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"Agarwal, Ravi P."
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Fractional calculus in the sky
Fractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L’Hospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering. In this short manuscript, we briefly review the tremendous effect that the main ideas of fractional calculus had in science and engineering and briefly present just a point of view for some of the crucial problems of this interdisciplinary field.
Journal Article
Interpolative Rus-Reich-Ćirić Type Contractions via Simulation Functions
In this paper, we introduce the notion of interpolative Rus-Reich-Ćirić type - contractions in the setting of complete metric space. We also consider some immediate consequences of our main results.
Journal Article
Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions
The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.
Journal Article
On complete monotonicity for several classes of functions related to ratios of gamma functions
Let Γ(x)\\(\\varGamma (x)\\) denote the classical Euler gamma function. The logarithmic derivative ψ(x)=[lnΓ(x)]′=Γ′(x)Γ(x)\\(\\psi (x)=[\\ln \\varGamma (x)]'=\\frac{\\varGamma '(x)}{ \\varGamma (x)}\\), ψ′(x)\\(\\psi '(x)\\), and ψ″(x)\\(\\psi ''(x)\\) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [ψ′(x)]2+ψ″(x)\\([\\psi '(x)]^{2}+ \\psi ''(x)\\), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.
Journal Article
Dynamical behaviors of a food-chain model with stage structure and time delays
Incorporating two delays (τ1 represents the maturity of predator, τ2 represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov–LaSalle’s principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if τ1∈(m1,m2) and τ2∈(m4,+∞), then the boundary equilibrium E2(x0,y10,y20,0,0) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if τ1∈(m2,+∞), then the axial equilibrium E1(k,0,0,0,0) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results.
Journal Article
An Analytical Technique, Based on Natural Transform to Solve Fractional-Order Parabolic Equations
This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.
Journal Article
Stability and numerical solutions of higher-order nonlinear time-dependent delay differential equations using Haar wavelet collocation method
In this paper, the authors present qualitative results for the solutions of nonlinear higher-order time-dependent delay differential equations. Proof of existence and uniqueness theorem for
n
th order time-dependent delay differential equations using the contraction mapping theorem is presented. Stability is verified through Hyers-Ulam and Hyers-Ulam-Rassias stability theorems, using the Picard operator with the Chebyshev norm and Gronwall’s inequality. The Haar wavelet collocation method is used to find numerical solutions of higher-order time dependent delay differential equations. Two numerical examples of higher-order nonlinear delay differential equations with time-dependent delays are discussed to show efficiency and reliability of the method. Numerical results are benchmarked against existing exact solutions to validate precision and accuracy. It is observed that inaccuracies decrease with higher resolution level. Error analysis including maximum absolute error, relative error, and root mean square error, are calculated to demonstrate the robustness of the method. Further, convergence rate is calculated for each example. A detailed investigation of computational complexity, including both time and space complexity is also discussed.
Journal Article
Stability of Delay Hopfield Neural Networks with Generalized Riemann–Liouville Type Fractional Derivative
The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann–Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included.
Journal Article
Exploring the Landscape of Fractional-Order Models in Epidemiology: A Comparative Simulation Study
Mathematical models play a crucial role in evaluating real-life processes qualitatively and quantitatively. They have been extensively employed to study the spread of diseases such as hepatitis B, COVID-19, influenza, and other epidemics. Many researchers have discussed various types of epidemiological models, including deterministic, stochastic, and fractional order models, for this purpose. This article presents a comprehensive review and comparative study of the transmission dynamics of fractional order in epidemiological modeling. A significant portion of the paper is dedicated to the graphical simulation of these models, providing a visual representation of their behavior and characteristics. The article further embarks on a comparative analysis of fractional-order models with their integer-order counterparts. This comparison sheds light on the nuances and subtleties that differentiate these models, thereby offering valuable insights into their respective strengths and limitations. The paper also explores time delay models, non-linear incidence rate models, and stochastic models, explaining their use and significance in epidemiology. It includes studies and models that focus on the transmission dynamics of diseases using fractional order models, as well as comparisons with integer-order models. The findings from this study contribute to the broader understanding of epidemiological modeling, paving the way for more accurate and effective strategies in disease control and prevention.
Journal Article