Second-order Differential Equations Research Articles

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.

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.

Ulam type stability for the second-order linear Hahn difference equations

Ulam type stability for the second-order linear Hahn difference equations

Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings.

Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.

Formation of WNL Stars in the MW and LMC Based on the k − ω Model

We adopt a set of second-order differential equations (k − ω model) to handle core convective overshooting in massive stars, simulate the evolution of nitrogen sequence Wolf–Rayet (WNL) stars with different metallicities and initial masses, both rotating and nonrotating models, and compare the results with the classical overshooting model. The results indicate that, under the same initial conditions, the k − ω model generally produces larger convective cores and wider overshooting regions, thereby increasing the mass ranges and extending the lifetimes of WNL stars, as well as the likelihood of forming WNL stars. The masses and lifetimes of WNL stars both increase with higher metallicities and initial masses. Under higher-metallicity conditions, the two overshooting schemes significantly differ in their impacts on the lifetimes of WNL stars, but are insignificant in the mass ranges of the WNL stars. Rotation may drive the formation of WNL stars in low-mass, metal-poor counterparts, with this effect being more pronounced in the overshooting model. The surface nitrogen of metal-rich WNL stars formed during the main-sequence phase is likely primarily from the CN cycle, while it may come from both the CN and NO cycles for relatively metal-poor counterparts. Our model can effectively explain the distribution of WNL stars in the Milky Way, but appears to have inadequacies in explaining the WNL stars in the LMC.

On the problem of controlling a second-order nonlinear system by means of discrete control under disturbance

The problem of bringing a trajectory to a neighborhood of zero under disturbance is considered in terms of a differential pursuit game. The dynamics are described by a nonlinear autonomous system of second-order differential equations. The set of values of the pursuer's controls is finite, and that of the evader (disturbance) is compact. The goal of the control, that is, the goal of the pursuer, is to bring, within a finite time, the trajectory to any predetermined neighborhood of zero, regardless of the actions of the disturbance. To construct the control, the pursuer knows only the phase coordinates and the value of the velocity at some discrete moments of time and the choice of the disturbance control is unknown. Conditions are obtained for the existence of a set of initial positions, from each point of which a capture occurs in the specified sense. Moreover, this set contains a certain neighborhood of zero. The winning control is constructed constructively and has an additional property specified in the theorem. In addition, an estimate of the time required to bring the speed from one given point to the neighborhood of another given point under disturbance conditions was obtained.

Some new results for a second-order differential equation arising from a mathematical model related to the Antarctic Circumpolar Current

Some new results for a second-order differential equation arising from a mathematical model related to the Antarctic Circumpolar Current

Leveraging feed-forward neural networks to enhance the hybrid block derivative methods for system of second-order ordinary differential equations

This study introduces an innovative method combining discrete hybrid block techniques and artificial intelligence to enhance the solution of second-order Ordinary Differential Equations (ODEs). By integrating feed-forward neural networks (FFNN) into the hybrid block derivative method (HBDM), the modified approach shows improved accuracy and efficiency compared to traditional methods. Through comprehensive comparisons with exact and existing solutions, the study demonstrates the effectiveness of the proposed approach. The evaluation, utilizing root mean square error (RMSE), confirms its superior performance, robustness, and applicability in diverse scenarios. This research sets a new standard for solving complex ODE systems, offering promising avenues for future research and practical implementations.