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Keywords = generalised Hyers–Ulam–Rassias

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15 pages, 355 KB  
Article
An Effective Approach Based on Generalized Bernstein Basis Functions for the System of Fourth-Order Initial Value Problems for an Arbitrary Interval
by Muhammad Basit, Komal Shahnaz, Rida Malik, Samsul Ariffin Abdul Karim and Faheem Khan
Mathematics 2023, 11(14), 3076; https://doi.org/10.3390/math11143076 - 12 Jul 2023
Viewed by 1766
Abstract
The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary [...] Read more.
The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective. Full article
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26 pages, 1514 KB  
Article
Deterministic and Fractional-Order Co-Infection Model of Omicron and Delta Variants of Asymptomatic SARS-CoV-2 Carriers
by Waqas Ali Faridi, Muhammad Imran Asjad, Shabir Ahmad, Adrian Iftene, Magda Abd El-Rahman and Mohammed Sallah
Fractal Fract. 2023, 7(2), 192; https://doi.org/10.3390/fractalfract7020192 - 14 Feb 2023
Cited by 5 | Viewed by 2666
Abstract
The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both [...] Read more.
The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)
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24 pages, 337 KB  
Article
Intuitionistic Fuzzy Stability of an Euler–Lagrange Symmetry Additive Functional Equation via Direct and Fixed Point Technique (FPT)
by P. Agilan, K. Julietraja, Nabil Mlaiki and Aiman Mukheimer
Symmetry 2022, 14(11), 2454; https://doi.org/10.3390/sym14112454 - 19 Nov 2022
Cited by 11 | Viewed by 1767
Abstract
In this article, a new class of real-valued Euler–Lagrange symmetry additive functional equations is introduced. The solution of the equation is provided, assuming the unknown function to be continuous and without any regularity conditions. The objective of this research is to derive the [...] Read more.
In this article, a new class of real-valued Euler–Lagrange symmetry additive functional equations is introduced. The solution of the equation is provided, assuming the unknown function to be continuous and without any regularity conditions. The objective of this research is to derive the Hyers–Ulam–Rassias stability (HURS) in intuitionistic fuzzy normed spaces (IFNS) by applying the classical direct method and fixed point techniques (FPT). Furthermore, it is proven that the Euler–Lagrange symmetry additive functional equation and the control function, which is the IFNS of the sums and products of powers of norms, is stable. In addition, a few examples where the solution of this equation can be applied in Fourier series and Fourier transforms are demonstrated. Full article
(This article belongs to the Section Mathematics)
28 pages, 360 KB  
Article
Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces
by Agilan Pasupathi, Julietraja Konsalraj, Nahid Fatima, Vallinayagam Velusamy, Nabil Mlaiki and Nizar Souayah
Symmetry 2022, 14(8), 1700; https://doi.org/10.3390/sym14081700 - 16 Aug 2022
Cited by 10 | Viewed by 1723
Abstract
Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability [...] Read more.
Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers–Ulam–Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples. Full article
(This article belongs to the Special Issue The Qualitative Theory of Functional Differential Equations and their Applications)
14 pages, 485 KB  
Article
Stability of a Nonlinear ML-Nonsingular Kernel Fractional Langevin System with Distributed Lags and Integral Control
by Kaihong Zhao
Axioms 2022, 11(7), 350; https://doi.org/10.3390/axioms11070350 - 21 Jul 2022
Cited by 25 | Viewed by 2278
Abstract
The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation [...] Read more.
The fractional Langevin equation has more advantages than its classical equation in representing the random motion of Brownian particles in complex viscoelastic fluid. The Mittag–Leffler (ML) fractional equation without singularity is more accurate and effective than Riemann–Caputo (RC) and Riemann–Liouville (RL) fractional equation in portraying Brownian motion. This paper focuses on a nonlinear ML-fractional Langevin system with distributed lag and integral control. Employing the fixed-point theorem of generalised metric space established by Diaz and Margolis, we built the Hyers–Ulam–Rassias (HUR) stability along with Hyers–Ulam (HU) stability of this ML-fractional Langevin system. Applying our main results and MATLAB software, we have carried out theoretical analysis and numerical simulation on an example. By comparing with the numerical simulation of the corresponding classical Langevin system, it can be seen that the ML-fractional Langevin system can better reflect the stationarity of random particles in the statistical sense. Full article
(This article belongs to the Special Issue Numerical Methods for Fractional and Integer PDEs)
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