
where P−lim represent the limit in the space of probabilities.
● Continuity in mean square sense over interval I, if ∀ Jo∈I, one has
limJ1→JoE[(F(J1,.)−F(Jo,.))2]=0, |
where E[F(J1,⋅)] represent the random variable's expected value.
● Differentiability in mean square sense at any arbitrary point J1, if one has random variable F′:I×Δ→R, then this holds
F′(J1,⋅)=P−limJ1→JoF(J1,⋅)−F(Jo,⋅)J1−Jo. |
● Mean-square integral over I, if ∀ J1∈I, with E[F(J1,⋅)]<∞. Let [J1,J2]⊆I, J1=bo<b1<b2...<bk=J2 is a partition of [J1,J2]. Let Fn∈[bn−1,bn],∀n=1,...,k. A random variable S:Δ→R is mean square integral of the stochastic process F over interval [J1,J2], if this holds
limk→∞E[(k∑n=1F(Fn,.)(bn−bn−1)−R(.))2]=0. |
In that case, it is written as
R(⋅)=∫J2J1F(a,⋅)da (a.e.). |
We can immediately deduce the following implication from the definition of mean-square integral. If only for everyone a∈[J1,J2] the inequality F(a,⋅)≤R(a,⋅) (a.e.) holds, then
∫J2J1F(a,⋅)da≤∫J2J1R(a,⋅)da (a.e.). |
Afzal et al. [25] developed following results using interval calculus for stochastic process.
Theorem 3.1. (See [25]) Let h:[0,1]→R+ and h≠0. A function F:I×Δ→R+I is h-Godunova-Levin stochastic process for mean square integrable IVFS. For each J1,J2∈[J1,J2]⊆I, if F∈SGPX(h,[J1,J2],R+I) and F∈ R+I. Almost everywhere, the following inequality is satisfied
h(12)2F(J1+J22,⋅)⊇1J2−J1∫J2J1F(a,⋅)da⊇[F(J1,⋅)+F(J2,⋅)]∫10dah(a). | (3.1) |
Theorem 3.2. (See [25]) Let gi∈R+ with k≥2. If h is non-negative and F:I×Δ→R is non-negative h-Godunova-Levin stochastic process for IVFS almost everywhere the following inclusion valid
F(1Gkk∑i=1giai,.)⊇k∑i=1[F(ai,.)h(giGk)]. | (3.2) |
Definition 3.2. (See [25]) Let h:[0,1]→R+. Then F:I×Δ→R+ is known as h-convex stochastic process, or that F∈SPX(h,I,R+), if ∀ J1,J2∈I and a∈[0,1], we have
F(aJ1+(1−a)J2,⋅)≤ h(a)F(J1,⋅)+h(1−a)F(J2,⋅). | (3.3) |
In (3.3), if "≤" is reverse with "≥", then we call it h-concave stochastic process or F∈SPV(h,I,R+).
Definition 3.3. (See [25]) Let h:[0,1]→R+. Then the stochastic process F=[F_,¯F]:I×Δ→R+I where [J1,J2]⊆I is known as h-Godunova-Levin stochastic process for IVFS or that F∈SGPX(h,[J1,J2],R+I), if ∀ J1,J2∈[J1,J2] and a∈[0,1], we have
F(aJ1+(1−a)J2,⋅)⊇ F(J1,⋅)h(a)+F(J2,⋅)h(1−a). | (3.4) |
In (3.4), if "⊇" is reverse with "⊆", then we call it h-Godunova-Levin concave stochastic process for IVFS or F∈SGPV(h,[J1,J2],R+I).
Definition 3.4. (See [25]) Let h:[0,1]→R+. Then the stochastic process F=[F_,¯F]:I×Δ→R+I where [J1,J2]⊆I is known as h-convex stochastic process for IVFS or that F∈SPX(h,[J1,J2],R+I), if ∀ J1,J2∈[J1,J2] and a∈[0,1], we have
F(aJ1+(1−a)J2,⋅)⊇ h(a)F(J1,⋅)+h(1−a)F(J2,⋅). | (3.5) |
In (3.5), if "⊇" is reverse with "⊆", then we call it h-concave stochastic process for IVFS or F∈SPV(h,[J1,J2],R+I).
Taking inspiration from the preceding literature and definitions, we are now in a position to define a new class of convex stochastic process.
Definition 4.1. Let h1,h2:[0,1]→R+. Then the stochastic process F=[F_,¯F]:I×Δ→R+I where [J1,J2]⊆I is called (h1,h2)-convex stochastic process for IVFS or that F∈SPX((h1,h2),[J1,J2],R+I), if ∀ J1,J2∈[J1,J2] and a∈[0,1], we have
F(aJ1+(1−a)J2,⋅)⊇ h1(a)h2(1−a)F(J1,⋅)+h1(1−a)h2(a)F(J2,⋅). | (4.1) |
In (4.1), if "⊇" is replaced with "⊆", then it is called (h1,h2)-concave stochastic process for IVFS or F∈SPV((h1,h2),[J1,J2],R+I).
Remark 4.1. (i) If h1=h2=1, subsequently, Definition 4.1 turns into a stochastic process for P-function.
(ii) If h1(a)=1h(a),h2=1, subsequently, Definition 4.1 turns into a stochastic process for h-Godunova-Levin function.
(iii) If h1(a)=a,h2=1, subsequently, Definition 4.1 turns into a stochastic process for general convex function.
(iv) If h1=as,h2=1, subsequently, Definition 4.1 turns into a stochastic process for s-convex function.
Theorem 4.1. Let h1,h2:[0,1]→R+ and H(12,12)≠0. A function F:I×Δ→R+I is (h1,h2)-convex stochastic process as well as mean square integrable for IVFS. For every J1,J2∈[J1,J2]⊆I, if F∈SPX((h1,h2),[J1,J2],R+I) and F∈ R+I. Almost everywhere, the following inclusion is satisfied
12[H(12,12)]F(J1+J22,⋅)⊇1J2−J1∫J2J1F(e,⋅)de⊇[F(J1,⋅)+F(J2,⋅)]∫10H(a,1−a)da, | (4.2) |
where H(a,1−a)=h1(a)h2(1−a).
Proof. Since F∈SPX((h1,h2),[J1,J2],R+I), we have
1[H(12,12)]F(J1+J22,⋅)⊇F(aJ1+(1−a)J2,⋅)+F((1−a)J1+aJ2,⋅)1[H(12,12)]F(J1+J22,⋅)⊇[∫10F(aJ1+(1−a)J2,⋅)da+∫10F((1−a)J1+aJ2,⋅)da]=[∫10F_(aJ1+(1−a)J2,⋅)da+∫10F_((1−a)J1aJ2,⋅)da,∫10¯F(aJ1+(1−a)J2,⋅)da+∫10¯F((1−a)J1aJ2,⋅)da]=[2J2−J1∫J2J1F_(e,⋅)de,2J2−J1∫J2J1¯F(e,⋅)de]=2J2−J1∫J2J1F(e,⋅)de. | (4.3) |
By Definition 4.1, one has
F(aJ1+(1−a)J2,⋅)⊇h1(a)h2(1−a)F(J1,⋅)+h1(1−a)h2(a)F(J2,⋅). |
Integrating one has
∫10F(aJ1+(1−a)J2,⋅)da⊇F(J1,⋅)∫10h1(a)h2(1−a)da+F(J2,⋅)∫10h1(1−a)h2(a)da. |
Accordingly,
1J2−J1∫J2J1F(e,⋅)de⊇[F(J1,⋅)+F(J2,⋅)]∫10H(a,1−a)da. | (4.4) |
Now, collect (4.3) and (4.4), we achieve the desired outcome.
12[H(12,12)]F(J1+J22,⋅)⊇1J2−J1∫J2J1F(e,⋅)de⊇[F(J1,⋅)+F(J2,⋅)]∫10H(a,1−a)da. |
Example 4.1. Consider [J1,J2]=[−1,1],h1(a)=a,h2=1, ∀ a∈ [0,1]. If F:[J1,J2]→RI+ is defined as
F(e,⋅)=[e2,5−ee]. |
Then,
12[H(12,12)]F(J1+J22,⋅)=[0,4],1J2−J1∫J2J1F(e,⋅)de=[13,10e−e2+12e],[F(J1,⋅)+F(J2,⋅)]∫10H(a,1−a)da=[2,10−e−1e]. |
As a result,
[0,4]⊇[13,10e−e2+12e]⊇[2,10−e−1e]. |
This verifies the above theorem.
Theorem 4.2. Let h1,h2:[0,1]→R+ and H(12,12)≠0. A function F:I×Δ→R+I is (h1,h2)-convex stochastic process as well as mean square integrable for IVFS. For every J1,J2∈[J1,J2]⊆I, if F∈SPX((h1,h2),[J1,J2],R+I) and F∈ R+I. Almost everywhere, the following inequality is satisfied
14[H(12,12)]2F(J1+J22,⋅)⊇△1⊇1J2−J1∫J2J1F(e,⋅)de⊇△2 |
⊇{[F(J1,⋅)+F(J2,⋅)][12+H(12,12)]}∫10H(a,1−a)da, |
where
△1=14H(12,12)[F(3J1+J24,⋅)+F(3J2+J14,⋅)], |
△2=[F(J1+J22,⋅)+F(J1,⋅)+F(J2,⋅)2]∫10H(a,1−a)da. |
Proof. Take [J1,J1+J22], we have
F(3J1+J24,⋅)⊇H(12,12)F(aJ1+(1−a)J1+J22,⋅)+H(12,12)F((1−a)J1+aJ1+J22,⋅). |
Integrating one has
F(3J1+J22,⋅)⊇H(12,12)[∫10F(aJ1+(1−a)J1+J22,⋅)da+∫10F(aJ1+J22+(1−a)J2,⋅)da]=H(12,12)[2J2−J1∫J1+J22J1F(e,⋅)de+2J2−J1∫J1+J22J1F(e,⋅)de]=H(12,12)[4J2−J1∫J1+J22J1F(e,⋅)de]. | (4.5) |
Accordingly,
14H(12,12)F(3J1+J22,⋅)⊇1J2−J1∫J1+J22J1F(e,⋅)de. | (4.6) |
Similarly for interval [J1+J22,J2], we have
14H(12,12)F(3J2+J12,⋅)⊇1J2−J1∫J2J1+J22F(e,⋅)de. | (4.7) |
Adding inclusions (4.6) and (4.7), we get
△1=14H(12,12)[F(3J1+J24,⋅)+F(3J2+J14,⋅)]⊇[1J2−J1∫J2J1F(e,⋅)de].Now14[H(12,12)]2F(J1+J22,⋅)=14[H(12,12)]2F(12(3J1+J24,⋅)+12(3J2+J14,⋅))⊇14[H(12,12)]2[H(12,12)F(3J1+J24,⋅)+H(12,12)F(3J2+J14,⋅)]=14H(12,12)[F(3J1+J24,⋅)+F(3J2+J14,⋅)]=△1⊇14H(12,12){H(12,12)[F(J1,⋅)+F(J1+J22,⋅)]+H(12,12)[F(J2,⋅)+F(J1+J22,⋅)]}=12[F(J1,⋅)+F(J2,⋅)2+F(J1+J22,⋅)]⊇[F(J1,⋅)+F(J2,⋅)2+F(J1+J22,⋅)]∫10H(a,1−a)da=△2⊇[F(J1,⋅)+F(J2,⋅)2+H(12,12)F(J1,⋅)+H(12,12)F(J2,⋅)]∫10H(a,1−a)da⊇[F(J1,⋅)+F(J2,⋅)2+H(12,12)[F(J1,⋅)+F(J2,⋅)]]∫10H(a,1−a)da⊇{[F(J1,⋅)+F(J2,⋅)][12+H(12,12)]}∫10H(a,1−a)da. |
Example 4.2. Recall the Example 4.1, we have
14[H(12,12)]2F(J1+J22,⋅)=[0,4],△1=[14,10−√e−1√e],△2=[12,92−e+e−14], |
and
{[F(J1,⋅)+F(J2,⋅)][12+H(12,12)]}∫10H(a,1−a)da=[2,10−e−1e]. |
Thus, we obtain
[0,4]⊇[14,10−√e−1√e]⊇[13,10e−e2+12e]⊇[12,92−e+e−14]⊇[2,10−e−1e]. |
This verifies the above theorem.
Theorem 4.3. Suppose h1,h2:(0,1)→R+ such that h1,h2≠0. A mappings F,A:I×Δ→R+I are (h1,h2)-convex stochastic process as well as mean square integrable for IVFS. For every J1,J2∈I, if F∈SPX((h1,h2),[J1,J2],R+I), A∈SPX((h1,h2),[J1,J2],R+I) and F,A∈ IRI. Almost everywhere, the following inclusion is satisfied
1J2−J1∫J2J1F(e,⋅)A(e,⋅)de⊇M(J1,J2)∫10H2(a,1−a)da+N(J1,J2)∫10H(a,a)H(1−a,1−a)da. |
Proof. Consider F∈SPX((h1,h2),[J1,J2],R+I), A∈SPX((h1,h2),[J1,J2],R+I) then, we have
F(J1a+(1−a)J2,⋅)⊇h1(a)h2(1−a)F(J1,⋅)+h1(1−a)h2(a)F(J2,⋅), |
A(J1a+(1−a)J2,⋅)⊇h1(1−a)h2(a)A(J1,⋅)+h1(1−a)h2(a)A(J2,⋅). |
Then,
F(J1a+(1−a)J2,⋅)A(J1a+(1−a)J2,⋅)⊇H2(a,1−a)F(J1,⋅)A(J1,⋅)+H2(1−a,a)[F(J1,⋅)A(J2,⋅)+F(J2,⋅)A(J1,⋅)]+H(a,a)H(1−a,1−a)F(J2,⋅)A(J2,⋅). |
Integrating, one has
∫10F(J1a+(1−a)J2,⋅)A(J1a+(1−a)J2,⋅)da=[∫10F_(J1a+(1−a)J2,⋅)A_(J1a+(1−a)J2,⋅)da,∫10¯F(J1a+(1−a)J2,⋅)¯A(J1a+(1−a)J2,⋅)da]=[1J2−J1∫J2J1F_(e,⋅)A_(e,⋅)de,1J2−J1∫J2J1¯F(e,⋅)¯A(e,⋅de]=1J2−J1∫J2J1F(e,⋅)A(e,⋅)de⊇M(J1,J2)∫10H2(a,1−a)da+N(J1,J2)∫10H(a,a)H(1−a,1−a)da. |
Thus, it follows
1J2−J1∫J2J1F(e,⋅)A(e,⋅)de⊇M(J1,J2)∫10H2(a,1−a)da+N(J1,J2)∫10H(a,a)H(1−a,1−a)da. |
Theorem is proved.
Example 4.3. Let [J1,J2]=[0,1],h1(a)=a, h2(a)=1 for all a∈ (0,1). If F,A:[J1,J2]⊆I→RI+ are defined as
F(e,⋅)=[e2,4−ee]andA(e,⋅)=[e,3−e2]. |
Then, we have
1J2−J1∫J2J1F(e,⋅)A(e,⋅)de=[14,35−6e3],M(J1,J2)∫10H2(a,1−a)da=[13,17−2e3] |
and
N(J1,J2)∫10H(a,a)H(1−a,1−a)da=[0,6−e2]. |
Since
[14,35−6e3]⊇[13,52−7e6]. |
Consequently, Theorem 4.3 is verified.
Theorem 4.4. Suppose h1,h2:(0,1)→R+ such that h1,h2≠0. A mappings F,A:I×Δ→R+I are (h1,h2)-convex stochastic process as well as mean square integrable for IVFS. For every J1,J2∈I, if F∈SPX((h1,h2),[J1,J2],R+I), A∈SPX((h1,h2),[J1,J2],R+I) and F,A∈ IRI. Almost everywhere, the following inequality is satisfied
12[H(12,12)]2F(J1+J22,⋅)A(J1+J22,⋅)⊇1J2−J1∫J2J1F(e,⋅)A(e,⋅)de+M(J1,J2)∫10H(a,a)H(1−a,1−a)da+N(J1,J2)∫10H2(a,1−a)da. |
Proof. Since F,A∈SPX((h1,h2),[J1,J2],R+I), one has
F(J1+J22,⋅)⊇H(12,12)F(J1a+(1−a)J2,⋅)+H(12,12)F(J1(1−a)+aJ2,⋅),A(J1+J22,⋅)⊇H(12,12)A(J1a+(1−a)J2,⋅)+H(12,12)A(J1(1−a)+aJ2,⋅). | (4.8) |
F(J1+J22,⋅)A(J1+J22,⋅)⊇[H(12,12)]2[F(J1a+(1−a)J2,⋅)A(J1a+(1−a)J2,⋅)+F(J1(1−a)+aJ2,⋅)A(J1(1−a)+aJ2,⋅)]+[H(12,12)]2[F(J1a+(1−a)J2,⋅)A(J1(1−a)+aJ2,⋅)+F(J1(1−a)+aJ2,⋅)A(J1a+(1−a)J2,⋅)]⊇[H(12,12)]2[F(J1a+(1−a)J2,⋅)A(J1a+(1−a)J2,⋅)+F(J1(1−a)+aJ2,⋅)A(J1(1−a)+aJ2,⋅)]+[H(12,12)]2[(h1(a)F(J1,⋅)+h1(1−a)F(J2,⋅))(h2(1−a)A(J1,⋅)+h2(a)A(J2,⋅))]+[(H(1−a,a)A(f)+H(a,1−a)A(g))(H(a,1−a)A(f)+H(1−a,a)A(g))]⊇[H(12,12)]2[F(J1a+(1−a)J2,⋅)A(J1a+(1−a)J2,⋅)+F(J1(1−a)aJ2,⋅)A(J1(1−a)aJ2,⋅)]+[H(12,12)]2[(2H(a,a)H(1−a,1−a))M(J1,J2)+(H2(a,1−a)+H2(1−a,a))N(J1,J2)]. Integration over (0,1) , we have∫10F(J1+J22,⋅)A(J1+J22,⋅)da=[∫10F_(J1+J22,⋅)A_(J1+J22,⋅)da,∫10¯F(J1+J22,⋅)¯A(J1+J22,⋅)da]⊇2[H(12,12)]2[1J2−J1∫J2J1F(e,⋅)A(e,⋅)de]+2[H(12,12)]2[M(J1,J2)∫10H(a,a)H(1−a,1−a)da+N(J1,J2)∫10H2(a,1−a)da].Divide both sides by 12[H(12,12)]2 in above inclusion, we have 12[H(12,12)]2F(J1+J22,⋅)A(J1+J22,⋅)⊇1J2−J1∫J2J1F(e,⋅)A(e,⋅)de+M(J1,J2)∫10H(a,a)H(1−a,1−a)da+N(J1,J2)∫10H2(a,1−a)da. |
Accordingly, the above theorem can be proved.
Example 4.4. Recall the Example 4.3, we have
12[H(12,12)]2F(J1+J22,⋅)A(J1+J22,⋅)=[14,44−11√e2],1J2−J1∫J2J1F(e,⋅)A(e,⋅)de=[512,22712],M(J1,J2)∫10H2(a,1−a)da=[13,17−2e3] |
and
N(J1,J2)∫10H(a,a)H(1−a,1−a)da=[0,6−e2]. |
This implies
[14,44−11√e2]⊇[512,412−10e3]. |
This verifies the above theorem.
The following lemma aids in achieving our goal [56].
Lemma 4.1. Consider F:I×Δ⊆R→R be a stochastic process that can be mean squared differentiated on the interior of an interval I. Likewise, if the derivative of F is mean square integrable, on [J1,J2], and J1,J2∈I, then this holds:
F(b,⋅)−1J2−J1∫J2J1F(a,⋅)da=(b−J1)2J2−J1∫10sF′(ab+(1−a)J1,⋅)da−(J2−b)2J2−J1∫10sF′(ab+(1−a)J2,⋅)da,∀b∈[J1,J2]. |
Theorem 4.5. Let h,h1,h2:(0,1)→R are non-negative and multiplicative functions and a⊆h(a) for each a∈(0,1). Let F:I×Δ⊆R→R+I be a stochastic process that can be mean squared differentiated on the interior of an interval I. Likewise, if the derivative of F is mean square integrable, on [J1,J2], and J1,J2∈I. If |F′| is (h1,h2)-convex stochastic process for IVFS on I and satisfying |F′(b,⋅)|≤γ for each b, then
D([F_(b,⋅),¯F(b,⋅)],[1J2−J1∫J2J1F_(a,⋅)da,1J2−J1∫J2J1¯F(a,⋅)da])=max{|F_(b,⋅)−1J2−J1∫J2J1F_(a,⋅)da|,|¯F(b,⋅)−1J2−J1∫J2j1¯F(a,⋅)da|}⊇γ[(b−J1)2+(J2−b)2]J2−J1∫10[h(a)H(a,1−a)+h(a)H(1−a,a)]da |
∀ b∈[J1,J2].
Proof. By Lemma 4.1, we have |F′| is (h1,h2)-convex stochastic process for IVFS, then
max{|F_(b,⋅)−1J2−J1∫J2J1F_(a,⋅)da|,|¯F(b,⋅)−1J2−J1∫J2j1¯F(a,⋅)da|}⊇(b−J1)2J2−J1∫10a|F′(ab+(1−a)J1,⋅)|ds+(J2−b)2J2−J1∫10a|F′(ab+(1−a)J2,⋅)|ds⊇(b−J1)2J2−J1∫10a[h1(a)h2(1−a)|F′(b,⋅)|+h1(1−a)h2(a)|F′(J1,⋅),⋅|]da+(J2−b)2J2−J1∫10a[h1(a)h2(1−a)|F′(b,⋅)|+h1(1−a)h2(a)|F′(J2,⋅),⋅|]da⊇γ(b−J1)2J2−J1∫10[h(a)h1(a)h2(1−a)+h(a)h1(1−a)h2(a)]da+(J2−b)2J2−J1∫10[h(a)h1(a)h2(1−a)+h(a)h1(1−a)h2(a)]da⊇γ[(b−J1)2+(J2−b)2]J2−J1∫10[h(a)h1(a)h2(1−a)+h(a)h1(1−a)h2(a)]da=γ[(b−J1)2+(J2−b)2]J2−J1∫10[h(a)H(a,1−a)+h(a)H(1−a,a)]da. |
The proof is completed.
Theorem 4.6. Let gi∈R+. If h1,h2 are non-negative and multiplicative functions where F:I×Δ→R+I is non-negative (h1,h2)-convex stochastic process for IVFS then this stochastic inclusion holds
F(1Gdd∑i=1giai,⋅)⊇d∑i=1H(giGd,Gd−1Gd)F(ai,⋅), | (4.9) |
where Gd=∑di=1gi.
Proof. When d=2, then (4.9) detain. Assume that (4.9) is also word for d−1, then
F(1Gdd∑i=1giai,⋅)=F(gdGdad+d−1∑i=1giGdai,⋅)⊇h1(gdGd)h2(Gd−1Gd)F(ad,⋅)+h1(Gd−1Gd)h2(gdGd)F(d−1∑i=1giGdai,⋅)⊇h1(gdGd)h2(Gd−1Gd)F(ad,⋅)+h1(Gd−1Gd)h2(gdGd)d−1∑i=1[H(giGd,Gd−2Gd−1)F(ai,⋅)]⊇h1(gdGd)h2(Gd−1Gd)F(ad,⋅)+d−1∑i=1H(giGd,Gd−2Gd−1)F(ai,⋅)⊇d∑i=1H(giGd,Gd−1Gd)F(ai,⋅). |
The conclusion is correct based on mathematical induction.
In this article, the results of the h-convex stochastic process are extended from pseudo order to a more generalized class (h1,h2)-convex stochastic process with set inclusions for interval-valued functions. By utilizing this, we were able to develop Hermite-Hadamard, Ostrowski, and Jensen inequalities. Furthermore our main findings are verified by providing some examples that are not trivial. The results of our research can be applied in a variety of situations to provide a variety of new as well as well-known outcomes. Further refinements and improvements to previously published findings are provided in this paper. A future development of this study will enable its use in a variety of modes, including time scale calculus, coordinates, interval analysis, fractional calculus, quantum calculus, etc. This paper's style and current developments should pique readers' interest and encourage further research.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R8). Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
The authors declare that they have no competing interests.
[1] |
V. Prajapati, R. Meher, A robust analytical approach to the generalized Burgers-Fisher equation with fractional derivatives including singular and non-singular kernels, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.06.035 doi: 10.1016/j.joes.2022.06.035
![]() |
[2] |
L. Verma, R. Meher, Z. Avazzadeh, O. Nikan, Solution for generalized fuzzy fractional Kortewege-de Varies equation using a robust fuzzy double parametric approach, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.04.026 doi: 10.1016/j.joes.2022.04.026
![]() |
[3] |
M. Alqhtani, K. Saad, W. Weera, W. Hamanah, Analysis of the fractional-order local Poisson equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
![]() |
[4] |
M. Al-Sawalha, R. Agarwal, R. Shah, O. Ababneh, W. Weera, A reliable way to deal with fractional-order equations that describe the unsteady flow of a polytropic gas, Mathematics, 10 (2022), 2293. https://doi.org/10.3390/math10132293 doi: 10.3390/math10132293
![]() |
[5] |
S. Mukhtar, S. Noor, The numerical investigation of a fractional-order multi-dimensional model of Navier-Stokes equation via novel techniques, Symmetry, 14 (2022), 1102. https://doi.org/10.3390/sym14061102 doi: 10.3390/sym14061102
![]() |
[6] |
N. Shah, Y. Hamed, K. Abualnaja, J. Chung, A. Khan, A comparative analysis of fractional-order Kaup-Kupershmidt equation within different operators, Symmetry, 14 (2022), 986. https://doi.org/10.3390/sym14050986 doi: 10.3390/sym14050986
![]() |
[7] |
N. Shah, H. Alyousef, S. El-Tantawy, J. Chung, Analytical investigation of fractional-order Korteweg-De-Vries-Type equations under Atangana-Baleanu-Caputo operator: Modeling nonlinear waves in a plasma and fluid, Symmetry, 14 (2022), 739. https://doi.org/10.3390/sym14040739 doi: 10.3390/sym14040739
![]() |
[8] |
N. Aljahdaly, A. Akgul, R. Shah, I. Mahariq, J. Kafle, A comparative analysis of the fractional-order coupled Korteweg-De Vries equations with the Mittag-Leffler law, J. Math., 2022 (2022), 1–30. https://doi.org/10.1155/2022/8876149 doi: 10.1155/2022/8876149
![]() |
[9] |
H. Ismael, H. Bulut, H. Baskonus, W-shaped surfaces to the nematic liquid crystals with three nonlinearity laws, Soft Comput., 25 (2020), 4513–4524. https://doi.org/10.1007/s00500-020-05459-6 doi: 10.1007/s00500-020-05459-6
![]() |
[10] |
M. Yavuz, T. Sulaiman, A. Yusuf, T. Abdeljawad, The Schrodinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel, Alex. Eng. J., 60 (2021), 2715–2724. https://doi.org/10.1016/j.aej.2021.01.009 doi: 10.1016/j.aej.2021.01.009
![]() |
[11] |
K. Safare, V. Betageri, D. Prakasha, P. Veeresha, S. Kumar, A mathematical analysis of ongoing outbreak COVID-19 in India through nonsingular derivative, Numer. Meth. Part. D. E., 37 (2020), 1282–1298. https://doi.org/10.1002/num.22579 doi: 10.1002/num.22579
![]() |
[12] |
H. Ismael, H. Baskonus, H. Bulut, Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model, Discrete Cont. Dyn.-S., 14 (2021), 2311. https://doi.org/10.3934/dcdss.2020398. doi: 10.3934/dcdss.2020398
![]() |
[13] |
H. Yasmin, N. Iqbal, A comparative study of the fractional coupled Burgers and Hirota-Satsuma KdV equations via analytical techniques, Symmetry, 14 (2022), 1364. https://doi.org/10.3390/sym14071364 doi: 10.3390/sym14071364
![]() |
[14] |
R. Alyusof, S. Alyusof, N. Iqbal, M. Arefin, Novel evaluation of the fractional acoustic wave model with the exponential-decay kernel, Complexity, 2022 (2022), 1–14. https://doi.org/10.1155/2022/9712388 doi: 10.1155/2022/9712388
![]() |
[15] |
M. Yavuz, T. Sulaiman, A. Yusuf, T. Abdeljawad, The Schrodinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel, Alex. Eng. J., 60 (2021), 2715–2724. https://doi.org/10.1016/j.aej.2021.01.009 doi: 10.1016/j.aej.2021.01.009
![]() |
[16] |
H. Triki, A. Biswas, Dark solitons for a generalized nonlinear Schrodinger equation with parabolic law and dual-power law nonlinearities, Math. Meth. Appl. Sci., 34 (2011), https://doi.org/10.1002/mma.1414 doi: 10.1002/mma.1414
![]() |
[17] |
L. Zhang, J. Si, New soliton and periodic solutions of (1+2)-dimensional nonlinear Schrodinger equation with dual-power law nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2747–2754. https://doi.org/10.1016/j.cnsns.2009.10.028 doi: 10.1016/j.cnsns.2009.10.028
![]() |
[18] |
Q. Xu, S. Songhe, Numerical analysis of two local conservative methods for two-dimensional nonlinear Schrodinger equation, SCIENTIA SINICA Math., 48 (2017), 345. https://doi.org/10.1360/scm-2016-0308 doi: 10.1360/scm-2016-0308
![]() |
[19] |
J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential, Phys. Rev. Lett., 86 (2001), 1402–1405. https://doi.org/10.1103/PhysRevLett.86.1402 doi: 10.1103/PhysRevLett.86.1402
![]() |
[20] |
A. Trombettoni, A. Smerzi, Discrete solitons and breathers with dilute Bose-Einstein condensates, Phys. Rev. Lett., 86 (2001), 2353–2356. https://doi.org/10.1103/physrevlett.86.2353 doi: 10.1103/physrevlett.86.2353
![]() |
[21] |
A. Biswas, Quasi-stationary non-Kerr law optical solitons, Opt. Fiber Technol., 9 (2003), 224–259. https://doi.org/10.1016/s1068-5200(03)00044-0 doi: 10.1016/s1068-5200(03)00044-0
![]() |
[22] |
M. Eslami, M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrödinger equation with dual-power law nonlinearity in nonlinear optical fibers, Eur. Phys. J. Plus, 128 (2013). https://doi.org/10.1140/epjp/i2013-13140-y doi: 10.1140/epjp/i2013-13140-y
![]() |
[23] |
A. Hyder, The influence of the differential conformable operators through modern exact solutions of the double Schrödinger-Boussinesq system, Phys. Scripta, 2021. https://doi.org/10.1088/1402-4896/ac169f doi: 10.1088/1402-4896/ac169f
![]() |
[24] |
G. Adomian, R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl., 91 (1983), 39–46. https://doi.org/10.1016/0022-247x(83)90090-2 doi: 10.1016/0022-247x(83)90090-2
![]() |
[25] |
S. Alinhac, M. Baouendi, A counterexample to strong uniqueness for partial differential equations of Schrodinger's type, Commun. Part. Diff. Eq., 19 (1994), 1727–1733. https://doi.org/10.1080/03605309408821069 doi: 10.1080/03605309408821069
![]() |
[26] |
J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178 (1999), 257–262. https://doi.org/10.1016/s0045-7825(99)00018-3 doi: 10.1016/s0045-7825(99)00018-3
![]() |
[27] |
J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., 35 (2000), 743. https://doi.org/10.1016/s0020-7462(98)00085-7 doi: 10.1016/s0020-7462(98)00085-7
![]() |
[28] |
D. D. Ganji, M. Rafei, Solitary wave solutions for a generalized Hirota Satsuma coupled KdV equation by homotopy perturbation method, Phys. Lett. A, 356 (2006), 131–137. https://doi.org/10.1016/j.physleta.2006.03.039 doi: 10.1016/j.physleta.2006.03.039
![]() |
[29] |
A. M. Siddiqui, R. Mahmood, Q. K. Ghori, Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A, 352 (2006), 404–410. https://doi.org/10.1016/j.physleta.2005.12.033 doi: 10.1016/j.physleta.2005.12.033
![]() |
[30] |
Y. Qin, A. Khan, I. Ali, M. Al Qurashi, H. Khan, R. Shah, D. Baleanu, An efficient analytical approach for the solution of certain fractional-order dynamical systems, Energies, 13 (2020), 2725. https://doi.org/10.3390/en13112725 doi: 10.3390/en13112725
![]() |
[31] |
M. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 1–21. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376
![]() |
[32] |
J. H. He, Y. O. El-Dib, Homotopy perturbation method for Fangzhu oscillator, J. Math. Chem., 58 (2020), 2245–2253. https://doi.org/10.1007/s10910-020-01167-6 doi: 10.1007/s10910-020-01167-6
![]() |
[33] |
J. Honggang, Z. Yanmin, Conformable double Laplace-Sumudu transform decomposition method for fractional partial differential equations, Complexity, 2022 (2022), 1–8. https://doi.org/10.1155/2022/7602254 doi: 10.1155/2022/7602254
![]() |
[34] |
L. Akinyemi, O. S. Iyiola, Exact and approximate solutions of time-fractional models arising from physics via Shehu transform, Math. Meth. Appl. Sci., 43 (2020), 7442–7464. https://doi.org/10.1002/mma.6484 doi: 10.1002/mma.6484
![]() |
[35] |
L. Ali, R. Shah, W. Weera, Fractional view analysis of Cahn-Allen equations by new iterative transform method, Fractal Fract., 6 (2022), 293. https://doi.org/10.3390/fractalfract6060293 doi: 10.3390/fractalfract6060293
![]() |
[36] |
S. Maitama, W. Zhao, Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences, J. Appl. Math. Comput. Mech., 20 (2021), 71–82. https://doi.org/10.17512/jamcm.2021.1.07 doi: 10.17512/jamcm.2021.1.07
![]() |
[37] |
M. Liaqat, A. Khan, M. Alam, M. Pandit, S. Etemad, S. Rezapour, Approximate and Closed-Form solutions of Newell-Whitehead-Segel eEquations via modified conformable Shehu transform decomposition method, Math. Probl. Eng., 2022 (2022), 1–14. https://doi.org/10.1155/2022/6752455 doi: 10.1155/2022/6752455
![]() |
1. | Fei Yu, Yiya Wu, Xuqi Wang, Ting He, ShanKou Zhang, Jie Jin, New discrete memristive hyperchaotic map: modeling, dynamic analysis, and application in image encryption, 2025, 13, 2296-424X, 10.3389/fphy.2025.1617964 |