Second-order Differential Equations Research Articles

Application of Artificial Bee Colony Algorithm in Solving Second-Order Differential Equations

Application of Artificial Bee Colony Algorithm in Solving Second-Order Differential Equations

Applications of Catalan Neural Network (CaNN) model

In this paper, we construct Catalan neural network (CaNN) model and examine the nonlinear solvers of the ordinary nonlinear differential equation using the CaNN model with linear and nonlinear activation functions. Using a single-layer functional linking neural network (FLANN) architecture, we extend input models using the first five Catalan polynomials, update network parameters that we initially randomly select, and use error backpropagation algorithms. The CaNN trial solution for solving nonlinear differential equations is in excellent agreement with the analytical solution. Also, the validity of the CaNN method is investigated by treating second-order differential equations of Lane–Emden type as boundary and initial value problems.

Theoretical Study of Position-Dependent Mass Particle in Cosmic String Space-Time and External Magnetic Field

In this work, we discuss the case of a non-relativistic quantum particle in the cosmic string space-time under the influence of the presence of external magnetic and exponential potential. It’s shown with the help of the Laplace transform method the second-order differential equation of the Schrodinger equation is solvable where it is reduced to a first-order differential equation. We obtain the eigenfunction and the non-relativistic energy spectrum. In addition, these results may be useful in the future study of thermodynamic, optic, and magnetic properties of the atomic interaction of a system particle. Keywords: Schrodinger equation; Laplace transform method; Position dependent mass; External magnetic field; Cosmic string

Asymptotic and Oscillatory Analysis of Second-Order Differential Equations with Distributed Deviating Arguments

This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria.

New Oscillation Conditions Test for Neutral Differential Equations

In this work, we investigate new oscillation conditions of delay differential equations of second-order. We start by finding new relations and inequalities between the corresponding function and its solution, also with its derivatives. The new inequalities allow us to improve the asymptotic and oscillation conditions of the solutions of the this equation. By using an improved method and approach, we obtain new criteria that test the all solutions oscillation. The article provides some examples that highlight the significance of new conditions and properties.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

On a discrete version of Fejér inequality for α-convex sequences without symmetry condition

Abstract In this study, we introduce the notion of α \alpha -convex sequences which is a generalization of the convexity concept. For this class of sequences, we establish a discrete version of Fejér inequality without imposing any symmetry condition. In our proof, we use a new approach based on the choice of an appropriate sequence, which is the unique solution to a certain second-order difference equation. Moreover, we obtain a refinement of the standard (right) Fejér inequality for convex sequences.

Performance analysis of passive heave compensator operated with inclined piston using analytical method.

When performing operations such as drilling in the ocean, floating platforms undergo vertical motion due to waves, which negatively impacts these operations. So passive heave compensator(PHC) is used to counteract it. In many studies, the analysis of PHC with pneumatic springs has been done. Some of these papers analyzed it using simple second-order differential equations, but there was no case of modeling considering the pneumatic spring installed tilted in the PHC. On the other hand, in some papers, the characteristics were analyzed by numerical methods using multi-body dynamics equations, and it is time consuming. Therefore, in this paper, a second-order ordinary differential equation modeling method for PHC with an inclined pneumatic spring was presented and verified by comparison with the numerical method. Using the model presented in this paper, it is possible to analyze as accurate as conventional analytical methods, and to analyze characteristics faster than numerical methods.

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments

Abstract Optimal control of non-autonomous second-order stochastic differential equations with delayed arguments is indispensable for managing systems exposed to uncertainty, time-dependent dynamics, and historical influences. These equations underpin a wide range of applications, including finance, engineering, and biology, where it’s imperative to make informed decisions that mitigate risks or maximize returns while considering the inherent randomness, evolving conditions, and the impact of past states. By employing optimal control techniques, we can devise strategies that are resilient to uncertainty, adaptable to changing circumstances, and capable of accounting for the memory effects of previous events. This empowers us to optimize system performance, bolster stability, and attain desired objectives in intricate and dynamic environments. So, the goal of this article is to introduce a novel model of second-order perturbed stochastic differential equations incorporating non-local finite delay and deviated arguments in the setting of Hilbert spaces. Moreover, essential criteria are presented to examine the existence of a mild solution and evaluate the potential for approximate and optimal control of the proposed system. These results have been obtained by using evolution operators, fixed point techniques, random analytic methods, and compact semigroup theory. Further, to support the theoretical results, the optimal controllability of our model was studied by considering the Lagrange problem. Finally, the results were applied to discuss the approximate controllability of a partial differential equation. These models have the potential to advance the understanding and application of optimal control techniques for a wider range of complex systems.

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Efficient computation of the sinc matrix function for the integration of second-order differential equations

Understanding the nonlinear reactive transport model in porous catalysts

This paper discusses the steady-state nonlinear reaction-diffusion processes in porous catalysts. This model is based on a nonlinear second-order differential equation that includes a nonlinear term associated with the Michaelis-Menten and non-Michaelis-Menten kinetics of the reaction. The nonlinear equations can be approximately solved using the Taylors series, modified Taylors series and Akabri-Ganji technique to obtain the concentration of dissolved species. The influence of the half-saturation parameter and the characteristic reaction rate on concentration is explored. Sensitivity analysis of parameters is discussed. Our analytical findings were compared with numerical solutions.

On the solutions of the Lane-Emden model via non-dyadic wavelet scheme to unravel astrophysical phenomena

The research focuses on exploring the characteristics of the Lane Emden equation, which is a second-order ordinary differential equation that has its roots in astrophysics. To approximate this equation, a technique that employs non-dyadic Haar wavelets in conjunction with quasi-linearization is utilized. By utilizing non-dyadic wavelets, the ordinary differential equation can be simplified into a system of algebraic equations. Error estimates are provided to assess the accuracy of the produced data. A comparison between the existing solutions and the numerical results obtained using the suggested approach is performed to showcase its efficiency and advantages. The non-dyadic Haar wavelet approach is found to be a rich structure for numerous solutions that span a wide range of physical parameter.

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

. We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering, it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature, only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.

Localised gravity and resolved braneworlds

Deriving an effective massless field theory for fluctuations about a braneworld spacetime requires analysis of the transverse-space-wavefunction’s second-order differential equation. There can be two strikingly different types of effective theory. For a supersymmetric braneworld, one involves a technically consistent embedding of a supergravity theory on the worldvolume; the other can produce, in certain situations, a genuine localisation of gravity near the worldvolume but not via a technically consistent embedding. So, in the latter situation, the theory’s dynamics remains higher-dimensional but there can still be a lower-dimensional effective-theory interpretation of the dynamics at low worldvolume momenta / large worldvolume distances.This paper examines the conditions for such a gravity localisation to be possible. Localising gravity about braneworld spacetimes requires finding solutions to transverse-space self-adjoint Sturm-Liouville problems admitting a normalisable zero mode in the noncompact transverse space. This in turn requires analysis of Sturm-Liouville problems with radial singular endpoints following a formalism originating in the work of Hermann Weyl. Examples of such gravity-localising braneworld systems are found and analysed in this formalism with underlying “skeleton” braneworlds of Salam-Sezgin, resolved D3-brane and Randall-Sundrum II types.

Convergence of Graph-Based Fixed Point Results with Application to Fredholm Integral Equation

In this manuscript, we present a novel concept termed graphical Θc-Kannan contraction within the context of graphically controlled metric-type spaces. Unlike traditional Kannan contraction, this novel concept presents a modified method of contraction mapping. We discuss the significance and the existence of fixed point results within the framework of this novel contraction. To strengthen the credibility of our theoretical remarks, we provide a comparison example demonstrating the efficiency of our suggested framework. Our study not only broadens the theoretical foundations inside graphically controlled metric-type spaces by introducing and examining visual Θc-Kannan contraction, but it also demonstrates the practical significance of our innovations through significant examples. Furthermore, applying our findings to second-order differential equations by constructing integral equations into the domain of Fredholm sheds light on the broader implications of our research in the field of mathematical analysis and contributes to the advancement of this field.

On the Oscillatory Behavior of Solutions of Second-Order Damped Differential Equations with Several Sub-Linear Neutral Terms

The oscillation and asymptotic behavior of solutions of a general class of damped second-order differential equations with several sub-linear neutral terms is considered. New sufficient conditions are established to fulfill a part of the gap in the oscillation theory for the case of sub-linear neutral equations. Our main results improve and generalize some of those recently published in the literature. Several examples are given to support our results.

Loss of the Sturm–Liouville Property of Time-Varying Second-Order Differential Equations in the Presence of Delayed Dynamics

This paper considers a nominal undelayed and time-varying second-order Sturm–Liouville differential equation on a finite time interval which is a nominal version of another perturbed differential equation subject to a delay in its dynamics. The nominal delay-free differential equation is a Sturm–Liouville system in the sense that it is subject to prescribed two-point boundary conditions. However, the perturbed differential system is not a Sturm–Liouville system, in general, due to the presence of delayed dynamics. The main objective of the paper is to investigate the loss of the boundary values of the Sturm–Liouville nominal undelayed system in the presence of the delayed dynamics. Such a delayed dynamics is considered a perturbation of the nominal dynamics related to the Sturm–Liouville system with given two-point boundary values. In particular, this loss of the Sturm–Liouville exact tracking of the prescribed two-point boundary values might happen because the nominal boundary values may become lost by the state trajectory solution in the presence of delays, related to the undelayed case, due to the presence of the delayed dynamics. The worst-case error description of the deviation of the two-point boundary values of the current perturbed differential with respect to those of the nominal Sturm–Liouville system is characterized in terms of error norms related to the nominal system. Under sufficiently small deviations of the parameterization of the perturbed system with respect to the nominal one, such a worst-error characterization makes the current perturbed system an approximate Sturm–Liouville system of the nominal undelayed one.

Stability and boundedness criteria for certain second-order nonlinear neutral stochastic functional differential equations

This paper presents stochastic stability and stochastic boundedness to certain second-order nonlinear neutral stochastic differential equations. The second-order differential equation is weakened to a neutral stochastic system of first-order equations and used together with a second-order quadratic function to obtain perfect Lyapunov-Krasovskii functional. This functional is adapted and applied to obtain criteria on the nonlinear functions to ensure novel results on stochastic stability and stochastic asymptotic stability of the zero solution. Furthermore, when the forcing term is nonzero, fresh results on stochastic boundedness and uniform stochastic boundedness of solutions are obtained. The results of this paper are original, new, essentially improving, complementing, and simplifying several related ones in the literature. Two special cases of the theoretical results are supplied to demonstrate the applicability of the hypothetical results.

An exact solution of the lubrication equations for the Oldroyd-B model in a hyperbolic pipe

An exact analytical solution of the lubrication equations for the steady, isothermal, incompressible flow of a viscoelastic Oldroyd-B fluid in a hyperbolic cylindrical contracting pipe is derived. The solution is valid for values of the Deborah number, De, up to order unity (De is defined as the ratio of the longest relaxation time of the polymer to the characteristic residence time of the fluid in the pipe), all values of the ratio of the polymer viscosity to the total viscosity of the fluid, η, and typical values of the contraction ratio, Λ, encountered in experiments and practical applications. It is provided in terms of the streamfunction only and is used in the momentum balance to derive a strongly non-linear ordinary differential equation of second order with unknown a function which corresponds to a modified fluid velocity along the main flow direction. The final equation is solved semi-numerically using a fully spectral (Legendre)-Galerkin approach to resolve the unknown function almost down to machine accuracy. The exact solution for the polymer extra-stresses, which is emphasized is not the full solution of the complete lubrication equations, allows for the derivation of a variety of theoretical expressions for the average pressure-drop along the pipe. In all cases, a decrease in the pressure drop compared to the Newtonian value with increasing De, η and/or Λ is predicted. The differences between the corresponding analytical solution for the planar geometrical configuration are also identified and discussed.