Category type | Value of parameters | The bound |
I | no parameter | 20.6350 |
II | no parameter | 3.3678e+08 |
III | λ=0.1 | 11.2124 |
III | λ=1 | 3.1070 |
III | λ=2 | 1.9214 |
IV | ρ=0.1 | 1.3997e+13 |
IV | ρ=0.5 | 425.8013 |
IV | ρ=10 | 1.3535 |
This manuscript introduces novel rough approximation operators inspired by topological structures, which offer a more flexible approach than existing methods by extending the scope of applications through a reliance on a general binary relation without constraints. Initially, four distinct types of neighborhoods, termed basic-minimal neighborhoods, are generated from any binary relation. The relationships between these neighborhoods and their properties are elucidated. Subsequently, new rough set models are constructed from these neighborhoods, outlining the main characteristics of their lower and upper approximations. These approximations are applied to classify the subset regions and to compute the accuracy measures. The primary advantages of this approach include its ability to achieve the highest accuracy values compared to all approaches in the published literature and to maintain the monotonicity property of the accuracy and roughness measures. Furthermore, the efficacy of the proposed technique is demonstrated through the analysis of heart failure diagnosis data, showcasing a 100% accuracy rate compared to previous methods, thus highlighting its clinical significance. Additionally, the topological properties of the proposed approaches and the topologies generated from the suggested neighborhoods are discussed, positioning these methods as a bridge to more topological applications in the rough set theory. Finally, an algorithm and flowchart are developed to illustrate the determination and utilization of basic-minimal exact sets in decision-making problems.
Citation: D. I. Taher, R. Abu-Gdairi, M. K. El-Bably, M. A. El-Gayar. Decision-making in diagnosing heart failure problems using basic rough sets[J]. AIMS Mathematics, 2024, 9(8): 21816-21847. doi: 10.3934/math.20241061
[1] | Muhammad Tariq, Sotiris K. Ntouyas, Hijaz Ahmad, Asif Ali Shaikh, Bandar Almohsen, Evren Hincal . A comprehensive review of Grüss-type fractional integral inequality. AIMS Mathematics, 2024, 9(1): 2244-2281. doi: 10.3934/math.2024112 |
[2] | Abd-Allah Hyder, Mohamed A. Barakat, Doaa Rizk, Rasool Shah, Kamsing Nonlaopon . Study of HIV model via recent improved fractional differential and integral operators. AIMS Mathematics, 2023, 8(1): 1656-1671. doi: 10.3934/math.2023084 |
[3] | Mustafa Gürbüz, Yakup Taşdan, Erhan Set . Ostrowski type inequalities via the Katugampola fractional integrals. AIMS Mathematics, 2020, 5(1): 42-53. doi: 10.3934/math.2020004 |
[4] | Jun Moon . The Pontryagin type maximum principle for Caputo fractional optimal control problems with terminal and running state constraints. AIMS Mathematics, 2025, 10(1): 884-920. doi: 10.3934/math.2025042 |
[5] | Ismail Gad Ameen, Dumitru Baleanu, Hussien Shafei Hussien . Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations. AIMS Mathematics, 2024, 9(6): 15819-15836. doi: 10.3934/math.2024764 |
[6] | Pinghua Yang, Caixia Yang . The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative. AIMS Mathematics, 2023, 8(5): 11837-11850. doi: 10.3934/math.2023599 |
[7] | Iyad Suwan, Mohammed S. Abdo, Thabet Abdeljawad, Mohammed M. Matar, Abdellatif Boutiara, Mohammed A. Almalahi . Existence theorems for Ψ-fractional hybrid systems with periodic boundary conditions. AIMS Mathematics, 2022, 7(1): 171-186. doi: 10.3934/math.2022010 |
[8] | 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 |
[9] | Wei Zhang, Jifeng Zhang, Jinbo Ni . New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative. AIMS Mathematics, 2022, 7(1): 1074-1094. doi: 10.3934/math.2022064 |
[10] | Muneerah AL Nuwairan, Ahmed Gamal Ibrahim . The weighted generalized Atangana-Baleanu fractional derivative in banach spaces- definition and applications. AIMS Mathematics, 2024, 9(12): 36293-36335. doi: 10.3934/math.20241722 |
This manuscript introduces novel rough approximation operators inspired by topological structures, which offer a more flexible approach than existing methods by extending the scope of applications through a reliance on a general binary relation without constraints. Initially, four distinct types of neighborhoods, termed basic-minimal neighborhoods, are generated from any binary relation. The relationships between these neighborhoods and their properties are elucidated. Subsequently, new rough set models are constructed from these neighborhoods, outlining the main characteristics of their lower and upper approximations. These approximations are applied to classify the subset regions and to compute the accuracy measures. The primary advantages of this approach include its ability to achieve the highest accuracy values compared to all approaches in the published literature and to maintain the monotonicity property of the accuracy and roughness measures. Furthermore, the efficacy of the proposed technique is demonstrated through the analysis of heart failure diagnosis data, showcasing a 100% accuracy rate compared to previous methods, thus highlighting its clinical significance. Additionally, the topological properties of the proposed approaches and the topologies generated from the suggested neighborhoods are discussed, positioning these methods as a bridge to more topological applications in the rough set theory. Finally, an algorithm and flowchart are developed to illustrate the determination and utilization of basic-minimal exact sets in decision-making problems.
The inverse problem for differential equations is part of the fascinating branches of mathematics. It is developed in connection with diverse branches of applied science [1]. The type of boundary condition determines the identity of the problem. If the boundary is given for an initial time the problem is recognized as an initial value problem (IVP); if the boundary is described for the final, it is called a terminal value problem (TVP). If the boundary is described at both times, it is a Sturm-Liouville problem [2].
For ordinary differential equations (ODEs), the terminal value problem can be transformed into an initial value problem, and they are generally well-posed problems. Surprisingly, this is not true for fractional differential equations (FDEs). Not only can a TVP for FDEs not be transformed into an IVP, but also such problems are not well-posed in general [3,4]. It has been recently noticed that a TVP for FDEs is well-posed in a small neighborhood of the boundary [5,6,7,8,9].
In recent decades, literature on fractional operators has been extensively increased for describing non-local dynamical processes [10]. The non-local property of such processes is usually captured by an integral with a memory kernel. Recently, in the notable paper [11], high dimensional problems are reduced to an integral equation with a memory kernel. This is a green light for replacing ordinary derivatives with fractional derivatives in complex processes. Thus, fractional differential equations can describe abundant models well.
There is no unified definition for a fractional derivative. However, if we restrict some properties (such as being singular or non-singular, local or non-local) we can obtain some completely separate classifications. Among them, the Caputo derivative has received more interest and attention in the literature, mainly for well-interpreted boundary conditions and more similarity to the ordinary derivative (the Caputo derivative of the constant function is zero) [12].
Similarly, fractional integrals have diverse definitions. The route toward the generalized fractional operator will be clear if we monitor some selected classes of fractional integrals. The Riemann-Liouville fractional integral is a generalization of the integer order integral operator
Inf(x)=∫xadt1∫t1adt2…∫tn−1af(tn)dtn=1(n−1)!∫xa(x−t)n−1f(t)dt, |
and the Hadamard fractional integral is a generalization of the nth order integral operator
HInf(x)=∫xadt1t1∫t1adt2t2…∫tn−1af(tn)dtntn =1(n−1)!∫xa(ln(xt))n−1f(t)dtt |
where n∈N [13]. Katugampola [14,15] introduced his fractional derivative by replacing ti with tρi, ρ>0. Fu et al. have replaced exponential functions to introduce exponential fractional derivatives [16]. It is tempting to replace ti with a general function g(ti), (where g is a weighted function) to study generalized fractional derivatives. Such unified generalization of fractional operators with respect to another function has been studied in [17,18].
It is a common aspect of pure mathematics to see an application of a seemingly not applicable concept later. Analogously, for generalized fractional derivatives, we see some noticeable applications that have been published just very recently. For example, the subdiffusion equation with a generalized fractional derivative has been effectively invoked to describe subdiffusion in a medium with an evolving structure over time [19]. The application of generalized fractional operators for the Fokker-Planck equation has been noticed in [16,20]. We will show another importance of such generalization in inverse problems.
Considering the generality of the fractional derivative with respect to another derivative and its application, it is highly motivated to study fractional differential equations with respect to another derivative.
A ν-dimensional terminal value problem for a system of generalized fractional differential equations (GFDEs) is described by
GaDˉα,gty=f(t,y), t∈[a,b] | (1.2) |
and
y(b)=yb, | (1.3) |
where GaDˉα,gt is a generalized fractional derivative with respect to the vector function g=[g1,…,gν]T (gi:[a,b]→R are strictly increasing functions with continuous derivatives g′i on (a,b)), ˉα=[α1,…,αν]T (αi∈(0,1)), y:[a,b]→Rn is an unknown vector function, f:[a,b]×Rn→Rn is a given vector function describing an evolutionary processes, and yb∈Rn is a terminal value.
In a real dynamical process, we do not expect each separate component to have the same memory. Therefore, the study of FDEs with vector order is more important than that for fixed order. We note that the vector order introduced in [21] is an element-wise operator and defined by
GaDˉα,gty=[GaDα1,g1ty1,…,GaDαν,gνtyν]. |
This paper contributes the following achievements:
∗ Study of the TVP for systems of high dimensional nonlinear generalized FDEs,
∗ Computing generalizing fractional operators by classical fractional operators,
∗ Introducing suitable weighted space that relates previous studies to generalized fractional derivative,
∗ Introducing and analyzing a comprehensive collocation method that covers all generalized operators,
∗ Obtaining a lower bound for well-posedness of TVPs with various weight functions,
∗ Applying generalized derivatives for estimating the past population of diabetes cases.
To the best of our knowledge, these are investigated for the first time. The important achievement of this paper is that it gives us more analyzed options in modeling dynamical processes, to choose a better weight function. If we need an inverse problem with a longer interval, the result of this paper introduces a generalized derivative that guarantees the well-posedness in a study model.
A generalized fractional derivative can be transformed into a classical fractional derivative [13,22]. This fact was noticed and employed for solving generalized fractional differential equations, especially in [22]. In this respect, we introduced a weighted space with respect to another function to carry out our analysis from generalized fractional-order derivatives toward classical fractional derivatives.
Numerical methods in parallel to theoretical analysis for solving fractional differential equations have developed rapidly. The Jacobi spectral methods for solving related functional integral and fractional differential equations have been extensively studied in [23,24,25,26]. Applying this method for terminal value problems can be found in [27]. The collocation methods on piecewise polynomials spaces for fractional differential equations were studied in [6,9]. The spectral methods provide fast convergence methods that transform a related problem into a high-dimensional algebraic equation, while the collocation method provides a high order method with more options for control of the error. As far as we know, the collocation method for terminal value problems with generalized fractional differential equations is not yet studied and not analyzed. An important aim of this paper also is to propose and analyze such methods for these problems.
This paper is organized as follows. In Section 2, we review the generalized fractional operators (integral and derivative operators). In Section 3, we transform TVPs for systems of FDEs into weakly singular Fredholm-Volterra integral equations with vanishing delay. In Section 4, we obtain a lower bound for the well-posedness of such problems. In Section 5, we propose a numerical method, and in Section 6 we provide an error analysis for the proposed method. In Section 7, after validating the proposed method, we compare the effects of various choices of weight function on modeling with a TVP.
The generalized fractional operator is defined with respect to the weight function g:[a,b]→R with the following properties:
C1: g∈C1[a,b];
C2: g is a strictly increasing function;
C3: g−1:g([a,b])→[a,b] exists and is continuous (by C2, g([a,b])=[g(a),g(b)]).
Definition 2.1. [18] Let f∈C[a,b] (a,b∈R). The generalized fractional integral GaIα,gt of order α (α∈C, Re(α)>0) with the weight function g is defined by
GaIα,gtf(t)=1Γ(α)∫tag′(τ)f(τ)(g(t)−g(τ))1−αdτ, t∈[a,b]. | (2.1) |
Remark 1. Particular choices of g result famous generalizations and classical definitions of fractional operators:
(I). If g(x)=x, then GaIα,gt is the classical Riemann-Liouville fractional integral,
(II). If g(x)=ln(x) on [1,b], (b>1), then GaIα,gt is the Hadamard fractional integral, in this case g−1(x)=ex,
(III). If g(x)=eλxλ (λ>0) on [a,b], then GaIα,gt is the exponential fractional integral, in this case g−1(x)=ln(λx)λ, [16],
(IV). If g(x)=xρ, then GaIα,gt is the Katugampola fractional integral, in this case g−1(x)=x1ρ [14,15].
Definition 2.2. Let X[a,b] be a space of real valued functions on [a,b]. The weighted space with respect to the weight function g is defined by
Xg[a,b]={k:[a,b]→R∣kog−1∈X[g(a),g(b)]}, |
with the weighted norm
‖k‖g=‖k(g−1)‖, |
where ‖.‖ is a norm of the space X[g(a),g(b)].
For example, Lg[a,b] and Cg[a,b] are weighted spaces of the Lebesgue integrable functions and the continuous functions with Lebesgue norm and supremum norm, respectively.
Remark 2. Suppose k∈C[a,b]. Since g−1 is continuous, kog−1 is continuous. Thus, k∈Cg[a,b] and C[a,b]⊂Cg[a,b]. Similarly, we can prove that Cg[a,b]⊂C[a,b]. For supremum norm in this space, we have
sup[a,b]|k(x)|=sup[g(a),g(b)]|k(g−1(x))|. |
Thus, Cg[a,b]=C[a,b]. However, this type of equivalency can not happen for all spaces like L[a,b]. In this case, by substituting x=g(y) and dx=g′(y)dy, we have
‖k‖g=‖k(g−1)‖=∫g(b)g(a)|k(g−1(x))|dx=∫ba|k(y)|g′(y)dy≠‖k‖. |
Throughout the paper we generally use the supremum norm unless we mention it. The following theorem describes the generalized fractional integral by the Riemann-Liouville fractional integral.
Theorem 2.1. Suppose α∈C, Re(α)>0, and g:[a,b]→R has the properties C1–C3. Then, for h∈C[a,b],
GaIα,gth(g(t))=(RLg(a)Iαth(t))o(g(t)), | (2.2) |
where RLg(a)Iαt stands for the Riemann-Liouville fractional integral, and the notation "o" stands for the combination operator.
Proof. Setting k(t)=h(g(t)), we get
GaIα,gtk(t)=1Γ(α)∫tah(g(τ))g′(τ)(g(t)−g(τ))1−αdτ=1Γ(α)∫g(t)g(a)h(x)(g(t)−x)1−αdx =(1Γ(α)∫tg(a)h(x)(t−x)1−αdx)o(g(t)). | (2.3) |
Here, we integrated with substitution x=g(τ), (dx=g′(τ)dτ). Thus, we obtain
GaIα,gtk(t)=(RLg(a)Iαth(t))o(g(t)). |
This completes the proof.
Remark 3. Theorem 2.1 shows that RLg(a)Iαtk(g−1(t)) is well-defined if and only if GaIα,gtk(t) is well-defined. Let X[g(a),g(b)] be the space that the operator RLg(a)Iαt is well-defined. Then, the operator GaIα,gtk(t) is well-defined on the weighted space Xg[a,b].
Definition 2.3. [18] Let α∈C, Re(α)>0, and n=[Re(α)]+1. The generalized fractional derivatives on [a,b] (0≤a<b) are defined by
GaDα,gtf(t)=(GaIn−α,gt(1g′(t)ddt)nf(t))=1Γ(n−α)∫ta(1g′(τ)ddτ)nf(τ)(g(t)−g(τ))1−n+αg′(τ)dτ | (2.4) |
for α∉N and for a Borel function f:[a,b]→R in which the integral in Eq (2.4) is well-defined. For α∈N, it is the classical integer order definition, i.e., KaDα,ρtf(t)=dn−1dtn−1.
Theorem 2.2. Let α∈C, and 0<Re(α)<1. Then,
GaDα,gtf(g(t))=(Cg(a)Dαtf(t))o(g(t)). | (2.5) |
Proof. Setting h(t)=f(g(t)), we get
GaDα,gth(t)=1Γ(1−α)∫tah′(τ)(g(t)−g(τ))αdτ=1Γ(1−α)∫taf′(g(τ))g′(τ)(g(t)−g(τ))αdτ=1Γ(1−α)∫g(t)g(a)f′(y)(g(t)−y)αdy=(Cg(a)Dαtf(t))o(g(t)). | (2.6) |
This completes the proof.
Computations of generalized derivatives and integrals of functions (g(t)−g(a))v, for 0<Re(α)<1 are a direct consequence of Theorems 2.1 and 2.2. Thus, we have
GaDα,gt(g(t)−g(a))v=(Cg(a)Dαt(t−g(a))v)(g(t))={(Γ(v+1)Γ(v−α+1)(t−g(a))v−α)o(g(t)),v≠0,0,v=0,={(Γ(v+1)Γ(v−α+1)(g(t)−g(a))v−α),v≠0,0,v=0, | (2.7) |
and
GaIα,gt(g(t)−g(a))v=(RLg(a)Iαt(t−g(a))v)o(g(t))=(Γ(v+1)Γ(v+α+1)(t−g(a))v+α)o(g(t))=Γ(v+1)Γ(v+α+1)(g(t)−g(a))v+α. | (2.8) |
Now, we can use Theorem 2.2 to find a suitable space for defining a generalized fractional derivative. Regularly, the spaces ACn[a,b] and Cn[a,b] are the best choices for classical fractional derivatives with order α such that n−1<Re(α)<n, (Theorems 2.1 and 2.2 of [28], also see [13]). We recall that
ACn[a,b]={f:[a,b]→C,f(n−1)∈AC[a,b]}, n∈N, |
where AC[a,b] is the set of absolute continuous functions. Thus, we can state the following lemma.
Lemma 2.1. Let f∈ACng[a,b]. Then, GaDα,gtf(t) exists almost everywhere on [a, b]. Moreover, if f∈Cng[a,b], then GaDα,gtf∈C[a,b].
Proof. The proof is a straightforward result of Eq (2.5), and Theorems 2.1 and 2.2 of [28].
Remark 4. Equations (2.2) and (2.5) can be rewritten as
GaIα,gtf(t)=(RLg(a)Iαtf(g−1(t)))o(g(t)) | (2.9) |
and
GaDα,gtf(t)=(Cg(a)Dαtf(g−1(t)))o(g(t)). | (2.10) |
Lemma 2.2. Let α∈(0,1], ρ∈[0,1), and f∈AC[a,b].Then,
GaItα,gGaDα,gtf(t)=f(t)−f(a). | (2.11) |
Proof. It follows from Eq (2.9) that
GaItα,g(GaDα,gtf(t))=(RLg(a)Iαt(GaDα,gtf(t)o(g−1(t))))o(g(t)). |
Applying Eq (2.10), we get
GaItα,g(GaDα,gtf(t))=(RLg(a)Iαt((Cg(a)Dαtf(g−1(t)))o(g(t))o(g−1(t))))o(g(t)). |
Taking into account that (g(t))o(g−1(t))=t, we obtain
GaItα,g(GaDα,gtf(t))=(RLg(a)Iαt(Cg(a)Dαtf(g−1(t))))o(g(t)). |
Finally, from properties of classical fractional operators, we obtain
GaItα,g(GaDα,gtf(t))=(f(g−1(t))−f(g−1(g(a))))o(g(t))=(f(g−1(t))−f(a))o(g(t))=f(t)−f(a), | (2.12) |
which completes the proof.
Remark 5. All the mentioned results can be generalized to higher-dimensional spaces by the vector order notation. For vector order ˉα=[α1,…,αν]T (Re(αi)∈(0,1), i=1,…,ν) and vector weights g=[g1,…,gν]T we have
GaItˉα,gGaDˉα,gtf(t)=[GaItα1,g1GaDα1,g1tf1,…,GaIαν,gνGaDαν,gνtfν]T=[f1(t)−f1(a),…,fν(t)−fν(a)]T=[f1(t),…,fν(t)]T−[f1(a),…,fν(a)]T=f(t)−f(a), | (2.13) |
where f=[f1,…,fν]T belongs to the ν-dimensional space (AC[a,b])ν. It is a straightforward generalization of a one-dimensional case since all operators operate element-wise [21].
As an application of Eqs (2.7), (2.8)and (2.11), we can solve a linear initial value problem for FDEs:
GaDˉα,gty=Ay(t)+B, t∈(a,b), | (2.14) |
with the initial value y(0)=y0 and imposing αi=α, gi=g for i=1,⋯,ν. Applying GaItα,g to both sides of Eq (2.14) and using Eq (2.11), we have
y=y0+GaItˉα,gB+GaItˉα,gAy(t), t∈(a,b). | (2.15) |
Here, it is worth to mentioning that, for general ˉα and g, the commutative property
GaItˉα,gAyn=GaItˉα,gAyn |
does not hold for vector order fractional integrals as well as for vector order fractional derivatives.
The iterative method
yn+1=y0+GaItˉα,gB+GaItˉα,gAyn, n=1,2,..., | (2.16) |
with the initial value
y1=y0+GaItˉα,gB |
is utilized to obtain the exact solution y=limn→∞yn. Recursively, we obtain
y2=y0+GaItˉα,gB+AGaItˉα,gy1=y0+BΓ(ˉα+1).(g(t)−g(a))ˉα+AGaItˉα,g(y0+GaItˉˉα,gB)=y0+BΓ(ˉα+1).(g(t)−g(a))ˉα+Ay0Γ(ˉα+1).(g(t)−g(a))ˉα+ABΓ(2ˉα+1).(g(t)−g(a))2ˉα | (2.17) |
and similarly
y3=y0+GaItˉα,gB+AGaItˉα,gy2=y0+BΓ(ˉα+1).(g(t)−g(a))ˉα+Ay0Γ(ˉα+1).(g(t)−g(a))ˉα+ABΓ(2ˉα+1)(g(t)−g(a))2ˉα+A2y0Γ(2ˉα+1).(g(t)−g(a))2ˉα+A2BΓ(3ˉα+1).(g(t)−g(a))3ˉα. | (2.18) |
This pattern suggests
y(t)=∞∑n=0Any0Γ(nˉα+1).(g(t)−g(a))nˉα+AnBΓ((n+1)ˉα+1).(g(t)−g(a))(n+1)ˉα | (2.19) |
where the dot (".'') operator stands for element-wise multiplication.
Applying the fractional integral to both sides of the system (1.2), it follows from Eq (2.2) that
y(t)=y(a)+GaItˉα,gf(t,y(t)), t∈(a,b), | (3.1) |
or equivalently,
y(t)=y(a)+RLg(a)Iˉαtf(g−1(t),y(g−1(t)))og(t). | (3.2) |
Computationally, it is important to note that
f(g−1(t),y(g−1(t)))=[f1(g−11(t),[y1(g−11(t)),…,yν(g−11(t))])⋮fν(g−1ν(t),[y1(g−1ν(t)),…,yν(g−1ν(t))])]. |
Putting t=b in Eq (3.2) and using the terminal condition, we obtain
y(a)=y(b)−RLg(a)Iˉαtf(g−1(t),y(g−1(t)))og(t)∣t=b. | (3.3) |
Substituting y(a) from Eq (3.3) into Eq (3.2), we obtain
y(t)=y(b)−RLg(a)Iˉαtf(g−1(t),y(g−1(t)))og(t)∣t=b+RLg(a)Iˉαtf(g−1(t),y(g−1(t)))og(t). | (3.4) |
Equation (3.4) can be represented by a weakly singular Fredholm-Volterra integral equation with vanishing delay g(t)
y(t)=yb−1Γ(ˉα)∫g(b)g(a)f(g−1(τ),y(g−1(τ)))(g(b)−τ)1−ˉαdτ+1Γ(ˉα)∫g(t)g(a)f(g−1(τ),y(g−1(τ)))(g(t)−τ)1−ˉαdτ, | (3.5) |
or equivalently,
y(t)=yb−1Γ(ˉα)∫bag′(x)f(x,y(x))(g(b)−g(x))1−ˉαdx+1Γ(ˉα)∫tag′(x)f(x,y(x))(g(t)−g(x))1−ˉαdx. | (3.6) |
Remark 6. If g(t)=pt, the corresponding system has proportional delay [29]. Also, with g(t)=t32 and the non-local boundary condition
y(b)=1+1Γ(ˉα)∫bag′(x)f(x,y(x))(g(b)−g(x))1−ˉαdx, |
the system (3.6) is a Lighthill system. This system is used for describing the temperature distribution of the surface of a projectile moving through a laminar layer [30].
We suppose the vector function f=[f1,…,fν]T satisfies the following conditions.
(H1) The vector function f is continuous with respect to its variables on [0,T]×Rν,
(H2) Functions fi, i=1,…,ν, are Lipschitz functions with Lipschitz constants Li, i.e.,
‖fi(t,w)−fi(t,y)‖≤Li‖w−y‖ |
for y,w∈Rν, and we set
LM=maxi=1,…,ν‖Li‖. |
It is a straightforward conclusion of Theorem 2.1 and Remark 2 that the operator
GaItˉα,gf(t,.):(C[a,b])ν→(C[a,b])ν |
transforms a continuous vector function into a continuous vector function, and thus it is well-defined.
We use the norm
‖y‖=maxi=1,…,ν‖yi‖∞,‖yi‖∞=supt∈[a,b]‖y(t)‖ |
to state continuity of the operator GaItˉα,gf(t,.).
Theorem 4.1. Let assumptions (H1) and (H2) hold for the vector function f. Then, the operators GaItˉα,gf(t,.) and GaIbˉα,gf(b,.) are continuous, i.e.,
‖GaItˉα,gf(t,u)−GaItˉα,gf(t,v)‖≤M‖u−v‖ | (4.1) |
and
‖GaIbˉα,gf(b,u)−GaIbˉα,gf(b,v)‖≤M‖u−v‖, | (4.2) |
where the vector functions u and v belong to (C[a,b])ν and
M=maxi=1,…,νLi(gi(b)−gi(a))ˉαΓ(ˉα+1). | (4.3) |
Proof. The supremum norm of the ith component of the generalized fractional integral operator can be estimated by
‖(GaItˉα,gf(t,u))i−(GaItˉα,gf(t,v))i‖∞=supt∈[a,b]1Γ(αi)∫ta|g′i(x)(fi(x,u(x)−fi(x,v(x))|(gi(t)−gi(x))1−αidx. |
By assumptions C1–C3, gi(x) and (gi(t)−gi(x))1−αi for x∈[0,t] are non-negative. Thus,
‖(GaItˉα,gf(t,u))i−(GaItˉα,gf(t,v))i‖∞=supt∈[a,b]LiΓ(αi)∫tag′i(x)(gi(t)−gi(x))1−αidx‖u−v‖=LiΓ(αi)supt∈[a,b](gi(t)−gi(a))αiαi‖u−v‖=Li(gi(b)−gi(a))αiΓ(αi+1)‖u−v‖. | (4.4) |
Consequently, the inequalities (4.1) and (4.2) follow from Eq (4.4).
By Theorem 4.1, the operator T:(C[a,b])ν→(C[a,b])ν described by
T(y)=yb−GaIbˉα,gf(b,y)+GaItˉα,gf(t,y) | (4.5) |
is well-defined.
Theorem 4.2. Let assumptions (H1) and (H2) hold for f:[0,T]×Rν→Rν.Let
M:=maxi=1,…,ν2Li(gi(b)−gi(a))αiΓ(αi+1)<1. | (4.6) |
Then, the system (3.6) has a unique solution on (C[a,b])ν.
Proof. Systems (3.5) and (3.6) can be rewritten in the operator form
y=T(y). | (4.7) |
By using Theorem 4.1 for the vector functions u and v∈(C[a,b])ν, we have
‖T(u)−T(v)‖≤M‖u−v‖. | (4.8) |
From condition (4.6), M<1, and hence
‖T(u)−T(v)‖≤‖u−v‖. | (4.9) |
Thus, T is a contracting operator, and by the Banach fixed point theorem, it has a unique solution on (C[a,b])ν. This completes the proof.
Let us adopt notations of [29]. Let Ih={tn: a=t0<t1<…<tN=b} be a partition of [a,b], hn=tn+1−tn, σn=(tn,tn+1] (n=0,…,N−1) and h=maxn=0,…,N−1hn. We recall that the grading mesh points of the form tn:=(b−a)(n/N)r are an appropriate choice for integral equations with weakly singular kernels[29]. The piecewise polynomial space can be defined by
S−1m−1(Ih)={v:v∣σn∈Πm−1}, |
where Πm−1 is the space of polynomials of degree less than m. An approximate solution of the system (1.2) has the dense representation
uN(t0)=y0,uN(tn+vhn)=m∑i=1Li(v)Un,i,v∈(0,1], n=0,…,N−1, | (5.1) |
where Li are Lagrange interpolation formulas, and Un,i:=uN(tn,i) and collocation points are defined by tn,i:=tn+cihn for given collocation parameters 0≤c1<…<cm≤1. The operator PN that projects (C[0,T])ν into (S−1m−1(Ih))ν is described by
PN(u(tn,i))=u(tn,i), i=1,…,m, n=0,…,N−1. |
Thus, we can explicitly and uniquely define PN by
PN(uN(tn+vhn))=m∑i=1Li(v)Un,i, n=0,…,N−1. | (5.2) |
The quadrature approximation of the operator GaItˉα,gf(t,.) is computed by
QNuN(t)=GaItˉα,gPNf(t,uN(t)). |
Let t:=tn+vhn∈[tn,tn+1]. Then,
QNuN(t)=n−1∑l=0m∑j=1λn,l,j(v)f(tl,j,Ul,j)+m∑j=1λn,n,j(v)f(tn,j,Un,j), | (5.3) |
where the weights of the quadrature are determined by
λn,l,j(v)={hlΓ(ˉα)∫10g′(tl+zhl)Lj(z)(g(tn+vhn)−g(tl+zhl))1−ˉαdz,l=0,…,n−1,hnΓ(ˉα)∫v0g′(tn+zhn)Lj(z)(g(tn+vhn)−g(tn+zhn))1−ˉαdz,l=n, | (5.4) |
for j=1,…,m and n=0,…,N−1. Similarly, a quadrature method for GaIbˉα,gf(t,.) can be computed by
˜QNuN(t)=GaIbˉα,gPNf(t,uN(t)) |
or, equivalently,
˜QNuN=N−1∑l=0m∑j=1λN−1,l,j(1)f(tl,j,Ul,j). | (5.5) |
Setting t=to,k for o=0,…,N−1 and k=1,…,m, and using the introduced quadrature method, the unknown Uo,k can be obtained by solving the system of nonlinear equations described by
Uo,k=yb−˜QNuN+QNuN(to,k). | (5.6) |
After solving this system, we can obtain the dense approximate solution by Eq (5.1). The nonlinear system (5.6) is solved by the recursive formula
Uio,k=yb−˜QNui−1N+QNUi−1o,k, i=1,...,nt, | (5.7) |
where nt is the constant number of iterations (can be chosen by a user), or can be adopted by
‖uiN−ui−1N‖≤Tol |
with a given tolerance Tol.
In this research, we also need to introduce a numerical method for solving related initial value problems. Let us consider the system (1.2) with a given initial condition
y(a)=ya. | (5.8) |
A numerical approach by recursively solving the νm dimensional algebraic systems
Uo,k=ya+QNuN(to,k), k=1,…,m, | (5.9) |
for o=0,…,N−1 is utilized to obtain the corresponding dense solution. However, for the IVP in each interval, we need to solve only a nonlinear equation of dimension ν×m, while for the TVP we need to solve a nonlinear equation of dimension ν×m×N.
Does the nonlinear system (5.6) have a solution? Is it unique? What is the convergence of the proposed iterative method (5.7)? This subsection answers these questions.
Theorem 5.1. Let assumptions (H1) and (H2) hold for the vector function f:[0,T]×Rν→Rν. Let
Λ:=maxi=1,…,ν2Li(gi(b)−gi(a))αi‖PN‖2Γ(αi+1)<1 | (5.10) |
for a given N. Then, the approximate solution of the system (5.6) exists and is unique, and the iterative method (5.7) converges to the solution of the system (5.6) on collocation points.
Proof. Since uN∈(S−1m−1(Ih))ν we have PNuN=uN, and the dense approximate solution (5.1) satisfies
uN(t)=PNTNuN, | (5.11) |
where
TNu:=yb−˜QNu+QNu. |
Let u,v∈(S−1m−1(Ih))ν. The ith component of the QNu(t)−QNv(t) satisfies
|(QNu(t))i−(QNv(t))i|=|(GaItˉα,gPNf(t,u(t)))i−(GaItˉα,gPNf(t,v(t)))i|≤|1Γ(αi)∫ta|g′i(x)(PN(f(x,u(x)−f(x,v(x)))i|(gi(t)−gi(x))1−αidx|≤|1Γ(αi)∫tag′i(x)dx(gi(t)−gi(x))1−αi|‖(PN(f(x,u−f(x,v))i‖∞≤(gi(t)−gi(a))αiΓ(αi+1)‖Li(PN)i(u−v)i‖∞≤Li(gi(t)−gi(a))αiΓ(αi+1)‖PN‖‖u−v‖. | (5.12) |
Therefore,
‖QNu−QNv‖≤M‖PN‖‖u−v‖, | (5.13) |
where M is defined as in Eq (4.3). Similarly,
‖˜QNu−˜QNv‖≤M‖PN‖‖u−v‖. | (5.14) |
The operator PNTN is a contractor since
‖PNTN‖≤2‖PN‖2M(‖u−v‖)≤‖u−v‖ |
by the hypothesis of the theorem. It follows from the Banach fixed point theorem that the operator
PNTN:(S−1m−1(Ih))ν→(S−1m−1(Ih))ν |
has a unique solution, and the recursive formula
uiN=PNTNui−1N, i=1,2,⋯. |
converges to the fixed point of PNTN with any initial solution
u0N∈(S−1m−1(Ih))ν. |
We recall that (S−1m−1(Ih))ν is a finite dimensional space and thus a Banach space.
In this section, we show that the error of the discretized collocation method is proportional to the error of the quadrature method for the exact solution:
R(y):=(T−PNTN)(y). | (6.1) |
Theorem 6.1. Let assumptions (H1) and (H2) hold for f:[0,T]×Rν→Rν. Let Λ defined by (5.10) satisfy
Λ<1. | (6.2) |
Then, the approximate solution described by (5.1) and (5.6) exists, and
‖uN−y‖≤11−C‖R(y)‖. | (6.3) |
Proof. Since uN∈(S−1m−1(Ih))ν, it follows that PNuN=uN. Hence, the dense approximate solution (5.1) satisfies
uN(t)=PNTNuN, | (6.4) |
where
TNu=yb−˜QNu+QNu(t). |
By using Eq (6.1),
y=PNTN(y)+R(y). | (6.5) |
Setting
e:=uN−y |
and subtracting Eq (6.5) from Eq (5.10), we obtain
e=PNTNuN−PNTN(y)−R(y). | (6.6) |
The first difference of Eq (6.6) satisfies
‖PNTNuN−PNTN(y)‖≤‖PN‖‖TNuN−TN(y)‖≤‖PN‖‖−˜QNuN+QNuN(t)+˜QNy−QNy(t)‖≤‖PN‖(‖˜QNuN−˜QNy‖+‖QNuN(t)−QNy(t)‖). | (6.7) |
Let us consider the the ith component of the QNuN(t)−QNy(t), and we have
|(QNuN(t))i−(QNy(t))i|=|(GaItˉα,gPNf(t,uN(t)))i−(GaItˉα,gPNf(t,y(t)))i|≤|1Γ(αi)∫ta|g′i(x)(PN(f(x,uN(x)−f(x,y(x)))i|(gi(t)−gi(x))1−αidx|≤|1Γ(αi)∫tag′i(x)dx(gi(t)−gi(x))1−αi|‖(PN(f(x,uN(x)−f(x,y(x)))i‖∞≤(gi(t)−gi(a))αiΓ(αi+1)‖Li(PN)i(uN−y)i‖∞≤Li(gi(t)−gi(a))αiΓ(αi+1)‖PN‖‖uN−y‖. | (6.8) |
It follows from Eq (6.8) that
‖QNuN−QNy‖≤M‖PN‖‖uN−y‖. | (6.9) |
Similarly, we can prove that
‖˜QNuN−˜QNy‖≤M‖PN‖‖uN(x)−y(x)‖. | (6.10) |
Substituting Eqs (6.9) and (6.10) into Eq (6.7), we obtain
‖PNTNuN−PNTN(y)‖≤2M‖PN‖2‖uN(x)−y(x)‖=2M‖PN‖2‖e‖. | (6.11) |
Taking the norm from Eq (6.6) and using Eq (6.11), we obtain
‖e‖≤‖PNTNuN−PNTN(y)‖+‖R(y)‖≤2M‖PN‖2‖e‖+‖R(y)‖. | (6.12) |
This leads to an interesting bound for the error,
‖e‖≤11−Λ‖R(y)‖. | (6.13) |
The following lemma supports Theorem 6.1 by a bound for ‖PN‖.
Lemma 6.1. Let PN:(C[0,T])ν→(S−1m−1(Ih))ν be a projection defined by Eq (5.2). Let ‖.‖ be the induced norm of projection. Then, there exist N0∈N and a constant C such that
‖PN‖≤C |
for all N>N0.
Proof. We prove this statement for the 1-D case, and the proof of the ν-dimensional case is similar. Let ‖y‖<1.
‖PNy‖=maxn=0,…,N−1‖PNy∣σn‖={maxt∈[tn,tn+1], n=0,…,N−1∑mj=1∏mi=1,i≠j|y(tn,j)|,m=1,maxt∈[tn,tn+1], n=0,…,N−1∑mj=1∏mi=1,i≠jt−tn,itn,j−tn,i|y(tn,j)|,m>1. |
Setting t=tn+vhn and |y(z)|=max|y(tn,j)|, we obtain
‖PNy‖≤C|y(z)|C‖y‖ |
where
C={1,m=1,maxv∈[0,1]∑mj=1∏mi=1,i≠j|v−ci||cj−ci|,m>1, |
is independent of y and t.
Remark 7. We note that for m=1, we have ‖PN‖≤1. Thus, Λ in Eqs (6.2) and (5.10) will be equal to M defined in (4.6). This means that the conditions of (6.1), (5.1) and (4.2) coincide.
To start numerical implementation, we compute the weight coefficients for the given parameters m and c. Let m=1 and c1=θ, L1(t)=1. The corresponding methods are known as the θ method. The θ method approximates the solution with a constant function in each interval. In uniform mesh, it is similar to Haar scale functions. The weight of the proposed approximate quadrature can be computed by
λn,l,1(v)={(g(tn+vhn)−g(tl))ˉαΓ(ˉα+1)−(g(tn+vhn)−g(tl+1))ˉαΓ(ˉα+1),l=0,…,n−1,(g(tn+vhn)−g(tl))ˉαΓ(ˉα+1),l=n. |
Consider a system of GFDEs described by Eq (1.2) with the vector order ˉα=[0.5,0.5]T, the weight function g=[g,g]T, the source function
f(t,y)=[0.1sin(y21+y22)−0.1sin(2(g(t)−g(a)))+√π20.1cos(y1−y2)+0.1√π2−0.1] |
and the terminal value
yb=[√(g(b)−g(a)),√(g(b)−g(a))]T. |
Before giving the exact solution, we notice that f satisfies the condition (H1) and (H2). According to Theorem 4.2, for an appropriately chosen value of b, the solution falls into (C[a,b])2. The exact solution of this system is
y=[√(g(t)−g(a)),√(g(t)−g(a))]T. |
This is in agreement with the assertion of Theorem 4.2.
We notice that we could not choose a=0 for case II, since the function ln is not defined at zero. For a better comparison, we choose the initial terminal a=1, and we find b such that
b≤g−1(π0.16). | (7.1) |
Equation (7.1) guarantees the condition (4.6) of Theorem 4.1. Thus, the corresponding system has a unique solution with an arbitrary value of yb. Table 1 shows lower bounds of b for various choices of weight functions g obtained by Eq (7.1).
Category type | Value of parameters | The bound |
I | no parameter | 20.6350 |
II | no parameter | 3.3678e+08 |
III | λ=0.1 | 11.2124 |
III | λ=1 | 3.1070 |
III | λ=2 | 1.9214 |
IV | ρ=0.1 | 1.3997e+13 |
IV | ρ=0.5 | 425.8013 |
IV | ρ=10 | 1.3535 |
Remark 8. Table 1 highlights case II (Hadamard fractional operator) as well as case IV (Katugampola type) as candidates for having bigger bounds of well-posedness. The parameters of cases III and IV change b with different rates. The most rapid change occurs in the Katugampola type. We recall that the terminal value problem may not be well-posed [4], at all.
Remark 8 guides us toward available options for modeling inverse problems in applied science. Actually, we need well-posedness for larger intervals in general.
In Figure 1, we illustrate the exact and the approximate solutions for various weight functions g, with the upper bound b=2, the collocation parameter c=0.1, the graded mesh exponent r=2 and the number of steps N=20. The maximum errors and estimated orders of convergence for both components of the solution are compared in Tables 2 and 3, respectively. The maximum error is computed by
EN=maxn=0,…,N−1{uN(tn,1)−y(tn,1)}. |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |
It is known that, for y∈(C[a,b])2, R(y)=O(N−m)+O(Nrβ) (see for example [9]). In this example, m=1 and r=2; thus, according to Theorem 6.1,
e=O(N−1). |
Tables 2 and 3 show that the order of converge tends to 1. Thus, the reported results are in complete agreement with the theoretical results.
Diabetes is a silent epidemic of high blood sugar levels, mainly due to decreasing activities and an inappropriate diet. An epidemiology model using FADEs for simulating the population of diabetes cases is proposed in [31]. This model applies linear two-dimensional FDEs
GaD[α1,α2]T,[g1,g2]Tty(t)=Aλ(t)+B | (7.2) |
to describe the population of diabetes cases. Here,
A=[a11a12−a21a22] |
is a 2×2 matrix, and B is a 2-dimensional column vector. By some physical interpretation of the model [31], we impose 0≤a11<5, 0≤a12≤1 and 0≤a21<2. The interpretation of a22 is the rate of death in [31]. However, the population of the studied zone is not a constant parameter, and indeed it is increasing. Thus, we let a22 be positive, and we impose −1≤a22<8.
Let us use the system (7.2) as a black box for modeling the populations of diabetes cases with model parameters A and B. We use available data, which was reported in [32], for the population of diabetes cases from 1990–2017 to find the model parameters. Then, we use the obtained model for predicting the past dynamics by a terminal value problem.
We set ˉα=[0.9,0.9]T to involve history in our model. Our interest is to study the effect of g on the results of this model. Let the output of the black box be o(t)=y2(t), indicating the population of diabetes cases for an input value of t.
We use o for simulating the dynamics of the prevalence number of diabetes. Let D(t) show available data for the prevalence number of diabetes. Then, D(t)/DM is normalized data, where DM=max|D(t)|.
Setting A=[0,0;0,0], we obtain
y(t)=y0+(g(t)−g(a))αΓ(α+1)B. |
This means, by tending α→0, the matrix B can be regarded as an approximation for the slope of the curve y(t). Thus, a reasonable physical interpretation suggests choosing B as the prevalence rate of diabetes with some gain related to g.
According to [32], a prevalence rate is a number between 4500 and 5500 per 100k population. Thus, we use B=η[0,0.045]T to reduce the parameters in the related optimization problem, where the parameter η∈R is a model parameter.
Now, the inverse problem of modeling dictates to determining A and η such that minimizes the following objective function:
E=2017∑t=1990‖o(t)−D(t)/DM‖. | (7.3) |
The output o(t) will be calculated from the block box as an initial value problem on [1990,2017]. The modeled IVP can be solved by Eq (2.19) or a numerical method described by (5.9). We applied the latter one.
Let's apply the inverse problem to the obtained model. We emphasize that classical fractional-order TVPs (i.e., TVPs with Caputo fractional derivative) for high dimensional systems are not well-posed on the large intervals. A cure for such a modeling problem is the ultimate aim of this paper. To this end, we use appropriate generalized fractional derivatives with a larger well-posedness bound.
Let g(x)=[xρt,xρt]T. The case ρ=1 corresponds to the Caputo fractional derivative. In Figure 2(a) we depict the result of the fitted model for ρ=1. We apply (5.9) with N=200, c=1 and graded mesh of exponent r=2. The fitting error (7.3) in this case is E=0.24998. Since Λ=113.2473, the conditions (6.2) and (4.6) do not hold. Thus, Theorems 4.2, 5.1 and 6.1 do not guarantee the corresponding numerical and exact solutions exist. Also, there is no guarantee for convergence of the approximate solution. Figure 2(b) shows the approximate solution by the prescribed method in Eqs (5.1) and (5.7) (N=200, c=1 and r=2) for the corresponding TVP. It is not a surprise that the numerical approximation for the TVP does not converge to the corresponding solution of the IVP. Thus, this model cannot be used for predicting past dynamics.
Mathematically, the constraints for elements of A impose that the Lipschitz constant of f be less than 8. As a result, we can improve the domain of well-posedness by decreasing ρ in g(x)=[xρt,xρt]T. On the other hand, this restriction may increase the optimization error, i.e., E. Using the same procedure as we reported for the case ρ=1, we report the result for ρ=0.2,0.4,0.8 in Figures 3–5, respectively (the case ρ=0.6 is similar to case ρ=0.8 and it does not add more information).
For the case ρ=0.8, the fitting error E=0.1530 is decreased (Figure 3(a)–(c)). The λ corresponding to interval I1:=[1990,2017] is decreased to 13.1412. It is not in the zone of the predicted bound for well-posedness and convergence of the given TVP. However, it converges, and the solutions of the TVP and IVP coincide, as shown in Figure 3(b). The solution for prediction of past time on I2=[1940,2017] with the same TVP is illustrated in Figure 3(c). Obviously, increasing the interval length increases λ. For interval I2 we get λ=33.8248. There is no guarantee of well-posedness, and the results cannot be reliable (a similar pattern repeats for the case λ=0.6).
For the case ρ=0.4, the fitting error E=0.37360 is increased (Figure 4(a)–(c)). However, the values λ corresponding to I1 and I2 are decreased, and both are near the well-posedness and convergence zone. We can rely on these results.
The last case is ρ=0.2: see Figure 5(a)–(c). The optimization result will cross some boundaries of our restricted conditions on A. The values of λ for both intervals are less than 1. Well-posedness of the corresponding TVP and convergence of the proposed numerical method are ensured. However, the error decreases to E=0.9076. It behaves not better than a model with linear regression.
Conclusively, there is a contrast between the fitting error and the well-posedness condition. A compromise can be made by considering both conditions for a given model. In this case, we propose the model with ρ=0.4, for estimating past dynamics.
An extensive application of FDEs is involved in many branches of science, such as biology, epidemiology, medicine and chemistry. Always, these systems are higher dimensional and generally involve some terms of nonlinear Lotka Volterra dynamics with initial conditions. Therefore, as IVPs they are well-posed. However, the inverse problem is not well-posed for higher dimensional equations. It is useless and dangerous to use fractional derivatives for modeling such processes, especially when we have boundary conditions, without considering their well-posedness. Fortunately, this paper's results for modeling with general fractional derivatives open a new window to overcome this problem. However, we encounter another problem: increasing fitting error. For this aim, a compromising solution between well-posedness and less fitting error is considered.
The authors declare no conflict of interest.
[1] |
Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci., 11 (1982), 341–356. https://doi.org/10.1007/BF01001956 doi: 10.1007/BF01001956
![]() |
[2] | Z. Pawlak, Rough sets: Theoretical aspects of reasoning about data, Dordrecht: Springer, 1991. https://doi.org/10.1007/978-94-011-3534-4 |
[3] |
Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239–259. https://doi.org/10.1016/S0020-0255(98)10006-3 doi: 10.1016/S0020-0255(98)10006-3
![]() |
[4] |
A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fund. Inform., 27 (1996), 245–253. https://doi.org/10.3233/FI-1996-272311 doi: 10.3233/FI-1996-272311
![]() |
[5] |
E. A. Abo-Tabl, Rough sets and topological spaces based on similarity, Int. J. Mach. Learn. Cyber., 4 (2013), 451–458. https://doi.org/10.1007/s13042-012-0107-7 doi: 10.1007/s13042-012-0107-7
![]() |
[6] |
J. H. Dai, S. C. Gao, G. J. Zheng, Generalized rough set models determined by multiple neighborhoods generated from a similarity relation, Soft Comput., 22 (2018), 2081–2094. https://doi.org/10.1007/s00500-017-2672-x doi: 10.1007/s00500-017-2672-x
![]() |
[7] |
K. Qin, J. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 178 (2008), 4138–4141. https://doi.org/10.1016/j.ins.2008.07.002 doi: 10.1016/j.ins.2008.07.002
![]() |
[8] | A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, In: Rough sets, fuzzy sets, data mining, and granular computing, Berlin, Heidelberg: Springer, 13641 (2005), 64–73. https://doi.org/10.1007/11548669_7 |
[9] |
R. Abu-Gdairi, M. A. El-Gayar, M. K. El-Bably, K. K. Fleifel, Two different views for generalized rough sets with applications, Mathematics, 9 (2022), 2275. https://doi.org/10.3390/math9182275 doi: 10.3390/math9182275
![]() |
[10] |
R. Abu-Gdairi, M. A. El-Gayar, T. M. Al-shami, A. S. Nawar, M. K. El-Bably, Some topological approaches for generalized rough sets and their decision-making applications, Symmetry, 14 (2022), 95. https://doi.org/10.3390/sym14010095 doi: 10.3390/sym14010095
![]() |
[11] |
M. E. Abd El-Monsef, O. A. Embaby, M. K. El-Bably, Comparison between rough set approximations based on different topologies, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 292–305. https://doi.org/10.1504/IJGCRSIS.2014.068032 doi: 10.1504/IJGCRSIS.2014.068032
![]() |
[12] |
A. S. Nawar, M. K. El-Bably, A. E. F. El-Atik, Certain types of coverings based rough sets with application, J. Intell. Fuzzy Syst., 39 (2020), 3085–3098. https://doi.org/10.3233/JIFS-191542 doi: 10.3233/JIFS-191542
![]() |
[13] |
M. E. A. El-Monsef, A. M. Kozae, M. K. El-Bably, On generalizing covering approximation space, J. Egypt Math. Soc., 23 (2015), 535–545. https://doi.org/10.1016/j.joems.2014.12.007 doi: 10.1016/j.joems.2014.12.007
![]() |
[14] |
L. W. Ma, On some types of neighborhood-related covering rough sets, Int. J. Approx. Reason., 53 (2012) 901–911. https://doi.org/10.1016/j.ijar.2012.03.004 doi: 10.1016/j.ijar.2012.03.004
![]() |
[15] |
M. Atef, A. M. Khalil, S.-G. Li, A. A. Azzam, A. E. F. El Atik, Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space, J. Intell. Fuzzy Syst., 39 (2020), 4515–4531. https://doi.org/10.3233/JIFS-200482 doi: 10.3233/JIFS-200482
![]() |
[16] |
M. K. El-Bably, T. M. Al-shami, A. S. Nawar, A. Mhemdi, Corrigendum to "Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space", J. Intell. Fuzzy Syst., 41 (2021), 7353–7361. https://doi.org/10.3233/JIFS-211198 doi: 10.3233/JIFS-211198
![]() |
[17] |
M. K. El-Bably, T. M. Al-shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, Int. J. Biomath., 14 (2021), 2150086. https://doi.org/10.1142/S1793524521500868 doi: 10.1142/S1793524521500868
![]() |
[18] |
M. K. El-Bably, E. A. Abo-Tabl, A topological reduction for predicting of a lung cancer disease based on generalized rough sets, J. Intell. Fuzzy Syst., 41 (2021), 3045–3060. https://doi.org/10.3233/JIFS-210167 doi: 10.3233/JIFS-210167
![]() |
[19] |
M. El Sayed, M. A. El Safty, M. K. El-Bably, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Mathematics, 6 (2021), 7872–7894. https://doi.org/10.3934/math.2021457 doi: 10.3934/math.2021457
![]() |
[20] |
M. M. El-Sharkasy, Topological model for recombination of DNA and RNA, Int. J. Biomath., 11 (2018), 1850097. https://doi.org/10.1142/S1793524518500973 doi: 10.1142/S1793524518500973
![]() |
[21] |
M. A. El-Gayar, R. Abu-Gdairi, M. K. El-Bably, D. I. Taher, Economic decision-making using rough topological structures, J. Math., 2023 (2023), 4723233. https://doi.org/10.1155/2023/4723233 doi: 10.1155/2023/4723233
![]() |
[22] |
M. K. El-Bably, M. El-Sayed, Three methods to generalize Pawlak approximations via simply open concepts with economic applications, Soft Comput., 26 (2022), 4685–4700. https://doi.org/10.1007/s00500-022-06816-3 doi: 10.1007/s00500-022-06816-3
![]() |
[23] |
M. K. El-Bably, K. K. Fleifel, O. A. Embaby, Topological approaches to rough approximations based on closure operators, Granul. Comput., 7 (2022), 1–14. https://doi.org/10.1007/s41066-020-00247-x doi: 10.1007/s41066-020-00247-x
![]() |
[24] |
M. I. Ali, M. K. El-Bably, E. A. Abo-Tabl, Topological approach to generalized soft rough sets via near concepts, Soft Comput., 26 (2022), 499–509. https://doi.org/10.1007/s00500-021-06456-z doi: 10.1007/s00500-021-06456-z
![]() |
[25] |
M. K. El-Bably, R. Abu-Gdairi, M. A. El-Gayar, Medical diagnosis for the problem of Chikungunya disease using soft rough sets, AIMS Mathematics, 8 (2023), 9082–9105. https://doi.org/10.3934/math.2023455 doi: 10.3934/math.2023455
![]() |
[26] |
M. K. El-Bably, A. A. El Atik, Soft β-rough sets and their application to determine COVID-19, Turkish J. Math., 45 (2021), 1133–1148. https://doi.org/10.3906/mat-2008-93 doi: 10.3906/mat-2008-93
![]() |
[27] |
M. K. El-Bably, M. I. Ali, E. A. Abo-Tabl, New topological approaches to generalized soft rough approximations with medical applications, J. Math., 2021 (2021), 2559495. https://doi.org/10.1155/2021/2559495 doi: 10.1155/2021/2559495
![]() |
[28] |
M. A. El-Gayar, A. E. F. El Atik, Topological models of rough sets and decision making of COVID-19, Complexity, 2022 (2022), 2989236. https://doi.org/10.1155/2022/2989236 doi: 10.1155/2022/2989236
![]() |
[29] |
M. E. Abd El-Monsef, M. A. El-Gayar, R. M. Aqeel, On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 257–271. https://doi.org/10.1504/IJGCRSIS.2014.068022 doi: 10.1504/IJGCRSIS.2014.068022
![]() |
[30] |
M. E. Abd El Monsef, M. A. El-Gayar, R. M. Aqeel, A comparison of three types of rough fuzzy sets based on two universal sets, Int. J. Mach. Learn. Cyber., 8 (2017), 343–353. https://doi.org/10.1007/s13042-015-0327-8 doi: 10.1007/s13042-015-0327-8
![]() |
[31] |
M. Hosny, Idealization of j-approximation spaces, Filomat, 34 (2020), 287–301. https://doi.org/10.2298/FIL2002287H doi: 10.2298/FIL2002287H
![]() |
[32] | R. A. Hosny, M. K. El-Bably, A. S. Nawar, Some modifications and improvements to idealization of j-approximation spaces, J. Adv. Stud. Topol., 12 (2022), 1–7. |
[33] |
M. Kondo, W. A. Dudek, Topological structures of rough sets induced by equivalence relations, J. Adv. Computat. Intell. Intell. Inform., 10 (2006), 621–624. https://doi.org/10.20965/JACIII.2006.P0621 doi: 10.20965/JACIII.2006.P0621
![]() |
[34] |
W. Zhu, Topological approaches to covering rough sets, Inform. Sci., 177 (2007), 1499–1508. https://doi.org/10.1016/j.ins.2006.06.009 doi: 10.1016/j.ins.2006.06.009
![]() |
[35] |
M. E. Abd El-Monsef, A. M. Kozae, M. K. El-Bably, Generalized covering approximation space and near concepts with some applications, Appl. Comput. Inform., 12 (2016), 51–69. https://doi.org/10.1016/j.aci.2015.02.001 doi: 10.1016/j.aci.2015.02.001
![]() |
[36] |
Z. A. Ameen, R. A. Mohammed, T. M. Al-shami, B. A. Asaad, Novel fuzzy topologies formed by fuzzy primal frameworks, J. Intell. Fuzzy Syst., 2024, 1–10. https://doi.org/10.3233/JIFS-238408 doi: 10.3233/JIFS-238408
![]() |
[37] |
Z. A. Ameen, R. Abu-Gdairi, T. M. Al-shami, B. A. Asaad, M. Arar, Further properties of soft somewhere dense continuous functions and soft Baire spaces, J. Math. Comput. Sci., 32 (2023), 54–63. http://dx.doi.org/10.22436/jmcs.032.01.05 doi: 10.22436/jmcs.032.01.05
![]() |
[38] |
L. Ma, K. Jabeen, W. Karamti, K. Ullah, Q. Khan, H. Garg, et al., Aczel-Alsina power bonferroni aggregation operators for picture fuzzy information and decision analysis, Complex Intell. Syst., 10 (2024), 3329–3352. https://doi.org/10.1007/s40747-023-01287-x doi: 10.1007/s40747-023-01287-x
![]() |
[39] |
C. Zhang, D. Li, J. Liang, Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes, Inform. Sci., 507 (2020), 665–683. https://doi.org/10.1016/j.ins.2019.01.033 doi: 10.1016/j.ins.2019.01.033
![]() |
[40] |
P. Sivaprakasam, M. Angamuthu, Generalized Z-fuzzy soft β-covering based rough matrices and its application to MAGDM problem based on AHP method, Decis. Mak. Appl. Manag. Engrg., 6 (2023), 134–152. https://doi.org/10.31181/dmame04012023p doi: 10.31181/dmame04012023p
![]() |
[41] |
J. S. M. Donbosco, D. Ganesan, The Energy of rough neutrosophic matrix and its application to MCDM problem for selecting the best building construction site, Decis. Mak. Appl. Manag. Engrg., 5 (2022), 30–45. https://doi.org/10.31181/dmame0305102022d doi: 10.31181/dmame0305102022d
![]() |
[42] |
K. Y. Shen, Exploring the relationship between financial performance indicators, ESG, and stock price returns: A rough set-based bipolar approach, Decis. Mak. Adv., 2 (2024), 186–198. https://doi.org/10.31181/dma21202434 doi: 10.31181/dma21202434
![]() |
[43] |
T. M. Al-shami, An improvement of rough sets' accuracy measure using containment neighborhoods with a medical application, Inform. Sci., 569 (2021), 110–124. https://doi.org/10.1016/j.ins.2021.04.016 doi: 10.1016/j.ins.2021.04.016
![]() |
[44] |
T. M. Al-shami, D. Ciucci, Subset neighborhood rough sets, Knowledge Based Syst., 237 (2022), 107868. https://doi.org/10.1016/j.knosys.2021.107868 doi: 10.1016/j.knosys.2021.107868
![]() |
[45] |
R. Abu-Gdairi, M. K. El-Bably, The accurate diagnosis for COVID-19 variants using nearly initial-rough sets, Heliyon, 10 (2024), e31288. https://doi.org/10.1016/j.heliyon.2024.e31288 doi: 10.1016/j.heliyon.2024.e31288
![]() |
[46] | K. Dickstein, A. Cohen-Solal, G. Filippatos, J. J. V. Mcmurray, Developed in collaboration with the heart failure association of the ESC (HFA) and endorsed by the european society of intensive care medicine (ESICM), Eur. J. Heart Failure, 10 (2008), 933–989. |
[47] |
R. A. Hosny, R. Abu-Gdairi, M. K. El-Bably, Enhancing Dengue fever diagnosis with generalized rough sets: Utilizing initial-neighborhoods and ideals, Alexandria Eng. J., 94 (2024), 68–79. https://doi.org/10.1016/j.aej.2024.03.028 doi: 10.1016/j.aej.2024.03.028
![]() |
[48] |
O. Dalkılıç, N. Demirta, Algorithms for Covid-19 outbreak using soft set theory: Estimation and application, Soft Comput., 27 (2022), 3203–3211. https://doi.org/10.1007/s00500-022-07519-5 doi: 10.1007/s00500-022-07519-5
![]() |
[49] |
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
![]() |
[50] |
D. A. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
![]() |
[51] |
F. Feng, X. Liu, V. Leoreanu-Fotea, Y. B. Jun, Soft sets and soft rough sets, Inform. Sci., 181 (2011), 1125–1137. https://doi.org/10.1016/j.ins.2010.11.004 doi: 10.1016/j.ins.2010.11.004
![]() |
[52] |
R. Abu-Gdairi, A. A. Nasef, M. A. El-Gayar, M. K. El-Bably, On fuzzy point applications of fuzzy topological spaces, Int. J. Fuzz. Log. Intell, Syst., 23 (2023), 162–172. https://doi.org/10.5391/IJFIS.2023.23.2.162 doi: 10.5391/IJFIS.2023.23.2.162
![]() |
[53] |
I. M. Taha, Some new results on fuzzy soft r-minimal spaces, AIMS Mathematics, 7 (2022), 12458–12470. https://doi.org/10.3934/math.2022691 doi: 10.3934/math.2022691
![]() |
[54] |
I. M. Taha, Some new separation axioms in fuzzy soft topological spaces, Filomat, 35 (2021), 1775–1783. https://doi.org/10.2298/FIL2106775T doi: 10.2298/FIL2106775T
![]() |
[55] |
A. S. Nawar, M. A. El-Gayar, M. K. El-Bably, R. A. Hosny, θβ-ideal approximation spaces and their applications, AIMS Mathematics, 7 (2022), 2479–2497. https://doi.org/10.3934/math.2022139 doi: 10.3934/math.2022139
![]() |
[56] |
R. B. Esmaeel, M. O. Mustafa, On nano topological spaces with grill-generalized open and closed sets, AIP Conf. Proc., 2414 (2023), 040036. https://doi.org/10.1063/5.0117062 doi: 10.1063/5.0117062
![]() |
[57] |
R. B. Esmaeel, N. M. Shahadhuh, On grill-semi-P-separation axioms, AIP Conf. Proc., 2414 (2023), 040077. https://doi.org/10.1063/5.0117064 doi: 10.1063/5.0117064
![]() |
[58] |
R. Abu-Gdairi, A. A. El Atik, M. K. El-Bably, Topological visualization and graph analysis of rough sets via neighborhoods: A medical application using human heart data, AIMS Mathematics, 8 (2023), 26945–26967. https://doi.org/10.3934/math.20231379 doi: 10.3934/math.20231379
![]() |
[59] |
M. A. El-Gayar, R. Abu-Gdairi, Extension of topological structures using lattices and rough sets, AIMS Mathematics, 9 (2024), 7552–7569. https://doi.org/10.3934/math.2024366 doi: 10.3934/math.2024366
![]() |
[60] |
H. Lu, A. M. Khalil, W. Alharbi, M. A. El-Gayar, A new type of generalized picture fuzzy soft set and its application in decision making, Intell. Fuzzy Systs., 40 (2021), 12459–12475. https://doi.org/10.3233/JIFS-201706 doi: 10.3233/JIFS-201706
![]() |
1. | Guo-Cheng Wu, Babak Shiri, Qin Fan, Hua-Rong Feng, Terminal Value Problems of Non-homogeneous Fractional Linear Systems with General Memory Kernels, 2022, 1776-0852, 10.1007/s44198-022-00085-2 | |
2. | Kamal Shah, Thabet Abdeljawad, Study of a mathematical model of COVID-19 outbreak using some advanced analysis, 2022, 1745-5030, 1, 10.1080/17455030.2022.2149890 | |
3. | Li Tian, Ziqiang Wang, Junying Cao, A high-order numerical scheme for right Caputo fractional differential equations with uniform accuracy, 2022, 30, 2688-1594, 3825, 10.3934/era.2022195 | |
4. | Mohamed Obeid, Mohamed A. Abd El Salam, Jihad A. Younis, Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials, 2023, 31, 2769-0911, 10.1080/27690911.2023.2187388 | |
5. | Min Wang, Yeliz Karaca, Mati ur Rahman, Mi Zhou, A theoretical view of existence results by using fixed point theory for quasi-variational inequalities, 2023, 31, 2769-0911, 10.1080/27690911.2023.2167990 | |
6. | F. Mohammadizadeh, S.G. Georgiev, G. Rozza, E. Tohidi, S. Shateyi, Numerical solution of ψ-Hilfer fractional Black–Scholes equations via space–time spectral collocation method, 2023, 71, 11100168, 131, 10.1016/j.aej.2023.03.007 | |
7. | Tongke Wang, Sijing Liu, Zhiyue Zhang, Singular expansions and collocation methods for generalized Abel integral equations, 2023, 03770427, 115240, 10.1016/j.cam.2023.115240 | |
8. | Hamid Reza Marzban, Atiyeh Nezami, A hybrid of the fractional Vieta–Lucas functions and its application in constrained fractional optimal control systems containing delay, 2024, 1077-5463, 10.1177/10775463241273027 | |
9. | Hasib Khan, Jehad Alzabut, J.F. Gómez-Aguilar, Praveen Agarwal, Piecewise mABC fractional derivative with an application, 2023, 8, 2473-6988, 24345, 10.3934/math.20231241 | |
10. | Wadhah Al‐sadi, Zhouchao Wei, Tariq Q. S. Abdullah, Abdulwasea Alkhazzan, J. F. Gómez‐Aguilar, Dynamical and numerical analysis of the hepatitis B virus treatment model through fractal–fractional derivative, 2024, 0170-4214, 10.1002/mma.10348 | |
11. | Ruiqing Shi, Yihong Zhang, Stability analysis and Hopf bifurcation of a fractional order HIV model with saturated incidence rate and time delay, 2024, 108, 11100168, 70, 10.1016/j.aej.2024.07.059 | |
12. | Babak Shiri, Yong-Guo Shi, Dumitru Baleanu, The Well-Posedness of Incommensurate FDEs in the Space of Continuous Functions, 2024, 16, 2073-8994, 1058, 10.3390/sym16081058 | |
13. | A. M. Kawala, H. K. Abdelaziz, A hybrid technique based on Lucas polynomials for solving fractional diffusion partial differential equation, 2023, 9, 2296-9020, 1271, 10.1007/s41808-023-00246-4 | |
14. | Hui Fu, Wei Xie, Yonggui Kao, Adaptive sliding mode control for uncertain general fractional chaotic systems based on general Lyapunov stability, 2024, 05779073, 10.1016/j.cjph.2024.05.032 | |
15. | Bashir Ahmad, Ahmed Alsaedi, Areej S. Aljahdali, Sotiris K. Ntouyas, A study of coupled nonlinear generalized fractional differential equations with coupled nonlocal multipoint Riemann-Stieltjes and generalized fractional integral boundary conditions, 2023, 9, 2473-6988, 1576, 10.3934/math.2024078 | |
16. | Bakhtawar Pervaiz, Akbar Zada, Ioan‐Lucian Popa, Sana Ben Moussa, Hala H. Abd El‐Gawad, Analysis of fractional integro causal evolution impulsive systems on time scales, 2023, 46, 0170-4214, 15226, 10.1002/mma.9374 | |
17. | Chengdai Huang, Lei Fu, Shuang Liu, Jinde Cao, Mahmoud Abdel‐Aty, Heng Liu, Dynamical bifurcations in a delayed fractional‐order neural network involving neutral terms, 2024, 0170-4214, 10.1002/mma.10434 | |
18. | Leyla Soudani, Abdelkader Amara, Khaled Zennir, Junaid Ahmad, Duality of fractional derivatives: On a hybrid and non-hybrid inclusion problem, 2024, 0928-0219, 10.1515/jiip-2023-0098 | |
19. | Alireza Khabiri, Ali Asgari, Reza Taghipour, Mohsen Bozorgnasab, Ahmad Aftabi-Sani, Hossein Jafari, Analysis of fractional Euler-Bernoulli bending beams using Green’s function method, 2024, 106, 11100168, 312, 10.1016/j.aej.2024.07.023 | |
20. | Hassen Arfaoui, Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative, 2024, 47, 0170-4214, 10490, 10.1002/mma.10135 | |
21. | Lai Van Phut, Finite-time stability analysis of fractional fuzzy differential equations with time-varying delay involving the generalized Caputo fractional derivative, 2024, 35, 1012-9405, 10.1007/s13370-024-01201-9 | |
22. | Najat Almutairi, Sayed Saber, Hijaz Ahmad, The fractal-fractional Atangana-Baleanu operator for pneumonia disease: stability, statistical and numerical analyses, 2023, 8, 2473-6988, 29382, 10.3934/math.20231504 | |
23. | Yuequn Gao, Ning Li, Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays, 2023, 83, 11100168, 1, 10.1016/j.aej.2023.10.032 | |
24. | Saud Fahad Aldosary, Mohamed M. A. Metwali, Manochehr Kazemi, Ateq Alsaadi, On integrable and approximate solutions for Hadamard fractional quadratic integral equations, 2024, 9, 2473-6988, 5746, 10.3934/math.2024279 | |
25. | Farzaneh Safari, Juan J. Nieto, Numerical analysis with a class of trigonometric functions for nonlinear time fractional Wu-Zhang system, 2024, 86, 11100168, 194, 10.1016/j.aej.2023.11.065 | |
26. | Shabir Ahmad, Aman Ullah, Abd Ullah, Ngo Van Hoa, Fuzzy natural transform method for solving fuzzy differential equations, 2023, 27, 1432-7643, 8611, 10.1007/s00500-023-08194-w | |
27. | Zhiyao Ma, Ke Sun, Shaocheng Tong, Adaptive asymptotic tracking control of uncertain fractional-order nonlinear systems with unknown control coefficients and actuator faults, 2024, 182, 09600779, 114737, 10.1016/j.chaos.2024.114737 | |
28. | B. Radhakrishnan, T. Sathya, M. A. Alqudah, W. Shatanawi, T. Abdeljawad, Existence Results for Nonlinear Hilfer Pantograph Fractional Integrodifferential Equations, 2024, 23, 1575-5460, 10.1007/s12346-024-01069-x | |
29. | Mohammad Tavazoei, Mohammad Saleh Tavazoei, Fractional systems with commensurate orders inherit the monotonicity of magnitude-frequency response from their integer-order counterparts, 2024, 1077-5463, 10.1177/10775463241303209 | |
30. | Prakash Raj Murugadoss, Venkatesh Ambalarajan, Vinoth Sivakumar, Prasantha Bharathi Dhandapani, Dumitru Baleanu, Analysis of Dengue Transmission Dynamic Model by Stability and Hopf Bifurcation with Two-Time Delays, 2023, 28, 2768-6701, 10.31083/j.fbl2806117 | |
31. | Shazia Sadiq, Mujeeb ur Rehman, Solution of fractional Sturm–Liouville problems by generalized polynomials, 2024, 0264-4401, 10.1108/EC-04-2024-0356 | |
32. | Mingfang Lin, Zhonghui Ou, The analytical method of two-term time-fractional advection–dispersion–reaction models with sorption process, 2025, 114, 11100168, 702, 10.1016/j.aej.2024.11.112 | |
33. | Mohamed Rhaima, Lassaad Mchiri, Abdellatif Ben Makhlouf, Existence, Uniqueness, and Averaging Principle for a Class of Fractional Neutral Itô–Doob Stochastic Differential Equations, 2025, 0170-4214, 10.1002/mma.10850 | |
34. | Xin Chen, Bang‐Bang and Linear Quadratic Zero‐Sum Game for Noncausal and Causal Systems, 2025, 0170-4214, 10.1002/mma.10892 | |
35. | Md. Samshad Hussain Ansari, Muslim Malik, Controllability of multi-term fractional-order impulsive dynamical systems with φ-Caputo fractional derivative, 2025, 1311-0454, 10.1007/s13540-025-00393-6 |
Category type | Value of parameters | The bound |
I | no parameter | 20.6350 |
II | no parameter | 3.3678e+08 |
III | λ=0.1 | 11.2124 |
III | λ=1 | 3.1070 |
III | λ=2 | 1.9214 |
IV | ρ=0.1 | 1.3997e+13 |
IV | ρ=0.5 | 425.8013 |
IV | ρ=10 | 1.3535 |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |
Category type | Value of parameters | The bound |
I | no parameter | 20.6350 |
II | no parameter | 3.3678e+08 |
III | λ=0.1 | 11.2124 |
III | λ=1 | 3.1070 |
III | λ=2 | 1.9214 |
IV | ρ=0.1 | 1.3997e+13 |
IV | ρ=0.5 | 425.8013 |
IV | ρ=10 | 1.3535 |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |
Category type | E32 | E64 | E128 | log2E32/E64 | log2E64/E128 |
I | 0.0313 | 0.0156 | 0.0079 | 1.0000 | 0.9773 |
II | 0.0312 | 0.0156 | 0.0079 | 0.9997 | 0.9772 |
III, λ=0.1 | 0.0353 | 0.0177 | 0.0090 | 0.9981 | 0.9771 |
IV, ρ=0.1 | 0.0099 | 0.0049 | 0.0025 | 0.9998 | 0.9772 |