Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
-
Sobia Sultana
Abstract
Fractional stochastic differential equations (FSDEs) with fractional derivatives describe the anomalous diffusion processes by incorporating the memory effects and spatial heterogeneities of the porous medium. The stochastic component addresses the random nature of the fluid flow due to the variability in pore sizes and connectivity. The first objective of this research is to prove the well-posedness of a class of generalized proportional FSDEs, and we acquire the global existence and uniqueness of findings under certain settings that are coherent with the classic SDEs. The secondary purpose is to evaluate the continuity of findings in fractional-order formulations. The Carathéodory approximation is taken into account for a class of generalized proportional FSDEs, which is pivotal and provides well-known bounds on the norm of the solutions. Carathéodory’s approximation aids in approximating the FSDEs governing turbulent flows, ensuring the solutions are mathematically robust and physically meaningful. As is widely documented, the existence and uniqueness of solutions to certain types of differential equations can be formed under Lipschitz and linear growth conditions. Furthermore, a class of generalized proportional FSDEs with time delays is considered according to certain new requirements. With the aid of well-known inequalities and Itô isometry technique, the Ulam–Hyers stability of the analyzed framework is addressed utilizing Lipschitz and non-Lipschitz characteristics, respectively. Additionally, we provide two illustrative examples as applications to demonstrate the authenticity of our interpretations. The demonstrated outcomes will generalize some previously published findings. Finally, this deviation from fractional Brownian motion necessitates a model that can capture the subdiffusive or superdiffusive behavior.
1 Introduction
Fractional calculus is very pertinent in meaningful manifestations due to certain similar characteristics, including memory. Numerous sorts of kernels are used in fractional formulations [1,2]. Kochubei [3] investigated a very particular form of kernel known as a general fractional integral/derivative. Luchko [4] examined such generally designed integrals and derivatives in adequate functional spaces within the structure of applied mathematics. Luchko also probed the quantifiable attributes of multiple sorts of differential equations (DEs) solutions with broad sense fractional derivatives [5–7]. Several researchers have examined the stability and performance of fractional DEs (FDEs) [8,9]. For a framework to be reliable in the context of Lyapunov, its generalized vitality is not required to decay exponentially, and previously, the Mittag–Leffler (M–L) strength and the fractional Lyapunov communicative approach for multiple kinds of FDEs were initiated [10,11].
Recently, FDEs have grown in importance in both theoretical and practical aspects, attracting a considerable amount of interest from academics [12–14]. Numerous researchers concentrate on aspects of stochastic DEs (SDEs) including solution existence and uniqueness (E-U), continuous reliance of strategies on initialization and complex behavior [15–18]. Moreover, no scholars are interested in the continuity of findings on the fractional scaling factor of this type of formula; specifically, none of them are curious about the interaction between the alternatives of classical SDEs and the fractional ones. In this work, we continued our investigation of the proportional fractional derivatives and integrals revealed in the study by Jarad et al. [19], which was guided by the previously mentioned investigations. We show how fractional integral operators affect differential operators, and vice versa. Furthermore, this research examined this intriguing topic of SDEs and revealed the connection among them. If the fractional order and proportional index of the generalized proportional fractional (GPF) derivative converge to one, the findings of GPF SDEs reduces to the solutions of Caputo fractional SDEs and classic SDEs, respectively.
FDEs are an essential mechanism in many disciplines of scientific domains because of their non-local feature of GPF derivatives in time [20,21]. The kernel impact or nonlocal feature is represented by a convolution integral with an exponential function memory kernel that also gives the GPF-DEs a greater capability in dynamic structures; this is among the primary strengths of FDEs (non-local) in correlation with traditional (local) configurations [22–24]. The analysis of revolutionary calculus and FDEs is becoming increasingly popular (see [25–27], as well as the reference materials therein). In fluid physics, stochastic FDEs incorporating Carathéodory’s approximation offer a sophisticated framework for modeling complex phenomena such as turbulence, anomalous diffusion, viscoelastic fluid behavior, and particle dynamics. By ensuring the existence, uniqueness, and stability of solutions, Carathéodory’s approximation makes it feasible to apply these advanced mathematical tools to real-world fluid dynamics problems, leading to deeper insights and more accurate predictions. Moreover, stochastic FDEs tracking the movement of individual fluid particles (the Lagrangian approach) provide insights into mixing, dispersion, and turbulence. The random motion of particles is influenced by both the deterministic flow field and stochastic environmental factors for Lagrangian stochastic models of fluid particles.
However, because noisy perturbations are prevalent in complicated processes, stochastic modeling has played a significant role in a variety of research and industry. Plenty of efforts on SDEs or SPDEs have been made, with numerous scientific preferences having similar properties such as well-posedness, continuity, variability, transformed manifolds, and irreducible measure. Conga et al. [28] investigated the existence of reliable configurations for a particular class of stochastic differential equations. Xu et al. [29] presented the approximation concept for SDEs via the Caputo fractional derivative. Wang et al. [30] contemplated the asymptotic dynamics of stochastic lattice frameworks involving Caputo fractional time derivatives. Doan et al. [31] expounded a spatial and temporal weighted norm that is employed to evaluate the asymptotic distance between two distinguishable strategies. It is interesting to note that Doan et al. [32] founded the Euler–Maruyama form estimation outcome for Caputo FSDEs. Carahéodory initially regarded the Carathéodory approximate solution framework for ordinary DEs, and then Bell, Mohammad, and Mao lengthened it to include the specific instances of SDEs [33]. Wang et al. [34] examined the continuity, and Guo et al. [35] applied Carathéodory’s approximation for a Caputo-type FSDES.
In a presentation at Wisconsin University in 1940, Ulam suggested the reliability of systems of equations [36]. In 1941, Hyers [37] became the best person to respond to the inquiry. The Ulam–Hyers stability (U–Hs) was created as a result. In the meantime, an expanding number of individuals have been eager to look into the U–Hs. The characteristics of canonical and generalized M-L functions, as well as the U–Hs of sequential FDEs, had been demonstrated using fractional calculus and the Laplace transform technique in the study by Wang and Li [38]. Researchers examined the U–Hs, generalized U–Hs, U–H–Rassias stability, and U–H–Rassias stability of impulsive integrodifferential formulations incorporating Riemann–Liouville boundary assumptions in the study by Zada et al. [39]. For further investigation on U–Hs, we refer the readers to previous studies [40–42] and references cited therein.
To the highest potential of our expertise, no research has been conducted that focuses on the continuity of the findings of the GPF-SDEs in respect of fractional order, if it tends to the solution of the Caputo FSDEs and classical SDEs when the
Motivated by the aforesaid proclivity, in this work, we will investigate the existence and U–Hs of time-delayed generalized proportional FSDEs:
where
In comparison with the previous scientific studies [34,35], the significant achievements of this article encompass at least three components:
In this article, we only examine the scenario of
The methodologies we utilize to determine the E-U of generalized proportional FSDE solutions are quite revolutionary than those of previous studies [34, 35]. Krasnoselskii’s and Mönch’s fixed-point hypothesis were employed in previous studies [29, 30] to investigate the E-U. However, we use Carathéodory’s approximation in this article to examine the E-U.
Several previous studies [43,44] have employed a stronger Lipschitz assumption in the investigation of multiple stabilities, E-U of FSDEs. Even so, in this article, we discussed the U-Hs of GPF-SDEs via weak non-Lipschitz assumptions. This is a significant step forward in the investigation of the stability of GPF-SDEs.
2 Preliminaries
This portion outlines several terminologies, interpretations, and key formulas that will be employed throughout the remainder of the article. The viewer can refer to the article [19] for its explanation and verification.
Abdeljawad [45] and Khalil et al. [46] provided the following limit form description of the well-known conformable derivative:
It is self-evident that when a mapping
The major disadvantage of this derivative is that the mapping
Definition 2.1
[47] For
and
We recommend the viewer refer to previous studies [47,48] for more information on the control theory of the proportional derivative and its constituent mappings
It is simple to understand that
The integral and derivative of the GPF are described in the following.
Definition 2.2
[19] For
Definition 2.3
[19] For
Definition 2.4
[19] Let
is the left-sided GPF derivative of the function
By letting
Furthermore, we implement the respective approaches to ensure the solution’s global E-U. Assume that
(
(
To cope with FDEs, we require the following generalized Gronwall’s lemma for exponential function-type kernel (see [49]) as follows:
Lemma 2.1
For
then
where
3 Main results
In this section, we will present the well-posedness and the continuity of the solution of (1.1).
3.1 Well-posedness
First, we surmise the E-U of mild solutions for the subsequent equation using the aforementioned hypotheses
where
Definition 3.1
Suppose there is an
Theorem 3.1
Under the suppositions
Proof
The proof of the theorem will be classified into three cases.
Case I: Here, we intend to consider the simplistic form of arithmetic variant
We have
In view of Cauchy–Schwarz inequality and assumption
since
For
Thus, we have
letting
and
Using the fact of Lemma 2.1 yields
where
which predicts
Case II. In this case, we will illustrate that
Furthermore, we intend to prove that
For
Again, using the fact of the Cauchy–Schwarz inequality, we provide a bound to
It is not challenging to illustrate that
using the fact that
As a result of the foregoing analysis for
Case III. Here, we employ the Banach
Cases I and II demonstrate that the function
Therefore,
Choosing
using the fact that
For
If we make the assumption that (3.15) is true for
Using the fact that
Merging the aforesaid identity with (3.17), then we have
Then, for all
Now, simply to demonstrate the respective evidence to apply the Banach fixed point theorem and deduce the E-U of solutions, for this, we have
Then, all that remains is to demonstrate
Moreover, employing the D’Alembert discriminant technique, we simply have to prove the foregoing:
After simplification,
It is worth noting that when
Therefore,
For sufficiently large
Theorem 3.2
Suppose the hypothesis of Theorem
3.1
satisfies. Then, for each
where
Proof
Considering inequality (3.3) and implementing on the following expression:
Adopting the same technique of Theorem 3.1 and Lemma 2.1, we can establish the proof. Therefore, we omitted the details.□
3.2 Continuity via GPF
Here, we contemplate the continuity of the system of (1.1) on the GPF order for SDEs, i.e., when
and
Specifically, letting
Theorem 3.3
Under the hypothesis
Proof
By means of inequality (3.3), we have
Taking the following bound for
where we have used the fact that
For
where
Mingling the bounds of
Using the fact that
as
3.3 Carathéodory’s approximation
Here, the CA for SDE is discussed in this portion. We attempt to define Carathéodory’s approximate findings for SFDEs in the same way that we did for SDEs. For each integer
It is worth noting that
then for
and henceforth. Using this method, we can evaluate
Lemma 3.1
Under the hypothesis
where
Proof
By means of the hypothesis described in (3.3), we have
In view of Cauchy–Schwarz variant and supposition
Using the fact of
Taking into account (3.4) and (3.5), then using the terms
Observe that, for
Therefore, we have
Utilizing the fact that
which is the desired result.□
Lemma 3.2
Under the supposition
Proof
By the given hypothesis, we observe that
For
By means of the Cauchy–Schwarz inequality,
where
As a consequence, for
For
Mingling all the concluding mathematical expressions yields
This yields the desired outcome.□
Theorem 3.4
Under the hypothesis of
Proof
Observe that
So, applying the well-known inequality defined in (3.3), we have
For
Taking into consideration the Cauchy–Schwarz variant and assumption
Analogously, for
Furthermore, we can categorize
Utilizing the Itô’s isometry formulation, we have
and
Incorporating the assumptions for
Utilizing Lemma 3.2, if
On the other hand, if
From (3.25), we have
In view of Lemma 2.1, we find
This yields the desired outcome.□
Some remarkable results of the aforesaid findings are as follows:
Remark 3.1
(i) Letting
(ii) Letting
4 Ulam–Hyers stability
Definition 4.1
Suppose there is
For
(4.1)where
For every other solution
Definition 4.2
Assume that system (1.2) is U-Hs if
and
Remark 4.1
Suppose there is a mapping
where
where
(b) For each
where
In order to find the solution
Theorem 4.1
Under assumption
Proof
By means of Definition 4.1 and Remark 4.1, we write
In view of Definition 4.1 and utilizing (4.8), we have
Making use of Jensen’s variant, we find
Employing the Hölder inequality and assumption
where we have used the fact that
Now, utilizing the Itô isometry and the Hölder inequality, we find
where we have used the fact that
Under assumption
Thus, applying the Cauchy–Schwarz variant and Remark 4.1 to produce
where
It follows that
In contrast to the methodology of interacting with the delay [50,51], we acquire
also,
and then we attain
So that
Define a set
For
Therefore, we find
Thus,
According to Lemma 2.1, we find
Clearly, we see that
Finally, for every
As a result, this theorem is established.□
Theorem 4.2
Under the supposition
Proof
By means of the variant 4.9, we find
Attempting to take variant (4.9) and supposition
where
Combining (4.10)–(4.12) and assumption
It is simple to achieve employing supposition
Utilizing the variant (4.12), we have
Then,
Define a set
Evidently, we can draw the conclusion that
Choosing
For every
Furthermore, we have
Also,
According to Lemma 2.1, we find
Clearly, we see that
Finally, for every
As a result, this theorem is established.□
5 Examples
In this section, we will present illustrative examples for the previous findings.
Example 5.1
Assume the U-Hs and E-U of the result to the subsequent equation:
where
Now, we have
Thus, we have
Analogously, we have
Finally, we will perform a graphical illustration to ensure that the finding of (5.2) is U-H stable, as shown in Figure 1.
![Figure 1
X
(
0
)
=
1
{\bf{X}}\left(0)=1
,
ℏ
=
ε
1
⁄
2
\hslash ={\varepsilon }^{1/2}
,
E
(
sup
ζ
∈
[
0
,
8
]
‖
ℏ
(
ζ
)
‖
2
)
{\bf{E}}\left({\sup }_{\zeta \in \left[0,8]}\Vert \hslash \left(\zeta ){\Vert }^{2})
=
E
(
sup
ζ
∈
[
0
,
8
]
‖
ε
1
⁄
2
‖
2
)
{\bf{E}}\left({\sup }_{\zeta \in \left[0,8]}\Vert {\varepsilon }^{1/2}{\Vert }^{2})
= 0.001.](/document/doi/10.1515/phys-2024-0074/asset/graphic/j_phys-2024-0074_fig_001.jpg)
Example 5.2
Assume that U-H stability of the subsequent FSDEs having time delays
where
Now, we have
evidently,
Finally, we will perform a graphical illustration to ensure that the finding of (5.3) is U-Hs, as shown in Figure 2.
![Figure 2
X
(
0
)
=
3
{\bf{X}}\left(0)=3
,
ℏ
=
ε
1
⁄
2
⁄
2
\hslash ={\varepsilon }^{1/2}/2
,
E
(
sup
ζ
∈
[
0
,
8
]
‖
ℏ
(
ζ
)
‖
2
)
{\bf{E}}\left({\sup }_{\zeta \in \left[0,8]}\Vert \hslash \left(\zeta ){\Vert }^{2})
=
E
(
sup
ζ
∈
[
0
,
8
]
‖
ε
1
⁄
2
⁄
2
‖
2
)
{\bf{E}}\left({\sup }_{\zeta \in \left[0,8]}\Vert {\varepsilon }^{1/2}/2{\Vert }^{2})
= 0.007.](/document/doi/10.1515/phys-2024-0074/asset/graphic/j_phys-2024-0074_fig_002.jpg)
6 Conclusion
Carathéodory’s approximation has helped approximate the fractional derivatives, ensuring that the resulting FSDEs are mathematically well posed. It has provided a framework for analyzing the stability and uniqueness of solutions, which is crucial for understanding the fluid’s behavior under various conditions, especially in Lagrangian stochastic models of fluid particles.
Figuring out the descriptive characteristics of DEs is one of the most vital aspects of differential equation theory. Integral equations are valuable techniques for investigating such features. In this study, we first identified the well-posedness for generalized proportional FSDEs using various approximate techniques, and the restrictions implanted on the global E-U of the findings are coherent with the Caputo FSDEs and traditional SDEs. Then, we glanced at the continuity of findings with reverence for the fractional-order of such formulae, especially if the proportional index
-
Funding information: The author states no funding involved.
-
Author contribution: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The author states no conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
References
[1] Podlubny I. Fractional differential equations. San Diego CA: Academic Press; 1999. Search in Google Scholar
[2] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. Yverdon: Gordon and Breach; 1993. Search in Google Scholar
[3] Kochubei AN. General fractional calculus, evolution equations, and renewal processes. Integr Equ Oper Theo. 2011;71:583–600. 10.1007/s00020-011-1918-8Search in Google Scholar
[4] Luchko Y. General fractional integrals and derivatives with the Sonine kernels. Mathematics. 2021;9:594. 10.3390/math9060594Search in Google Scholar
[5] Khan H, Chen W, Khan A, Khan TS, Al-Madlal QM. Hyers-Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator. Adv Differ Equ. 2018;2018:1–6. 10.1186/s13662-018-1899-xSearch in Google Scholar
[6] Khan H, Tunç C, Khan A. Stability results and existence theorems for nonlinear delay-fractional differential equations with φp*-operator. J Appl Anal Comput. 2020;10(2):584–97. Search in Google Scholar
[7] Chu YM, Rashid S, Alzahrani T, Alhulayyil H, Alsagri H, Rehman S. Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviors. Sci Rep. 2024;13(1):22447. 10.1038/s41598-023-48873-0Search in Google Scholar PubMed PubMed Central
[8] Chu YM, Rashid S, QUA Asif, Abdalbagi M. On configuring new choatic behaviors for a variable fractional-order memristor-based circuit in terms of Mittag-Leffler kernel. Res Phy. 2023;53(1–4):106939. 10.1016/j.rinp.2023.106939Search in Google Scholar
[9] Khan H, Alzabut J, Shah A, He Z-Y, Etemad S, Rezapour S, Zada A. On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations. Fractals. 2023;31:2340055. 10.1142/S0218348X23400558Search in Google Scholar
[10] Li Y, Chen YQ, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica. 2009;45:2965–9. 10.1016/j.automatica.2009.04.003Search in Google Scholar
[11] Abed-Elhameed TM, Aboelenen T. Mittag-Leffler stability, control, and synchronization for chaotic generalized fractional-order systems. Adv Cont Discr Mod. 2022;2022:50. 10.1186/s13662-022-03721-9Search in Google Scholar
[12] Begum R, Tunç O, Khan H, Gulzar H, Khan A. A fractional order Zika virus model with Mittag-Leffler kernel. Chaos Solit Fract. 2021;146:110898. 10.1016/j.chaos.2021.110898Search in Google Scholar
[13] Chu YM, Alzahrani T, Rashid S, Alhulayyil H, Rashidah W, Rehman S. Complex dynamical analysis of fractional differences Willamowski-Rössler chemical reaction model in time-scale analysis, Res Phy. 2023;54(6):107023. 10.1016/j.rinp.2023.107023Search in Google Scholar
[14] Chu YM, Alzahrani T, Rashid S, Rashidah W, Rehman S, Alkhatib M. An advanced approach for the electrical responses of discrete fractional-order biophysical neural network models and their dynamical responses. Sci Rep. 2023;13(1):18180. 10.1038/s41598-023-45227-8Search in Google Scholar PubMed PubMed Central
[15] Tajadodi H, Khan A, Gómez-Aguilar JF, Khan H. Optimal control problems with Atangana-Baleanu fractional derivative. Optimal Cont Appl Meth. 2021;42(1):96–109. 10.1002/oca.2664Search in Google Scholar
[16] Khan H, Abdeljawad T, Gómez-Aguilar JF, Tajadodi H, Khan A. Fractional order Volterra integro-differential equation with Mittag-Leffler kernel. Fractals. 2021;29(6):2150154. 10.1142/S0218348X21501541Search in Google Scholar
[17] Shah K, Abdeljawad T, Study of a mathematical model of COVID-19 outbreak using some advanced analysis. Waves Rand Comp Media. 2022;1–18. 10.1080/17455030.2022.2149890Search in Google Scholar
[18] Shah K, Abdeljawad T, Alrabaiah H. On coupled system of drug therapy via piecewise equations. Fractals. 2022:30;2240206. 10.1142/S0218348X2240206XSearch in Google Scholar
[19] Jarad F, Abdeljawad T, Alzabut J, Generalized fractional derivatives generated by a class of local proportional derivatives. Eur Phys J Spec Top. 2017;226(16–18):3457–71. 10.1140/epjst/e2018-00021-7Search in Google Scholar
[20] Sk N, Mondal B, Thirthar AA, Alqudah MA, Abdeljawad T. Bistability and tristability in a deterministic prey-predator model: Transitions and emergent patterns in its stochastic counterpart. Chaos Solit Fract. 2023;176:114073. 10.1016/j.chaos.2023.114073Search in Google Scholar
[21] Gafel HS, Rashid S, Elagan SK. Novel codynamics of the HIV-1/HTLV-I model involving humoral immune response and cellular outbreak: A new approach to probability density functions and fractional operators. AIMS Math. 2023;8(12):28246–79. 10.3934/math.20231446Search in Google Scholar
[22] Rashid S, Hamidi SZ, Akram S, Aon M, Elagan SK, Alsubei SMT. Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional calculus and probabilistic approach. Sci Rep. 2024;14(1):8827. 10.1038/s41598-024-59261-7Search in Google Scholar PubMed
[23] Rashid S, Jarad F. Novel investigation of stochastic fractional differential equations measles model via the white noise and global derivative operator depending on Mittag-Leffler kernel. Comp Model Engin Sci. 2024;139(3):2289–327. 10.32604/cmes.2023.028773Search in Google Scholar
[24] Bouftouh O, Kabbaj S, Abdeljawad T, Khan A. Quasi controlled K metric spaces over C∗ algebras with an application to stochastic integral equations. Comput Model Eng Sci. 2023;135:2649–63. 10.32604/cmes.2023.023496Search in Google Scholar
[25] Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies. Vol. 207. Amsterdam, Netherlands: Elsevier; 2006. Search in Google Scholar
[26] Zhou Y, Wang JR, Zhang L. Basic: theory of fractional differential equations. Singapore: World Scientific; 2014. 10.1142/9069Search in Google Scholar
[27] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Hoboken, NJ, USA: Wiley-Interscience Publication; 1993. Search in Google Scholar
[28] Cong ND, Doan TS, Siegmund S, Tuan T. On stable manifolds for planar fractional differential equations. Appl Math Comput. 2014;226:157–68. 10.1016/j.amc.2013.10.010Search in Google Scholar
[29] Xu W, Xu W, Zhang S. The averaging principle for stochastic differential equations with Caputo fractional derivative. Appl Math Lett. 2019;93:79–84. 10.1016/j.aml.2019.02.005Search in Google Scholar
[30] Wang Y, Xu J, Kloeden P. Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative. Nonlinear Anal Theory Meth Appl. 2016;135:205–22. 10.1016/j.na.2016.01.020Search in Google Scholar
[31] Doan T, Kloeden P, Huong P, Tuan H. Asymptotic separation between solutions of Caputo fractional stochastic differential equations. Stoch Anal Appl. 2018;36:654–64. 10.1080/07362994.2018.1440243Search in Google Scholar
[32] Doan T, Huong P, Kloeden P. Euler-Maruyama scheme for Caputo stochastic fractional differential equations. J Comput Appl Math. 2020;112989:1–26. 10.1016/j.cam.2020.112989Search in Google Scholar
[33] Mao X, Stochastic differential equations and their applications. Chichester: Horwood; 2008. 10.1533/9780857099402Search in Google Scholar
[34] Wang W, Cheng S, Guo Z, Yan X. A note on the continuity for Caputo fractional stochastic differential equations. Chaos. 2020;30:073106. 10.1063/1.5141485Search in Google Scholar PubMed
[35] Guo Z, Hu J, Wang W. Carathéodoryas approximation for a type of Caputo fractional stochastic differential equations. Adv Diff Equ. 2020;2020:636. 10.1186/s13662-020-03020-1Search in Google Scholar
[36] Ulam S. A collection of mathematical problem. New York, NY, USA: Interscience; 1960. Search in Google Scholar
[37] Hyers DH, On the stability of the linear functional equation. Proc Nat Acad Sci. 1941;27:222–4. 10.1073/pnas.27.4.222Search in Google Scholar PubMed PubMed Central
[38] Wang J, Li X. A uniform method to Ulam-Hyers stability for some linear fractional equations, Mediterranean J Math. 2016;13:625–35. 10.1007/s00009-015-0523-5Search in Google Scholar
[39] Zada A, Alzabut J, Waheed H, Ioan-Lucian P. Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions. Adv Diff Equ. 2020;2020;64. 10.1186/s13662-020-2534-1Search in Google Scholar
[40] Luo D, Abdeljawad T, Luo Z. Ulam-Hyers stability results for a novel nonlinear Nabla Caputo fractional variable-order difference system. Turkish J Math. 2021;45:456–70. 10.3906/mat-2008-53Search in Google Scholar
[41] Wang J, Zhang Y. Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations. Optimization. 2014;63:1181–90. 10.1080/02331934.2014.906597Search in Google Scholar
[42] Zada A, Fatima S, Ali Z, Xu J, Cui Y. Stability results for a coupled system of impulsive fractional differential equations. Mathematics. 2019;7:927. 10.3390/math7100927Search in Google Scholar
[43] Luo DF, Shah K, Luo ZG. On the novel Ulam-Hyers stability for a class of nonlinear ψ Hilfer fractional differential equation with time-varying delays. Mediterranean J Math. 2019;16:112. 10.1007/s00009-019-1387-xSearch in Google Scholar
[44] Sathiyaraj T, Wang JR, Balasubramaniam P. Ulams stability of Hilfer fractional stochastic differential systems. European Phy J Plus. 2019;134:605. 10.1140/epjp/i2019-12952-ySearch in Google Scholar
[45] Abdeljawad T. On conformable fractional calculus. J Comput Appl Math. 2015;279:57–66. 10.1016/j.cam.2014.10.016Search in Google Scholar
[46] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70. 10.1016/j.cam.2014.01.002Search in Google Scholar
[47] Anderson DR, Ulness DJ. Newly defined conformable derivatives. Adv Dyn Sys Appl. 2015;10(2):109–37. Search in Google Scholar
[48] Anderson DR. Second-order self-adjoint differential equations using a proportional-derivative controller. Commun Appl Nonlin Anal. 2017;24:17–48. Search in Google Scholar
[49] Alzabut J, Abdeljawad T, Jarad F, Sudsutad W. A Gronwall inequality via the generalized proportional fractional derivative with applications. J Ineq Appl. 2019;2019:101. 10.1186/s13660-019-2052-4Search in Google Scholar
[50] Balasundaram K, Raja R, Zhu Q, Chandrasekaran S, Zhou H. New global asymptotic stability of discrete-time recurrent neural networks with multiple time-varying delays in the leakage term and impulsive effects. Neurocomputing. 2016;214:420–9. 10.1016/j.neucom.2016.06.040Search in Google Scholar
[51] Senthilraj S, Raja R, Jiang F, Zhu Q, Samidurai R. New delay-interval-dependent stability analysis of neutral type BAM neural networks with successive time delay components. Neurocomputing. 2016;171:1265–80. 10.1016/j.neucom.2015.07.060Search in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Regular Articles
- Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
- Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
- High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
- Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
- Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
- Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
- The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
- Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
- A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
- Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
- Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
- Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
- Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
- Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
- Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
- Chaotic control problem of BEC system based on Hartree–Fock mean field theory
- Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
- Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
- Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
- Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
- Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
- Model of conversion of flow from confined to unconfined aquifers with stochastic approach
- Study of fractional variable-order lymphatic filariasis infection model
- Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
- Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
- Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
- Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
- Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
- Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
- Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
- A two-stage framework for predicting the remaining useful life of bearings
- Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
- Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
- Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
- In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
- Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
- Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
- A comparative analysis of shielding effectiveness in glass and concrete containers
- Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
- Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
- Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
- Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
- Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
- Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
- Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
- Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
- Fractional view analytical analysis of generalized regularized long wave equation
- Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
- Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
- Estimation of monotone α-quantile of past lifetime function with application
- Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
- Study of fractional telegraph equation via Shehu homotopy perturbation method
- An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
- Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
- An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
- CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
- Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
- A short report on a probability-based interpretation of quantum mechanics
- Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
- Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
- Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
- Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
- A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
- Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
- Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
- Thermal analysis of extended surfaces using deep neural networks
- Steady-state thermodynamic process in multilayered heterogeneous cylinder
- Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
- Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
- Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
- The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
- Research on optimization of combustor liner structure based on arc-shaped slot hole
- Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
- Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
- Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
- Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
- Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
- Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
- A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
- New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
- Physical aspects of quantile residual lifetime sequence
- Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
- Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
- A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
- Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
- A novel tetra hybrid bio-nanofluid model with stenosed artery
- Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
- Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
- A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
- New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
- Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
- Inflation and the principle of equivalence
- The use of a supercontinuum light source for the characterization of passive fiber optic components
- Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
- A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
- Photon balance in the fiber laser model
- Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
- Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
- Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
- Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
- Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
- On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
- Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
- Review Article
- Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
- Special Issue on Predicting pattern alterations in nature - Part II
- A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
- On the existence and numerical simulation of Cholera epidemic model
- Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
- Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
- Prediction of COVID-19 spread with models in different patterns: A case study of Russia
- Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
- Accumulation process in the environment for a generalized mass transport system
- Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
- Special Issue on Nanomaterial utilization and structural optimization - Part II
- Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
- Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
- Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
- Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
- Special Issue on Nanofluids: Synthesis, Characterization, and Applications
- Exact solutions of a class of generalized nanofluidic models
- Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
- Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
- Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
- Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
- Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
- Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
- Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
- Comparative study of couple stress fluid flow using OHAM and NIM
- Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
- Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
- Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
- Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
- Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
- Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
- Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
- Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
- Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
- Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
- Pulsed excitation of a quantum oscillator: A model accounting for damping
- Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
- Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
- Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
- Effect of the gravitational field strength on the rate of chemical reactions
- The kinetic relativity theory – hiding in plain sight
- Special Issue on Advanced Energy Materials - Part III
- Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
Articles in the same Issue
- Regular Articles
- Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
- Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
- High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
- Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
- Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
- Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
- The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
- Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
- A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
- Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
- Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
- Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
- Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
- Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
- Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
- Chaotic control problem of BEC system based on Hartree–Fock mean field theory
- Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
- Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
- Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
- Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
- Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
- Model of conversion of flow from confined to unconfined aquifers with stochastic approach
- Study of fractional variable-order lymphatic filariasis infection model
- Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
- Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
- Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
- Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
- Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
- Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
- Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
- A two-stage framework for predicting the remaining useful life of bearings
- Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
- Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
- Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
- In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
- Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
- Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
- A comparative analysis of shielding effectiveness in glass and concrete containers
- Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
- Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
- Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
- Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
- Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
- Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
- Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
- Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
- Fractional view analytical analysis of generalized regularized long wave equation
- Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
- Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
- Estimation of monotone α-quantile of past lifetime function with application
- Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
- Study of fractional telegraph equation via Shehu homotopy perturbation method
- An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
- Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
- An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
- CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
- Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
- A short report on a probability-based interpretation of quantum mechanics
- Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
- Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
- Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
- Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
- A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
- Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
- Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
- Thermal analysis of extended surfaces using deep neural networks
- Steady-state thermodynamic process in multilayered heterogeneous cylinder
- Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
- Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
- Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
- The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
- Research on optimization of combustor liner structure based on arc-shaped slot hole
- Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
- Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
- Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
- Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
- Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
- Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
- A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
- New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
- Physical aspects of quantile residual lifetime sequence
- Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
- Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
- A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
- Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
- A novel tetra hybrid bio-nanofluid model with stenosed artery
- Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
- Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
- A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
- New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
- Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
- Inflation and the principle of equivalence
- The use of a supercontinuum light source for the characterization of passive fiber optic components
- Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
- A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
- Photon balance in the fiber laser model
- Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
- Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
- Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
- Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
- Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
- On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
- Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
- Review Article
- Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
- Special Issue on Predicting pattern alterations in nature - Part II
- A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
- On the existence and numerical simulation of Cholera epidemic model
- Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
- Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
- Prediction of COVID-19 spread with models in different patterns: A case study of Russia
- Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
- Accumulation process in the environment for a generalized mass transport system
- Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
- Special Issue on Nanomaterial utilization and structural optimization - Part II
- Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
- Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
- Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
- Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
- Special Issue on Nanofluids: Synthesis, Characterization, and Applications
- Exact solutions of a class of generalized nanofluidic models
- Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
- Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
- Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
- Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
- Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
- Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
- Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
- Comparative study of couple stress fluid flow using OHAM and NIM
- Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
- Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
- Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
- Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
- Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
- Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
- Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
- Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
- Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
- Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
- Pulsed excitation of a quantum oscillator: A model accounting for damping
- Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
- Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
- Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
- Effect of the gravitational field strength on the rate of chemical reactions
- The kinetic relativity theory – hiding in plain sight
- Special Issue on Advanced Energy Materials - Part III
- Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
