The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.
Citation: Muhammad Akram, Syed Muhammad Umer Shah, Mohammed M. Ali Al-Shamiri, S. A. Edalatpanah. Fractional transportation problem under interval-valued Fermatean fuzzy sets[J]. AIMS Mathematics, 2022, 7(9): 17327-17348. doi: 10.3934/math.2022954
[1] | Raina Ahuja, Meraj Ali Khan, Parul Tomar, Amit Kumar, S. S. Appadoo, Ibrahim Al-Dayel . Modified methods to solve interval-valued Fermatean fuzzy multi-criteria decision-making problems. AIMS Mathematics, 2025, 10(4): 9150-9170. doi: 10.3934/math.2025421 |
[2] | Muhammad Akram, Syed Muhammad Umer Shah, Mohammed M. Ali Al-Shamiri, S. A. Edalatpanah . Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets. AIMS Mathematics, 2023, 8(1): 924-961. doi: 10.3934/math.2023045 |
[3] | Naeem Jan, Jeonghwan Gwak, Juhee Choi, Sung Woo Lee, Chul Su Kim . Transportation strategy decision-making process using interval-valued complex fuzzy soft information. AIMS Mathematics, 2023, 8(2): 3606-3633. doi: 10.3934/math.2023182 |
[4] | Warud Nakkhasen, Teerapan Jodnok, Ronnason Chinram . Intra-regular semihypergroups characterized by Fermatean fuzzy bi-hyperideals. AIMS Mathematics, 2024, 9(12): 35800-35822. doi: 10.3934/math.20241698 |
[5] | Murugan Palanikumar, Nasreen Kausar, Harish Garg, Shams Forruque Ahmed, Cuauhtemoc Samaniego . Robot sensors process based on generalized Fermatean normal different aggregation operators framework. AIMS Mathematics, 2023, 8(7): 16252-16277. doi: 10.3934/math.2023832 |
[6] | Iqra Nayab, Shahid Mubeen, Rana Safdar Ali, Faisal Zahoor, Muath Awadalla, Abd Elmotaleb A. M. A. Elamin . Novel fractional inequalities measured by Prabhakar fuzzy fractional operators pertaining to fuzzy convexities and preinvexities. AIMS Mathematics, 2024, 9(7): 17696-17715. doi: 10.3934/math.2024860 |
[7] | Muhammad Akram, Naila Ramzan, Anam Luqman, Gustavo Santos-García . An integrated MULTIMOORA method with 2-tuple linguistic Fermatean fuzzy sets: Urban quality of life selection application. AIMS Mathematics, 2023, 8(2): 2798-2828. doi: 10.3934/math.2023147 |
[8] | Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad, Mohammed S. Abdo, Kamal Shah, Thabet Abdeljawad . Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative. AIMS Mathematics, 2022, 7(9): 15994-16016. doi: 10.3934/math.2022876 |
[9] | Snezhana Hristova, Antonia Dobreva . Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay. AIMS Mathematics, 2020, 5(4): 3809-3824. doi: 10.3934/math.2020247 |
[10] | Ali Ahmad, Humera Rashid, Hamdan Alshehri, Muhammad Kamran Jamil, Haitham Assiri . Randić energies in decision making for human trafficking by interval-valued T-spherical fuzzy Hamacher graphs. AIMS Mathematics, 2025, 10(4): 9697-9747. doi: 10.3934/math.2025446 |
The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.
The wave equation with internal and boundary damping, along with a source term, is described by the system:
{ωtt−Δω+ψ1(ωt)=F1(ω)onΩ×R+,ω=0onΓ0×R+,∂ωt∂η+ψ2(ωt)=F2(ω)onΓ1×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. | (1.1) |
In this problem, the functions F1 and F2 are nonlinear source terms on the domain Ω⊆Rn and the boundary ∂Ω=Γ0∪Γ1, respectively, where Γ0 and Γ1 are closed and disjoint and meas.(Γ0)>0. The vector η is the unit outer normal to ∂Ω. The functions ω0 and ω1 are given data. The functions ψ1 is a nonlinear damping acting on the domain Ω, while ψ2 is a nonlinear damping acting on the boundary ∂Ω.
The study of the existence, blow-up, and stability of solutions to wave equations has been extensively explored in previous research. For example, Lasiecka and Tataru [1] studied the following semilinear model of the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback:
{ωtt=Δω−χ0(ω),inΩ×R+,∂ω∂ν=−˜χ(ωt|Γ1)−χ1(ω|Γ1),onΓ1×R+,ω=0,onΓ0×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. | (1.2) |
Assuming that the velocity boundary feedback is dissipative and the other nonlinear terms are conservative, uniform decay rates for the solutions are derived. Georgiev and Todorova [2] studied system (1.1) with ψ1(ωt)=|ωt|ϑ−2ωt, ψ2(ωt)=F2(ω)=0 and F1(ω)=|ω|q−2ω, proving global existence for q≤ϑ and a blow-up result when q>ϑ. Levine and Serrin [3] expanded on this by investigating the case of negative energy with ϑ>1. Rivera and Andrade [4] examined a nonlinear wave equation with viscoelastic boundary conditions, showing the existence and uniform decay under certain initial data restrictions. Santos [5] focused on a one-dimensional wave equation with viscoelastic boundary feedback, demonstrating that under specific assumptions on g′ and g′′, sufficient dissipation leads to exponential or polynomial decay if the relaxation function follows the same pattern. Vitillaro [6] explored system (1.1) with ψ1(ωt)=F1(ω)=0 and ψ2(ωt)=|ωt|ϑ−2ωt and F2(ω)=|ω|q−2u, establishing local and global existence under appropriate conditions on the initial data and exponents. Cavalcanti et al. [7] studied the following problem
{utt−Δω+∫t0g(t−s)Δω(s)ds=0,inΩ×R+,ω=0,onΓ1×R+,∂ωt∂ν−∫t0g(t−s)∂ωt∂ν(s)ds+ψ(ωt)=0,onΓ0×R+, | (1.3) |
where ψ:R→R is a nondecreasing C1 function such that
ψ(s)s>0,for alls≠0 |
and there exist Ci>0,i=1,2,3,4, such that
{C1|s|p≤|ψ(s)|≤C2|s|1p,if|s|≤1,C3|s|≤|ψ(s)|≤C4|s|,if|s|>1, | (1.4) |
where p≥1. They proved global existence of both strong and weak solutions, along with uniform decay rates, under restrictive conditions on the damping function ψ and the kernel g. After that, Cavalcanti et al. [8] relaxed these conditions on ψ and g, demonstrating uniform stability based on their behavior. Al-Gharabli et al. [9] extended this work by considering a large class of relaxation functions and establishing general and optimal decay results. Messaoudi and Mustafa [10] focused on system (1.3), exploring more general relaxation functions, and achieved a general decay result without assuming growth conditions on ψ, with the results depending on both g and ψ. Cavalcanti and Guesmia [11] analyzed the following hyperbolic problem involving memory terms
{ωtt−Δω+F(x,t,∇ω)=0,inΩ×R+,ω=0,onΓ0,ω+∫t0g(t−s)∂ω∂μ(s)ds=0,onΓ1×R+, | (1.5) |
showing that under certain conditions, the memory term dissipation is sufficient to ensure system stability. Specifically, they demonstrated that if the relaxation function decays exponentially or polynomially, the solution follows the same decay rate.
Liu and Yu [12] investigated the following viscoelastic equation with nonlinear boundary damping and source terms
{ωtt−Δω+∫t0g(t−s)Δω(s)ds=0,inΩ×R+,ω=0,onΓ1×R+,∂ω∂ν−∫t0g(t−s)∂ω∂ν(s)ds+|ωt|m−2=|ω|p−2ω,onΓ0×R+, | (1.6) |
proving global existence and general decay of energy under suitable assumptions on the relaxation function and the initial data. Al-Mahdi et al. [13] extended this work by considering system (1.1) with modified terms: F1(u)=0, F2(ω)=|ω|q(x)−2ω, ψ1(ωt) is replaced by ∫t0g(t−s)Δω(s)ds, ψ2(ωt) is replaced by ∫t0g(t−s)∂ω∂nds+|ωt|ϑ(x)−2ωt, proving global existence and establishing general and optimal decay estimates under specific conditions on the relaxation function and variable exponents ϑ(x) and q(x). They also provided numerical tests to validate their theoretical decay results.
Zhang and Huang [14] studied a nonlinear Kirchhoff equation described by the system:
{ωtt−M(||∇ω||2)Δω+αωt+χ(ω)=0onΩ×R+,ω=0onΓ1×R+,∂ω∂η+ψ(ωt)=0onΓ0×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ, | (1.7) |
where Ω is a bounded domain of Rn with a smooth boundary ∂Ω=Γ0∪Γ1, and α is a positive real constant. The functions M(s),χ(ω),ψ(ωt) are satisfy some conditions, while η represents the unit outward normal vector. Using the Galerkin approximation, Zhang and Huang established the global existence and uniqueness of the solution. They also addressed challenges posed by the nonlinear terms M(s) and ψ(ωt) through a transformation to zero initial data and employed compactness, monotonicity, and perturbed energy method to resolve the problem. Zhang and Ouyang [15] examined a viscoelastic wave equation with a memory term, nonlinear damping, and a source term:
{|ωt|ρωtt−Δω+α|ωt|p−2ωt+∫t0g(t−s)Δω(s)ds=|ω|q−2ωonΩ×R+,ω=0onΓ×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ, | (1.8) |
where Ω is a bounded domain of Rn with a smooth boundary ∂Ω, ρ,α>0, p≥2, q>2, and g(t) is a positive function that represents the kernel of the memory term. Using the potential well method combined with the Galerkin approximation, they demonstrated the existence of global weak solutions. Additionally, under certain conditions on the damping coefficient and the relaxation function, they established the optimal decay of solutions via the perturbed energy method. They further showed that the solution can blow up for both positive and negative initial energy conditions.
For further results on wave equations, see the works of Aassila [16], Wang and Chen [17], Zuazua [18], Soufyane et al. [19], Zhang et al. [20].
There has been increasing interest among researchers in replacing constant exponents with variable exponents, driven by their practical applications [21] and related references. Variable exponents are commonly used in mathematical models and equations, particularly in damping terms, to better represent a system's diverse behaviors or properties. Damping, which helps dissipate energy and regulate a system's response to external forces, can be more accurately modeled using variable exponents. This allows for a more flexible representation of damping effects tailored to the specific characteristics of the system in question.
Inspired by these studies and the significance of mathematical models involving nonlinear damping and/or source terms with variable exponents, we consider problem (1.1) with ψ1(ωt)=ψ(ωt), F1(ω)=0 and ψ2(ωt)=|ωt|ϑ(x)−2ωt, and F2(ω)=|ω|θ(x)−2ω.
More precisely, we consider the following nonlinear wave equation with internal and boundary damping, along with a source term of variable exponent type:
{ωtt−Δω+ψ(ωt)=0onΩ×R+,ω=0onΓ0×R+,∂ω∂η+|ωt|ϑ(x)−2ωt=|ω|θ(x)−2ωonΓ1×R+,ω(x,0)=ω0(x),ωt(x,0)=ω1(x)inΩ. | (1.9) |
We aim to study the global existence and stability of solutions to problem (1.9). We investigate the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. Additionally, we derive general decay rates, including optimal exponential and polynomial decay rates as the special cases.
This paper is organized into five sections. In Section 2, we introduce the notation and necessary background material. In Section 3, we prove the global existence of the solution to the problem. In Sections 4 and 5, we present technical lemmas and decay results, respectively.
In this section, we outline some necessary materials for proving our results. Throughout the paper, we denote a generic positive constant by c. We consider the following assumptions:
(A1) ϑ:Γ1→[1,∞) is a continuous function such that
1<ϑ1≤ϑ(x)≤ϑ2<q:={2(n−1)n−2,n>2;∞,n=1,2, |
where
ϑ1:=essinfx∈Γ1ϑ(x),ϑ2:=esssupx∈Γ1ϑ(x). |
(A2) θ:Γ1→[1,∞) is a continuous function such that
1<θ1≤θ(x)≤ϑ2<q:={2(n−1)n−2, n > 2 ;∞, n = 1, 2 , |
where
θ1:=essinfx∈Γ1θ(x),θ2:=esssupx∈Γ1θ(x). |
Moreover, the variable functions ϑ(x) and θ(x) satisfy the log-Hölder continuity condition.
For more details about the Lebesgue and Sobolev spaces with variable exponents (see [22,23,24]).
(A3) ψ:R→R is a C0 nondecreasing function satisfying, for c1,c2>0,
s2+ψ2(s)|≤Ψ−1(sψ(s))for all|s|≤r,c1|s|≤|ψ(s)|≤c2|s|for all|s|≥r, |
where Ψ:(0,∞)→(0,∞) is C1 function which is a linear or strictly increasing and strictly convex C2 function on (0,r] with Ψ(0)=Ψ′(0)=0.
Remark 2.1. Condition (A3) was introduced for the first time in 1993 by Lasiecka and Tataru [1]. Examples of such functions satisfying Condition (A3) are the following:
(1) If ψ(s)=csq and q≥1, then Ψ(s)=csq+12 satisfies (A3).
(2) If ψ(s)=e−1s, then (A3) is satisfied for Ψ(s)=√s2e−√2s near zero.
We define the energy functional E(t) associated to system (1.9) as follows:
E(t):=12[‖ωt‖22+‖∇ω‖22]−∫Γ11θ(x)|ω|θ(x)dx. | (2.1) |
Lemma 2.1. The energy functional E(t) satisfies
ddtE(t)=−∫Γ1|ωt|ϑ(x)dx−∫Ωωtψ(ωt)dx≤0. | (2.2) |
Proof. Multiplying (1.9)1 by ωt integrating over the interval Ω, we have
∫Ωωωtt−∫ΩωtΔωdx+∫Ωωtψ(ωt)dx=0. |
Using integration by parts, we obtain
∫Ωωωtt+∫Ω∇ωt.∇ωdx−∫Γ1ωt∂ωt∂ηdx+∫Ωωtψ(ωt)dx=0. |
Now, using (1.9)3, and doing some modifications, we get
ddt(12∫Ωω2tdx+12∫Ω|∇ω|2dx−∫Γ11θ(x)|ω|θ(x)dx)=−∫Γ1|ωt|ϑ(x)dx−∫Ωωtψ(ωt)dx, |
which gives (2.2).
For completeness, we present the following existence result, which can be established using the Faedo-Galerkin method and the Banach fixed point theorem, similar to the approaches taken in [2,25,26] for analogous problems.
Theorem 2.1. (Local existence) Given (ω0,ω1)∈H1Γ0(Ω)×L2(Ω) and assume that (A1)−(A3) hold. Then, there exists T>0, such that problem (1.9) has a weak solution
ω∈L∞((0,T),H1Γ0(Ω))∩Lθ(.)(Γ1×(0,T)),ωt∈L∞((0,T),L2(Ω))∩Lϑ(.)(Γ1×(0,T)). |
In this section, we state and prove a global existence result under smallness conditions on the initial data (ω0,ω1). For this purpose, we define the following functionals:
J(t)=12‖∇ω‖22−1θ1∫Γ1|ωt|θ(x)dx | (3.1) |
and
I(t)=I(ω(t))=‖∇ω‖22−∫Γ1|ωt|θ(x)dx. | (3.2) |
Clearly, we have
E(t)≥J(t)+12‖ωt‖22. | (3.3) |
Lemma 3.1. Suppose that (A1)−(A3) hold and (ω0,ω1)∈H1Γ0(Ω)×L2(Ω), such that
cθ2eEθ2−22(0)+cθ2eEθ1−22(0)<1,I(ω0)>0, | (3.4) |
then
I(ω(t))>0,∀t>0. |
Proof. Since I is continuous and I(ω0)>0, then there exists Tm<T such that
I(ω(t))≥0, ∀t∈[0,Tm]; |
which gives
J(t)=1θ1I(t)+θ1−22θ1‖∇ω‖22≥θ1−22θ1‖∇ω‖22. | (3.5) |
Now,
‖∇ω‖22≤2θ1θ1−2J(t)≤2θ1θ1−2E(t)≤2θ1θ1−2E(0). | (3.6) |
Using Young's and Poincaré's inequalities and the trace theorem, we get ∀t∈[0,Tm],
∫Γ1|ω|θ(x)dx=∫Γ+1|ω|θ(x)dx+∫Γ−1|ω|θ(x)dx≤∫Γ+1|ω|θ2dx+∫Γ−1|ω|θ1dx≤∫Γ1|ω|θ2dx+∫Γ1|ω|θ1dx≤cθ2e‖∇ω‖θ22+cθ1e‖∇ω‖θ12≤(cθ2e‖∇ω‖θ2−22+cθ1e‖∇ω‖θ1−22)‖∇ω‖22<‖∇ω‖22, | (3.7) |
where
Γ−1={x∈Γ1:|ω(x,t)|<1}andΓ+1={x∈Γ1:|ω(x,t)|≥1}. |
Therefore,
I(t)=‖∇ω‖22−∫Γ1|ω|θ(x)>0. |
Proposition 3.1. Suppose that (A1)−(A3) hold. Let (ω0,ω1)∈H1Γ0(Ω)×L2(Ω) be given, satisfying (3.4). Then, the solution of (1.9) is global and bounded.
Proof. It suffices to show that ‖∇ω‖22+‖ωt‖22 is bounded independently of t. To achieve this, we use (2.2), (3.2) and (3.5) to get
E(0)≥E(t)=J(t)+12‖ωt‖22≥θ1−22θ1‖∇ω‖22+12‖ωt‖22+1θ1I(t)≥θ1−22θ1‖∇ω‖22+12‖ωt‖22. | (3.8) |
Since I(t) is positive, Therefore
‖∇ω‖22+‖ωt‖22≤CE(0), |
where C is a positive constant, which depends only on θ1 and the proof is completed.
Remark 3.1. Using (3.6), we have
‖∇ω‖22≤2θ1θ1−2E(0). | (3.9) |
In this section, we present and prove several essential lemmas for demonstrating the main results.
Lemma 4.1. The functional defined by
Δ(t)=∫Ωωωtdx | (4.1) |
satisfies, along the solutions of (1.9),
Δ′(t)≤−12∫Ω|∇ω|2dx+∫Ω|ω|θ(x)dx+c∫Ωω2tdx+c∫Ωψ2(ωt)dx+c∫Γ1|ωt|ϑ(x)dΓ+c∫Γ∗|ωt|2ϑ(x)−2dΓ, | (4.2) |
where Γ∗={x∈Γ1:ϑ(x)<2}.
Proof.
Δ′(t)=∫Ωω2tdx+∫ΩωΔωdx−∫Ωωψ(ωt)=∫Ωω2tdx−∫Ω|∇ω|2dx+∫Γ1ω∂ω∂ηdΓ−∫Ωωψ(ωt)=∫Ωω2tdx−∫Ω|∇ω|2dx−∫Γ1ω|ωt|ϑ(x)−2ωtdΓ+∫Γ1ω|ω|θ(x)−2ωdΓ−∫Ωωψ(ωt). | (4.3) |
The use of Young's and Poincaré's inequalities and choosing ε1=14cp give
−∫Ωωψ(ωt)dx≤ε1∫Ωω2dx+14ε1∫Ωψ2(ωt)dx≤cpε1∫Ω|∇ω|2dx+14ε1∫Ωψ2(ωt)dx≤14∫Ω|∇ω|2dx+cp∫Ωψ2(ωt)dx. | (4.4) |
Define the following partition of Γ1:
Γ∗={x∈Γ1:ϑ(x)<2},Γ∗∗={x∈Γ1:ϑ(x)≥2}. |
Now, using Young's and Poincaré's inequalities, we obtain
∫Γ∗ω|ωt|ϑ(x)−2ωtdΓ≤λcp||∇ω||22+14λ∫Γ∗|ωt|2ϑ(x)−2dΓ, | (4.5) |
choosing λ=18cp, then we have
∫Γ∗ω|ωt|ϑ(x)−2ωtdΓ≤18||∇ω||22+c∫Γ∗|ωt|2ϑ(x)−2dΓ. | (4.6) |
Using Young's inequality with p(x)=ϑ(x)ϑ(x)−1 and p′(x)=ϑ(x) so, for all x∈Ω, we have
|ωt|ϑ(x)−2ωtω≤ε2|ωt|ϑ(x)+Cε2(x)|ωt|ϑ(x), |
where
Cε2(x)=ε1−ϑ(x)2(ϑ(x))−ϑ(x)(ϑ(x)−1)ϑ(x)−1. |
Hence, Young's inequality gives
∫Γ∗∗ω|ωt|ϑ(x)−2ωtdΓ≤ε2∫Γ∗∗|ω|ϑ(x)dΓ+∫Γ∗∗Cε2(x)|ωt|ϑ(x)dΓ≤cε2(1+(2θ1θ1−2E(0))ϑ2−22)||∇ω||22+∫Γ∗∗Cε2(x)|ωt|ϑ(x)dΓ. | (4.7) |
Choosing ε2=18c(1+(2θ1θ1−2E(0))ϑ2−22), then Cε2(x) is bounded and noting that Γ∗∗⊂Γ1, then we have
∫Γ∗∗ω|ωt|ϑ(x)−2ωtdΓ≤18||∇ω||22+c∫Γ1|ωt|ϑ(x)dΓ. | (4.8) |
By combining the above estimates, the proof is completed.
Lemma 4.2. Let us introduce perturbed energy functional as follows:
M(t)=NE(t)+Δ(t) |
satisfies, for all t≥0 and for a positive constant N,
M′(t)≤−cE(t)−cE′(t)+c∫Ω(ω2t+ψ2(ωt))dx+c∫Γ∗|ωt|2ϑ(x)−2dΓ. | (4.9) |
Proof. We establish the proof by means of perturbed energy method. Taking the derivative of M with respect to t, and using the estimates in (4.2), and (2.2), we obtain
M′(t)≤−N∫Γ1|ωt|ϑ(x)dx−N∫Ωωtψ(ωt)dx−12∫Ω|∇ω|2dx+∫Ω|ω|θ(x)dx+c∫Ωω2tdx+c∫Ωψ2(ωt)dx+c∫Γ1|ωt|ϑ(x)dΓ+c∫Γ∗|ωt|2ϑ(x)−2dΓ. | (4.10) |
Choosing N large enough such that M∼E, and recalling (2.2), therefore the proof of (4.9) is completed.
Lemma 4.3. If 1<ϑ1<2, then the following estimate holds:
∫Γ∗|ωt|2ϑ(x)−2dΓ≤cE(t)−cE′(t)(E(t))2−ϑ12ϑ1−2−cE′(t). | (4.11) |
Proof. First, we define the following partition:
Γ∗1={x∈Γ∗:|ωt(t)|≤1},Γ∗2={x∈Γ∗:|ωt(t)|>1}, |
and use the fact that 2ϑ(x)−2ϑ(x)≥2ϑ1−2ϑ1, and Jensen's inequality to obtain
∫Γ∗|ωt|2ϑ(x)−2dΓ=∫Γ∗1|ωt|2ϑ(x)−2dΓ+∫Γ∗2|ωt|2ϑ(x)−2dΓ=∫Γ∗1[|ωt|ϑ(x)]2ϑ(x)−2ϑ(x)dΓ+∫Γ∗2|ωt|ϑ(x)+ϑ(x)−2dΓ≤c∫Γ∗1[|ωt|ϑ(x)]2ϑ1−2ϑ1dΓ+c∫Γ∗2|ωt|ϑ(x)dΓ≤c[−E′(t)]2ϑ1−2ϑ1−cE′(t). | (4.12) |
Using Young's inequality, we find that
[−E′(t)]2ϑ1−2ϑ1=(E(t))2−ϑ12ϑ1−2[−E′(t)]2ϑ1−2ϑ1(E(t))2−ϑ12ϑ1−2≤ε(E(t))ϑ12ϑ1−2−CεE′(t)(E(t))2−ϑ12ϑ1−2=εE(t)−CεE′(t)(E(t))2−ϑ12ϑ1−2. | (4.13) |
Choosing ε small enough, the proof of (4.11) is completed.
Remark 4.1. If ϑ1≥2 and since meas(Γ∗)=0 then
∫Γ∗|ωt|2ϑ(x)−2dΓ=0. | (4.14) |
Lemma 4.4. Under assumption (A3), the following estimates hold:
∫Ωωtψ(ωt)dx≤−cE′(t),if ψ is linear, | (4.15) |
∫Ωωtψ(ωt)dx≤cΨ−1(Λ(t))−cE′(t),if ψ is nonlinear, | (4.16) |
where Λ(t) is defined in the proof.
Proof. Case 1: ψ is linear, then
c∫Ω(ω2t+ψ2(ωt))dx≤−cE′(t). |
Case 2: ψ is nonlinear, we define the following partition of Ω
Ω1={x∈Ω:|ωt|≤r},Ω2={x∈Ω:|ωt|≥r}, |
where r is small enough such that
sψ(s)≤min{r,ψ(r)},|s|≤r. |
We also define
Λ(t)= ∫Ω1ωtψ(ωt)dx. |
Now, using hypothesis (A3) and Jensen's inequality, we get
∫Ω1(ω2t+ψ2(ωt))dx≤∫Ω1Ψ−1(ωtψ(ωt))dx≤cΨ−1(Λ). | (4.17) |
In this section, we state and prove the stability result of system (1.9).
Theorem 5.1. Assume that ϑ1≥2 and ψ is linear. Then
E(t)≤κ1e−κ2t, | (5.1) |
for some positive constants κ1 and κ2.
Proof. Combining (4.9), (4.15) with (4.14), we obtain,
M′(t)≤−cE(t)−cE′(t). |
Therefore, M+cE∼E and a simple integration over (0,t) yields, for some κ1,κ2>0,
E(t)≤κ1e−κ2t,t≥0. |
Theorem 5.2. Assume that 1<ϑ1<2 and ψ is linear. Then
E(t)≤c(1+t)−1α, | (5.2) |
where α=2−ϑ12ϑ1−2>0.
Proof. From (4.9), (4.11) and (4.15), we have
M′1(t)≤−cE(t)−cE′(t)(E(t))2−ϑ12ϑ1−2, | (5.3) |
where M1=M+cE∼E. Multiply both sides of (5.13) by (E(t))α where α=2−ϑ12ϑ1−2, to obtain
M′2(t)≤−cEα+1(t), | (5.4) |
where M2=(E(t))αM1+cE∼E. Integrating over (0,t) and using the equivalence relation lead to (5.2).
Theorem 5.3. Assume that ϑ1≥2 and ψ is nonlinear. Then, for some positive constants ϱ1 and ϱ2, we have
E(t)≤Ψ−11(ϱ1t+ϱ2),∀t≥0, | (5.5) |
where Ψ1(t)=∫1t1Ψ2(s)ds and Ψ2(t)=tΨ′(ε0t)
Proof. From (4.9), (4.1) and (4.15), we have
M′(t)≤−cE(t)+cΨ−1(Λ(t)). | (5.6) |
Now, for ε0<r, using the fact that E′≤0, Ψ′>0,Ψ′′>0 on (0,r], we find that the functional ˜M, by
˜M(t):=Ψ′(ε0E(t)E(0))M(t)+c0E(t), |
satisfies, for some α1,α1>0,
α1˜M(t)≤E(t)≤α2˜M(t), | (5.7) |
and
˜M′(t)=ε0E′(t)E(0)Ψ′′(ε0E(t)E(0))M(t)+Ψ′(ε0E(t)E(0))M′(t)+c0E′(t)≤−cE(t)Ψ′(ε0E(t)E(0))+cΨ′(ε0E(t)E(0))Ψ−1(Λ(t))+c0E′(t). | (5.8) |
Let Ψ∗ be the convex conjugate of Ψ in the sense of Young with A=Ψ′(ε0E(t)E(0)) and B=Ψ−1(Λ(t)), we arrive at
˜M′(t)≤−cE(t)Ψ′(ε0E(t)E(0))+cΨ∗(Ψ′(ε0E(t)E(0)))+cΛ(t)+c0E′(t)≤−cE(t)Ψ′(ε0E(t)E(0))+cε0E(t)E(0)Ψ′(ε0E(t)E(0))−cE′(t)+c0E′(t). |
Consequently, with a suitable choice of ε0 and c0, we obtain, for all t≥0,
˜M′(t)≤−cE′(t)E(0)Ψ′(ε0E(t)E(0))=−cΨ2(ε0E(t)E(0)), | (5.9) |
where Ψ2(t)=tΨ′(ε0t). Since Ψ′2(t)=Ψ′(ε0t)+ε0tΨ′′(ε0t), then, using the strict convexity of Ψ on (0,r], we find that Ψ′2(t),Ψ2(t)>0 on (0,1]. Thus, with
Φ(t)=εα1˜M(t)E(0),0<ε<1, |
taking in account (5.7) and (5.9), we have
Φ(t)∼E(t), | (5.10) |
and then
Φ′(t)≤−cΨ2(Φ(t)),∀t≥0. |
Then, a simple integration gives, for some ϱ1,ϱ2>0,
Φ(t)≤Ψ−11(ϱ1t+ϱ2),∀t≥0, | (5.11) |
where Ψ1(t)=∫1t1Ψ2(s)ds. A combination of (5.10) and (5.11) gives (5.5).
Theorem 5.4. Assume that 1<ϑ1<2 and ψ is nonlinear. Then, for some positive constants ϱ3 and ϱ4, we have
E(t)≤χ−11(ϱ3t+ϱ4),∀t≥0, | (5.12) |
where χ1(t)=∫1t1Ψ2(s)ds, χ2(t)=tχ′(ε0t), χ=(G−1+Ψ−1)−1 and G(t)=tϑ12ϑ1−2.
Proof. From (4.9) and (4.13), we have
M′(t)≤−cE(t)+(−E′(t))2ϑ1−2ϑ1+cΨ−1(Λ)(t), | (5.13) |
where M=EαM+cE∼E. Let G(t)=tϑ12ϑ1−2. Then the last inequality can be written as
M′(t)≤−cE(t)+G−1(−E′(t))+cΨ−1(Λ)(t). | (5.14) |
Therefore, (5.14) becomes
M′(t)≤−cE(t)+cχ−1(ξ(t)), | (5.15) |
where χ=(G−1+Ψ−1)−1 and ξ(t)=max{−E′(t),Λ(t)}. Define the following functional
K(t):=χ′(ε0E(t)E(0))M(t)+c0E(t), | (5.16) |
satisfies, for some α2,α3>0,
α2K(t)≤E(t)≤α3K(t). | (5.17) |
Combining (5.15) and (5.16), we obtain
K′(t)≤−cE(t)χ′(ε0E(t)E(0))+χ′(ε0E(t)E(0))χ−1(ξ(t))+c0E′(t). | (5.18) |
Let χ∗ be the convex conjugate of χ in the sense of Young, then
χ∗(s)=s(χ′)−1(s)−χ[(χ′)−1(s)],ifs∈(0,χ′(r)] | (5.19) |
and χ∗ satisfies the following generalized Young inequality
AB≤χ∗(A)+χ(B),ifA∈(0,χ′(r)],B∈(0,r]. | (5.20) |
Thus, with A=χ′(ε0E(t)E(0)) and B=χ−1(ξ(t)), we arrive at
K′(t)≤−cE(t)χ′(ε0E(t)E(0))+cε0E(t)E(0)χ′(ε0E(t)E(0))−cE′(t)+c0E′(t). |
Choosing c0,ε0 small enough, we get
K′(t)≤−cε0E(t)E(0)χ′(ε0E(t)E(0))=−cχ2(ε0E(t)E(0)), |
where χ2(t)=tχ′(ε0t). Letting
Y(t)=εα3K(t)E(0),0<ε<1, |
and taking in account (5.7) and (5.9), we have
Y(t)∼E(t), | (5.21) |
and then
Y′(t)≤−cχ2(Y(t)),∀t≥0. |
Then, a simple integration gives, for some ϱ3,ϱ4>0,
Y(t)≤χ−11(ϱ3t+ϱ4),∀t≥0, | (5.22) |
where χ1(t)=∫1t1χ2(s)ds, which finishes the proof.
Examples 5.1. The following examples illustrate our results:
(1) If ψ(t)=ct and ϑ(x)=2, then
E(t)≤c1e−c2t, | (5.23) |
which is an exponential decay.
(2) If ψ(t)=ct and ϑ(x)=2−34+x, then ϑ1=54 and ϑ2=75, then the energy functional satisfies
E(t)≤c(1+t)−23. | (5.24) |
(3) If ψ(t)=ct2 and ϑ(x)=2+11+x, then ϑ1=52, ϑ2=3 and ψ(t)=ct32. Then,
Ψ1−1(t)=(ct+1)−2. |
Therefore, we obtain
E(t)≤c(1+t)−2. | (5.25) |
(4) If ψ(t)=ct5 and ϑ(x)=2−34+x, then ϑ1=54, ϑ2=75 and ψ(t)=ct3. Then,
χ(s)=(G−1+Ψ−1)−1=(−1+√1+4s2)3 |
and
χ2(s)=3s√1+4s(−1+√1+4s2)2=3s2√1+4s+3s2√1+4s−3s2≤3s2+3s22√s−3s2=cs32. |
Therefore, we obtain
E(t)≤c(1+t)13. |
In this work, we consider a nonlinear wave equation with internal and boundary damping and a source term of variable exponent type. We prove the global existence and stability of solutions to this problem problem. We study the interaction between the internal nonlinear frictional damping and the nonlinear boundary damping of variable exponent type. In addition, we establish general decay rates, including optimal exponential and polynomial decay rates as the special cases.
Adel M. Al-Mahdi: Conceptualization, methodology, formal analysis, writing-original draft; Mohammad M. Al-Gharabli: Formal analysis, validation, writing-reviewing and editing; Mohammad Kafini: Conceptualization, methodology, formal analysis, reviewing. All authors have read and approved the final version of the manuscript for publication.
The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project (No. INCB2402), is also greatly acknowledged.
This work is funded by KFUPM, Grant No. INCB2402.
The authors declare no competing interests.
[1] |
L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
![]() |
[2] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
![]() |
[3] |
R. R. Yager, Pythagorean membership grades in multi-criteria decision making, IEEE T. Fuzzy Syst., 22 (2014), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
![]() |
[4] | R. R. Yager, Pythagorean fuzzy subsets, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 |
[5] | X. Han, P. Rani, Evaluate the barriers of blockchain technology adoption in sustainable supply chain management in the manufacturing sector using a novel Pythagorean fuzzy-CRITIC-CoCoSo approach, Oper. Manag. Res., 2022. https://doi.org/10.1007/s12063-021-00245-5 |
[6] |
T. Senapati, R. R. Yager, Fermatean fuzzy sets, J. Ambient Intell. Human. Comput., 11 (2020), 663–674. https://doi.org/10.1007/s12652-019-01377-0 doi: 10.1007/s12652-019-01377-0
![]() |
[7] |
T. Senapati, R. R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision making methods, Eng. Appl. Artif. Intel., 85 (2019), 112–121. https://doi.org/10.1016/j.engappai.2019.05.012 doi: 10.1016/j.engappai.2019.05.012
![]() |
[8] |
T. Senapati, R. R. Yager, Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making, Informatica, 30 (2019), 391–412. https://doi.org/10.15388/Informatica.2019.211 doi: 10.15388/Informatica.2019.211
![]() |
[9] | M. Akram, N. Ramzan, F. Feng, Extending COPRAS method with linguistic Fermatean fuzzy sets and hamy mean operators, J. Math., 2022 (2022), 8239263. https://doi.org/10.1155/2022/8239263 |
[10] |
M. Akram, R. Bibi M. A. Al-Shamiri, A decision-making framework based on 2-tuple linguistic Fermatean fuzzy hamy mean operators, Math. Probl. Eng., 2022 (2022), 1501880. https://doi.org/10.1155/2022/1501880 doi: 10.1155/2022/1501880
![]() |
[11] |
M. Akram, G. Muhiuddin, G. Santos-Garcia, An enhanced VIKOR method for multi-criteria group decision-making with complex Fermatean fuzzy sets, Math. Biosci. Eng., 19 (2022), 7201–7231. https://doi.org/10.3934/mbe.2022340 doi: 10.3934/mbe.2022340
![]() |
[12] | M. Akram, G. Shahzadi, B. Davvaz, Decision-making model for internet finance soft power and sportswear brands based on sine-trigonometric Fermatean fuzzy information, Soft Comput., 2022. https://doi.org/10.1007/s00500-022-07060-5 |
[13] |
S. Jeevaraj, Ordering of interval-valued Fermatean fuzzy sets and its applications, Expert Syst. Appl., 185 (2021), 115613. https://doi.org/10.1016/j.eswa.2021.115613 doi: 10.1016/j.eswa.2021.115613
![]() |
[14] |
P. Rani, A. R. Mishra, Interval-valued fermatean fuzzy sets with multi-criteria weighted aggregated sum product assessment-based decision analysis framework, Neural Comput. Appl., 34 (2022), 8051–8067. https://doi.org/10.1007/s00521-021-06782-1 doi: 10.1007/s00521-021-06782-1
![]() |
[15] |
D. Sergi, I. U. Sari, T. Senapati, Extension of capital budgeting techniques using interval-valued Fermatean fuzzy sets, J. Intell. Fuzzy Syst., 42 (2022), 365–376, https://doi.org/10.3233/jifs-219196 doi: 10.3233/jifs-219196
![]() |
[16] |
R. E. Bellman, L. A. Zadeh, Decision making in a fuzzy environment, Manage. Sci., 17 (1970), 141–164. https://doi.org/10.1287/mnsc.17.4.B141 doi: 10.1287/mnsc.17.4.B141
![]() |
[17] |
H. J. Zimmerman, Fuzzy programming and linear programming with several objective functions, Fuzzy Set. Syst., 1 (1978), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3 doi: 10.1016/0165-0114(78)90031-3
![]() |
[18] | S. K. Mahato, L. Sahoo, A. K. Bhunia, Effects of defuzzification methods in redundancy allocation problem with fuzzy valued reliabilities via genetic algorithm, Int. J. Inform. Comput. Secur., 2 (2013), 106–115. |
[19] | L. Sahoo, Effect of defuzzification methods in solving fuzzy matrix games, J. New Theory, 8 (2015), 51–64. |
[20] |
A. A. H. Ahmadini, F. Ahmad, Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophic environment, AIMS Mathematics, 6 (2021), 4556–4580, https://doi.org/10.3934/math.2021269 doi: 10.3934/math.2021269
![]() |
[21] |
J. Ahmed, M. G. Alharbi, M. Akram, S. Bashir, A new method to evaluate linear programming problem in bipolar single-valued neutrosophic environment, Comput. Model. Eng. Sci., 129 (2021), 881–906. https://doi.org/10.32604/cmes.2021.017222 doi: 10.32604/cmes.2021.017222
![]() |
[22] |
M. Akram, I. Ullah, T. Allahviranloo, S. A. Edalatpanah, Fully Pythagorean fuzzy linear programming problems with equality constraints, Comput. Appl. Math., 40 (2021), 120. https://doi.org/10.1007/s40314-021-01503-9 doi: 10.1007/s40314-021-01503-9
![]() |
[23] |
M. Akram, I. Ullah, T. Allahviranloo, S. A. Edalatpanah, LR-type fully Pythagorean fuzzy linear programming problems with equality constraints, J. Intell. Fuzzy Syst., 41 (2021), 1975–1992. https://doi.org/ 10.3233/JIFS-210655 doi: 10.3233/JIFS-210655
![]() |
[24] |
M. Akram, G. Shahzadi, A. A. H. Ahmadini, Decision-making framework for an effective sanitizer to reduce COVID-19 under Fermatean fuzzy environment, J. Math., 2020 (2020), 3263407. https://doi.org/10.1155/2020/3263407 doi: 10.1155/2020/3263407
![]() |
[25] |
M. Akram, I. Ullah, M. G. Alharbi, Methods for solving LR-type Pythagorean fuzzy linear programming problems with mixed constraints, Math. Probl. Eng., 2021 (2021), 4306058. https://doi.org/10.1155/2021/4306058 doi: 10.1155/2021/4306058
![]() |
[26] |
M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Solution of fully bipolar fuzzy linear programming models, Math. Probl. Eng., 2021 (2021), 9961891. https://doi.org/10.1155/2021/9961891 doi: 10.1155/2021/9961891
![]() |
[27] |
M. A. Mehmood, M. Akram, M. G. Alharbi, S. Bashir, Optimization of LR-type fully bipolar fuzzy linear programming problems, Math. Probl. Eng., 2021 (2021), 1199336. https://doi.org/10.1155/2021/1199336 doi: 10.1155/2021/1199336
![]() |
[28] |
F. L. Hitchcock, The distribution of product from several resources to numerous localities, J. Math. Phys., 20 (1941), 224–230. https://doi.org/10.1002/sapm1941201224 doi: 10.1002/sapm1941201224
![]() |
[29] |
A. Charnes, W. W. Cooper, Programming with linear fractional functionals, Naval Res. Logist. Q., 9 (1962), 181–186. https://doi.org/10.1002/nav.3800090303 doi: 10.1002/nav.3800090303
![]() |
[30] |
P. Pandey, A. P. Punnen, A simplex algorithm for piecewise-linear fractional programming problems, Eur. J. Oper. Res., 178 (2007), 343–358. https://doi.org/10.1016/j.ejor.2006.02.021 doi: 10.1016/j.ejor.2006.02.021
![]() |
[31] | K. Swarup, Transportation technique in linear fractional functional programming, J. Royal Naval Sci. Serv., 21 (1966), 256–260. |
[32] |
A. Gupta, S. Khanna, M. C. Puri, A paradox in linear fractional transportation problems with mixed constraints, Optimization, 27 (1993), 375–387, https://doi.org/10.1080/02331939308843896 doi: 10.1080/02331939308843896
![]() |
[33] | D. Monta, Some aspects on solving a linear fractional transportation problem, JAQM, 2 (2007), 343–348. |
[34] | M. Sivri, I. Emiroglu, C. Guler, F. Tasci, A solution proposal to the transportation problem with the linear fractional objective function, 2011 Fourth International Conference on Modeling, Simulation and Applied Optimization, 2011, 1–9. https://doi.org/10.1109/ICMSAO.2011.5775530 |
[35] | V. D. Joshi, N. Gupta, Linear fractional transportation problem with varying demand and supply, Le Mat., 66 (2011), 3–12. |
[36] | A. Kumar, A. Kaur, Application of classical transportation methods to find the fuzzy optimal solution of fuzzy transportation problems, Fuzzy Inform. Eng., 3 (2011), 81–99. https://doi.org/10.1007/s12543-011-0068-7 |
[37] |
B. Kaushal, R. Arora, S. Arora, An aspect of bilevel fixed charge fractional transportation problem, Int. J. Appl. Comput. Math., 6 (2020), 14. https://doi.org/10.1007/s40819-019-0755-3 doi: 10.1007/s40819-019-0755-3
![]() |
[38] | N. Guzel, Y. Emiroglu, F. Tapci, C. Guler, M. Syvry, A solution proposal to the interval fractional transportation problem, Appl. Math. Inf. Sci., 6 (2012), 567–571. |
[39] |
A. Ebrahimnejad, An improved approach for solving fuzzy transportation problem with triangular fuzzy numbers, J. Intell. Fuzzy Syst., 29 (2015), 963–974. https://doi.org/ 10.3233/IFS-151625 doi: 10.3233/IFS-151625
![]() |
[40] |
S. T. Liu, Fractional transportation problem with fuzzy parameters, Soft Comput., 20 (2016), 3629–3636. https://doi.org/ 10.1007/s00500-015-1722-5 doi: 10.1007/s00500-015-1722-5
![]() |
[41] |
A. Ebrahimnejad, New method for solving fuzzy transportation problems with LR flat fuzzy numbers, Inform. Sci., 357 (2016), 108–124. https://doi.org/10.1016/j.ins.2016.04.008 doi: 10.1016/j.ins.2016.04.008
![]() |
[42] | S. Mohanaselvi, K. Ganesan, A new approach for solving linear fuzzy fractinal transportational problem, Int. J. Civil Eng. Technol., 8 (2017), 1123–1129. |
[43] |
M. R. Safi, S. M. Ghasemi, Uncertainty in linear fractional transportation problem, Int. J. Nonlinear Anal. Appl., 8 (2017), 81–93. http://doi.org/10.22075/ijnaa.2016.504 doi: 10.22075/ijnaa.2016.504
![]() |
[44] |
S. K. Bharati, S. R. Singh, Transportation problem under interval-valued intuitionistic fuzzy environment, Int. J. Fuzzy Syst., 20 (2018), 1511–1522. https://doi.org/10.1007/s40815-018-0470-y doi: 10.1007/s40815-018-0470-y
![]() |
[45] |
A. Mahmoodirad, T. Allahviranloo, S. Niroomand, A new efective solution method for fully fuzzy transportation problem, Soft Comput., 23 (2019), 4521–4530. https://doi.org/10.1007/s00500-018-3115-z doi: 10.1007/s00500-018-3115-z
![]() |
[46] |
R. Kumar, S. A. Edalatpanah, S. Jha, R. Singh, A Pythagorean fuzzy approach to the transportation problem, Complex Intell. Syst., 5 (2019), 255–263. https://doi.org/10.1007/s40747-019-0108-1 doi: 10.1007/s40747-019-0108-1
![]() |
[47] | S. K. Bharati, Trapezoidal intuitionistic fuzzy fractional transportation problem, In: Soft computing for problem solving, Singapore: Springer, 2019,833–842. https://doi.org/10.1007/978-981-13-1595-4-66 |
[48] |
S. K. Bharati, Transportation problem with interval-valued intuitionistic fuzzy sets: Impact of a new ranking, Prog. Artif. Intell., 10 (2021), 129–145. https://doi.org/10.1007/s13748-020-00228-w doi: 10.1007/s13748-020-00228-w
![]() |
[49] |
J. Pratihar, R. Kumar, S. A. Edalatpanah, A. Dey, Modified Vogel's approximation method for transportation problem under uncertain environment, Complex Intell. Syst., 7 (2021), 29–40. https://doi.org/10.1007/s40747-020-00153-4 doi: 10.1007/s40747-020-00153-4
![]() |
[50] |
C. Veeramani, S. A. Edalatpanah, S. Sharanya, Solving the multi-objective fractional transportation problem through the neutrosophic goal programming approach, Discrete Dyn. Nat. Soc., 2021 (2021), 7308042. https://doi.org/10.1155/2021/7308042 doi: 10.1155/2021/7308042
![]() |
[51] |
L. Sahoo, A new score function based Fermatean fuzzy transportation problem, Res. Control Optim., 4 (2021), 100040. https://doi.org/10.1016/j.rico.2021.100040 doi: 10.1016/j.rico.2021.100040
![]() |
[52] |
M. K. Sharma, Kamini, N. Dhiman, V. N. Mishra, H. G. Rosales, A. Dhaka, et al., A fuzzy optimization technique for multi-objective aspirational level fractional transportation problem, Symmetry, 13 (2021), 1465. https://doi.org/10.3390/sym13081465 doi: 10.3390/sym13081465
![]() |
[53] |
M. A. El Sayed, M. A. Abo-Sinna, A novel approach for fully intuitionistic fuzzy multi-objective fractional transportation problem, Alex. Eng. J., 60 (2021), 1447–1463. https://doi.org/10.1016/j.aej.2020.10.063 doi: 10.1016/j.aej.2020.10.063
![]() |
[54] |
S. A. Bas, H. G. Kocken, B. A. Ozkok, A novel iterative method to solve a linear fractional transportation problem, Pak. J. Stat. Oper. Res., 18 (2022), 151–166. https://doi.org/10.18187/pjsor.v18i1.3889 doi: 10.18187/pjsor.v18i1.3889
![]() |
[55] |
A. Singh, R. Arora, S. Arora, Bilevel transportation problem in neutrosophic environment, Comput. Appl. Math., 41 (2022), 44. https://doi.org/10.1007/s40314-021-01711-3 doi: 10.1007/s40314-021-01711-3
![]() |
[56] | E. B. Bajalinov, Linear fractional programming theory methods applications and software, New York: Springer, 2003. https://doi.org/10.1007/978-1-4419-9174-4 |
1. | Muhammad Akram, Syed Muhammad Umer Shah, Mohammed M. Ali Al-Shamiri, S. A. Edalatpanah, Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets, 2023, 8, 2473-6988, 924, 10.3934/math.2023045 | |
2. | Gulfam Shahzadi, Anam Luqman, Mohammed M. Ali Al-Shamiri, Dragan Pamučar, The Extended MOORA Method Based on Fermatean Fuzzy Information, 2022, 2022, 1563-5147, 1, 10.1155/2022/7595872 | |
3. | Anam Luqman, Gulfam Shahzadi, Multi-attribute decision-making for electronic waste recycling using interval-valued Fermatean fuzzy Hamacher aggregation operators, 2023, 2364-4966, 10.1007/s41066-023-00363-4 | |
4. | Ruijuan Geng, Ying Ji, Shaojian Qu, Zheng Wang, Data-driven product ranking: A hybrid ranking approach, 2023, 10641246, 1, 10.3233/JIFS-223095 | |
5. | Muhammad Akram, Syed Muhammad Umer Shah, Tofigh Allahviranloo, A new method to determine the Fermatean fuzzy optimal solution of transportation problems, 2023, 44, 10641246, 309, 10.3233/JIFS-221959 | |
6. | Chuan-Yang Ruan, Xiang-Jing Chen, Li-Na Han, Fermatean Hesitant Fuzzy Prioritized Heronian Mean Operator and Its Application in Multi-Attribute Decision Making, 2023, 75, 1546-2226, 3203, 10.32604/cmc.2023.035480 | |
7. | Arunodaya Raj Mishra, Pratibha Rani, Muhammet Deveci, Ilgin Gokasar, Dragan Pamucar, Kannan Govindan, Interval-valued Fermatean fuzzy heronian mean operator-based decision-making method for urban climate change policy for transportation activities, 2023, 124, 09521976, 106603, 10.1016/j.engappai.2023.106603 | |
8. | Fethullah Göçer, A Novel Extension of Fermatean Fuzzy Sets into Group Decision Making: A Study for Prioritization of Renewable Energy Technologies, 2024, 49, 2193-567X, 4209, 10.1007/s13369-023-08307-5 | |
9. | Gülçin Büyüközkan, Deniz Uztürk, Öykü Ilıcak, Fermatean fuzzy sets and its extensions: a systematic literature review, 2024, 57, 1573-7462, 10.1007/s10462-024-10761-y | |
10. | Muhammad Akram, Sundas Shahzadi, Syed Muhammad Umer Shah, Tofigh Allahviranloo, An extended multi-objective transportation model based on Fermatean fuzzy sets, 2023, 1432-7643, 10.1007/s00500-023-08117-9 | |
11. | Daud Ahmad, Kiran Naz, Mariyam Ehsan Buttar, Pompei C. Darab, Mohammed Sallah, Extremal Solutions for Surface Energy Minimization: Bicubically Blended Coons Patches, 2023, 15, 2073-8994, 1237, 10.3390/sym15061237 | |
12. | Ritu Arora, Chandra K. Jaggi, An aspect of bilevel interval linear fractional transportation problem with disparate flows: a fuzzy programming approach, 2023, 14, 0975-6809, 2276, 10.1007/s13198-023-02069-x | |
13. | Nilima Akhtar, Sahidul Islam, Linear fractional transportation problem in bipolar fuzzy environment, 2024, 17, 26667207, 100482, 10.1016/j.rico.2024.100482 | |
14. | Sakshi Dhruv, Ritu Arora, Shalini Arora, 2023, Chapter 60, 978-3-031-39773-8, 540, 10.1007/978-3-031-39774-5_60 | |
15. | Ömer Faruk Görçün, Alptekin Ulutaş, Ayşe Topal, Fatih Ecer, Telescopic forklift selection through a novel interval-valued Fermatean fuzzy PIPRECIA–WISP approach, 2024, 255, 09574174, 124674, 10.1016/j.eswa.2024.124674 | |
16. | Awdhesh Kumar Bind, Deepika Rani, Ali Ebrahimnejad, J.L. Verdegay, New strategy for solving multi-objective green four dimensional transportation problems under normal type-2 uncertain environment, 2024, 137, 09521976, 109084, 10.1016/j.engappai.2024.109084 | |
17. | Revathy Aruchsamy, Inthumathi Velusamy, Prasantha Bharathi Dhandapani, Suleman Nasiru, Christophe Chesneau, Ghous Ali, Modern Approach in Pattern Recognition Using Circular Fermatean Fuzzy Similarity Measure for Decision Making with Practical Applications, 2024, 2024, 2314-4785, 1, 10.1155/2024/6503747 | |
18. | Muhammad Akram, Sundas Shahzadi, Syed Muhammad Umer Shah, Tofigh Allahviranloo, A fully Fermatean fuzzy multi-objective transportation model using an extended DEA technique, 2023, 8, 2364-4966, 1173, 10.1007/s41066-023-00399-6 | |
19. | Yuan Rong, Ran Qiu, Linyu Wang, Liying Yu, Yuting Huang, An integrated assessment framework for the evaluation of niche suitability of digital innovation ecosystem with interval-valued Fermatean fuzzy information, 2024, 138, 09521976, 109326, 10.1016/j.engappai.2024.109326 | |
20. | Chuanyang Ruan, Xiangjing Chen, Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Set and Its Application to Multi-Attribute Decision Making, 2023, 12, 2075-1680, 979, 10.3390/axioms12100979 | |
21. | Hongpeng Wang, Caikuan Tuo, Zhiqin Wang, Guoye Feng, Chenglong Li, Enhancing Similarity and Distance Measurements in Fermatean Fuzzy Sets: Tanimoto-Inspired Measures and Decision-Making Applications, 2024, 16, 2073-8994, 277, 10.3390/sym16030277 | |
22. | S. Niroomand, A. Mahmoodirad, A. Ghaffaripour, T. Allahviranloo, A. Amirteimoori, M. Shahriari, A bi-objective carton box production planning problem with benefit and wastage objectives under belief degree-based uncertainty, 2024, 9, 2364-4966, 10.1007/s41066-023-00423-9 | |
23. | Nurdan Kara, Fatma Tiryaki, SOLVING THE MULTI-OBJECTIVE FRACTIONAL SOLID TRANSPORTATION PROBLEM BY USING DIFFERENT OPERATORS, 2024, 1072-3374, 10.1007/s10958-024-07140-x | |
24. | Saliha Karadayi-Usta, Role of artificial intelligence and augmented reality in fashion industry from consumer perspective: Sustainability through waste and return mitigation, 2024, 133, 09521976, 108114, 10.1016/j.engappai.2024.108114 | |
25. | Gulfam Shahzadi, Anam Luqman, Faruk Karaaslan, A decision-making technique under interval-valued Fermatean fuzzy Hamacher interactive aggregation operators, 2023, 1432-7643, 10.1007/s00500-023-08479-0 | |
26. | Yuan Rong, Liying Yu, Yi Liu, Vladimir Simic, Harish Garg, The FMEA model based on LOPCOW-ARAS methods with interval-valued Fermatean fuzzy information for risk assessment of R&D projects in industrial robot offline programming systems, 2024, 43, 2238-3603, 10.1007/s40314-023-02532-2 | |
27. | Junjie Li, Kai Gao, Yuan Rong, A hybrid multi-criteria group decision methodology based on fairly operators and EDAS method under interval-valued Fermatean fuzzy environment, 2024, 9, 2364-4966, 10.1007/s41066-024-00463-9 | |
28. | Deepika Rani, 2024, Chapter 18, 978-981-97-3179-4, 277, 10.1007/978-981-97-3180-0_18 | |
29. | Said Broumi, S. Sivasankar, Assia Bakali, Mohamed Talea, 2024, Chapter 18, 978-981-97-6971-1, 413, 10.1007/978-981-97-6972-8_18 | |
30. | Aakanksha Singh, Ritu Arora, Shalini Arora, A new Fermatean fuzzy multi‐objective indefinite quadratic transportation problem with an application to sustainable transportation, 2024, 0969-6016, 10.1111/itor.13513 | |
31. | Anam Luqman, Gulfam Shahzadi, Multi-criteria group decision-making based on the interval-valued q-rung orthopair fuzzy SIR approach for green supply chain evaluation and selection, 2023, 8, 2364-4966, 1937, 10.1007/s41066-023-00411-z | |
32. | A Mohamed Atheeque, S Sharief Basha, Signless Laplacian energy aware decision making for electric car batteries based on intuitionistic fuzzy graphs, 2024, 107, 0036-8504, 10.1177/00368504241301813 | |
33. | Qiaosong Jing, Enhanced Decision Model for Optimize Operational Performance and Cost Efficiency Under Disc Interval-Valued Fermatean Fuzzy Acknowledge, 2024, 12, 2169-3536, 194423, 10.1109/ACCESS.2024.3520223 | |
34. | Dianwei Zou, Liping Pan, Evaluating Visual Satisfaction in Urban Sculpture Spaces: A Multi-Criteria Decision Making Approach, 2025, 13, 2169-3536, 9354, 10.1109/ACCESS.2025.3528258 | |
35. | Liu Peng, Fusing Emotion and Art Communication: A Disc Interval-Valued Fermatean Fuzzy Decision-Making Approach, 2025, 13, 2169-3536, 8723, 10.1109/ACCESS.2025.3527580 | |
36. | Wajahat Ali, Shakeel Javaid, A solution of mathematical multi-objective transportation problems using the fermatean fuzzy programming approach, 2025, 0975-6809, 10.1007/s13198-025-02716-5 | |
37. | Rabia Mazhar, Shahida Bashir, Muhammad Shabir, Mohammed Al-Shamiri, A soft relation approach to approximate the spherical fuzzy ideals of semigroups, 2025, 10, 2473-6988, 3734, 10.3934/math.2025173 | |
38. | Tarishi Baranwal, A. Akilbasha, Economical heuristics for fully interval integer multi-objective fuzzy and non-fuzzy transportation problems, 2024, 34, 0354-0243, 743, 10.2298/YJOR240115035B | |
39. | P. Anukokila, B. Radhakrishnan, 2025, Chapter 4, 978-981-96-0388-6, 45, 10.1007/978-981-96-0389-3_4 | |
40. | Aakanksha Singh, Ritu Arora, Shalini Arora, 2025, Chapter 9, 978-981-96-0388-6, 149, 10.1007/978-981-96-0389-3_9 | |
41. | Wei Xia, Yang Liu, Ya Pi, Optimize Decision Model for Urban Land Development Using Disc Fermatean Fuzzy Information With Z-Numbers, 2025, 13, 2169-3536, 68370, 10.1109/ACCESS.2025.3560685 |