A Study of Monotonicity Analysis for the Delta and Nabla Discrete Fractional Operators of the Liouville–Caputo Family

Mohammed, Pshtiwan Othman; Goodrich, Christopher S.; Srivástava, H. M.; Al-Sarairah, Eman; Hamed, Y. S.

doi:10.3390/axioms12020114

Axioms

2023

DOI: 10.3390/axioms12020114

|View full text |Cite

A Study of Monotonicity Analysis for the Delta and Nabla Discrete Fractional Operators of the Liouville–Caputo Family

Pshtiwan Othman Mohammed

Christopher S. Goodrich

H. M. Srivástava

et al.

Abstract: In the present article, we explore the correlation between the sign of a Liouville–Caputo-type difference operator and the monotone behavior of the function upon which the difference operator acts. Finally, an example is also provided to demonstrate the application and the validation of the results which we have proved herein.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Article3

Relationship

Self Cite0

Independent3

Authors

Journals

Cited by 3 publications

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Implementing the finite element numerical approach to optimize thermosolutal convection (TSC) energy imparted by discrete heat and solute sources in generalized Newtonian fluid flow in square enclosure

Bilal,

Ali,

Khan

et al. 2024

J.Umm Al-Qura Univ. Appll. Sci.

View full text Add to dashboard Cite

Mesmerizing utilizations of thermosolutal convection (TSC) have been found in metallurgical and petrochemical procedures, crystal growth, thermal exchangers, nuclear waste disposal, and building environment control. In the current study, we considered three heat and solute sources on the left boundary of the cavity, which raises the significance of studying multiple chemical and civil engineering phenomena. Therefore, the prime motivation to commence this work is to attain the optimization of thermosolutal attributes characterized by non-Newtonian (power–law) liquid in the presence of installed sources. For this purpose, the natural convective flow governed by the nonpersistent impact of buoyancy forces is encompassed in the square enclosure. In addition to the source location, the vertical boundaries are considered to be cold, whereas the horizontal boundaries are kept insulated. Owing to the free convective flow, no compositional and thermal gradients are induced by lid movement, and all walls are prescribed with no-slip constraints. To examine the transport mechanism, structuring of problem is attained in the form of dimensional governing expressions by employing conservation laws. Subsequently, constitutive equations in a dimensionless format are accessed through similar variables. The finite element approach is employed to find the solution using COMSOL Multiphysics software. The deviations in the thermal and solutal fields are examined against the involved physical parameters by illustrating graphs and tables. Quantities of interest, such as the mean Nusselt and Sherwood numbers, are also computed against sundry parameters. The assurance of the computed results is validated by constructing a comparison with published studies. A grid sensitivity test is also executed to ensure the credibility of the built-in software (COMSOL). It is observed that the averaged heat and mass flux coefficient diminishes with an elevation in the magnitude of the power-law index (n), irrespective of the change in other involved parameters. The average Sherwood number tends to increase for a positive magnitude of (N) in contrast to negative values of (N). The average heat flux and average mass flux coefficients increase until the Prandtl number (Pr) reaches a value of 5 in shear thinning case at $$Ra\ge {10}^{5},$$ R a ≥ 10 5 , whereas, for n > 1 there is no substantial effect on either the average heat flux coefficient or average mass flux coefficient.

show abstract

Implementing the finite element numerical approach to optimize thermosolutal convection (TSC) energy imparted by discrete heat and solute sources in generalized Newtonian fluid flow in square enclosure

Bilal,

Ali,

Khan

et al. 2024

J.Umm Al-Qura Univ. Appll. Sci.

View full text Add to dashboard Cite

show abstract

An analysis of exponential kernel fractional difference operator for delta positivity

Mohammed

2024

Nonlinear Engineering

View full text Add to dashboard Cite

Positivity analysis for a fractional difference operator including an exponential formula in its kernel has been examined. A composition of two fractional difference operators of order ( ν , μ ) \left(\nu ,\mu ) in the sense of Liouville–Caputo type operators has been analysed in cases when ν ≠ μ \nu \ne \mu and ν = μ \nu =\mu . Due to the kernel of the fractional difference operator being convergent, there has been a restriction in the domain of the solution. Incidentally, a negative lower bounded condition has been carried out through analysing the positivity results. For a better understanding, an increasing function has been considered as a test for the main results.

show abstract

Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications

Mohammed,

Lupas,

Agarwal

et al. 2024

Axioms

View full text Add to dashboard Cite

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size p0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

A Study of Monotonicity Analysis for the Delta and Nabla Discrete Fractional Operators of the Liouville–Caputo Family

Cited by 3 publications

References 39 publications

Implementing the finite element numerical approach to optimize thermosolutal convection (TSC) energy imparted by discrete heat and solute sources in generalized Newtonian fluid flow in square enclosure

Implementing the finite element numerical approach to optimize thermosolutal convection (TSC) energy imparted by discrete heat and solute sources in generalized Newtonian fluid flow in square enclosure

An analysis of exponential kernel fractional difference operator for delta positivity

Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications

Contact Info

Product

Resources

About