Second-order Differential Equations Research Articles

A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations

Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1–3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure. New version program summaryProgram Title: InSyDE – Invariants and Symmetries of (rational second order ordinary) Differential Equations.CPC Library link to program files:https://doi.org/10.17632/4ytft6zgk7.3Licensing provisions: CC by NC 3.0Programming language: MapleSupplementary material: Theoretical results and revision of the S-function method.Journal reference of previous version: Comput. Phys. Comm. Volume 234, January 2019, Pages 302-314 - https://doi.org/10.1016/j.cpc.2018.05.009Does the new version supersede the previous version?: Yes.Nature of problem: Determining first integrals of rational second order ordinary differential equations.Solution method: The method is explained in the Summary of revisions and Supplementary material.Reasons for the new version: The InSyDE package after determining the S-function still needs to solve a first-order ordinary differential equation (1ODE) associated with the nonlocal symmetry (the so-called associated 1ODE – see [2]). The problem is that, for very complicated 1ODEs, this may not be practically feasible. We have developed an new and more efficient method that uses the nonlocal symmetry to (for a large class of 1ODEs) determine the first integral in a linear way.Summary of revisions: In order to implement the new method just mentioned above we have made modifications to the command (Sfunction) and introduced a new one: command (Darlin).

Investigation of real-world second-order singular differential equations by optimal homotopy analysis technique

Investigation of real-world second-order singular differential equations by optimal homotopy analysis technique

New Criteria for Guaranteeing Oscillation of Second-Order Differential Equations with Several Delays

New Criteria for Guaranteeing Oscillation of Second-Order Differential Equations with Several Delays

New Fixed-Point Results in Controlled Metric Type Spaces with Applications

In this manuscript, we present several novel results in fixed-point theory for a complete controlled metric space. The first presented result is inspired from the Caristi contraction where we explore the existence and uniqueness of fixed points under specific conditions. Furthermore, we propose a graphical form of it by endowing the considered space with a graph and develop a new fixed-point theorem, which is illustrated by two examples. Also, we establish a theorem for the α-admissible mapping. To demonstrate its effectiveness, the last theorem proposes an approach to solve a second-order differential equation.

Oscillation Solutions for Nonlinear Second-Order Neutral Differential Equations

This research investigates the oscillation criteria of nonlinear second-order neutral differential equations with multiple delays, focusing on their noncanonical forms. By leveraging an innovative iterative technique, new relationships are established to enhance the monotonic properties of positive solutions. These advancements lead to the derivation of novel oscillation criteria that significantly extend and refine the existing body of knowledge in this domain. The proposed criteria address gaps in the literature, providing a robust framework for analyzing such differential equations. To demonstrate their practical implications, three illustrative examples are presented, showcasing the applicability and effectiveness of the results in solving real-world problems involving delay differential equations.

A fuzzy based computational model to analyze the influence of mitochondria, buffer, and ER fluxes on cytosolic calcium distribution in neuron cells.

A free calcium ion in the cytosol is essential for many physiological and physical functions. Also, it is known as a second messenger as the quantity of free calcium ions is an essential part of brain signaling. In this work, we have attempted to study calcium signaling in the presence of mitochondria, buffer, and endoplasmic reticulum fluxes. Small organelles called mitochondria are found in the nervous system and are involved in several cellular functions, including energy production, response to stress, calcium homeostasis regulation, and pathways leading to cell death. It has been discovered that buffer, endoplasmic reticulum, and mitochondria significantly affect calcium signaling. To investigate how various circumstances impact the quantity of calcium in the cytosol, a mathematical model ofa second-order linear partial differential equation with fuzzy boundary conditionshas been developed. Systems having ambiguous or imprecise boundaryvalues can be effectively modeled and simulated with the help of fuzzy boundary conditions. Models can provide more dependable and instructive outcomes and become adaptable to real-world circumstances by implementing fuzzy logic into boundary conditions. In this paper, we observed the Fuzzy Laplace Transform to solve variable coefficient fuzzy differential equations using triangular fuzzy numbers. It isnoted that maintaining the delicate calcium ion balance, which controls essential cellular functions, depends on the buffer affinity. Also, neurodegenerative illnesses like Alzheimer's, Parkinson's, etc. are linked to disruptions in the control of components such as buffers, mitochondria, and the endoplasmic reticulum.

PARALLEL PERFORMANCE OF THE EXPLICIT FDTD METHODS APPLIED ON 3D ACOUSTIC WAVE EQUATION

The 3D acoustic wave equation is a second-order linear hyperbolic partial differential equation. It is widely studied in acoustics, fluid dynamics, electromagnetism, and seismology within both science and engineering fields. Among the various numerical methods, the finite difference method is considered to be the simpler to understand and easy to implement. Based on the discretize schemes finite difference time domain method described by the explicit scheme in which spatial second order derivatives are evaluated at the previous time step. The solution of 3D acoustic wave equation may be required on the high-resolution mesh points consequently more computational resources are required. In this study parallel algorithm for the numerical solution of 3D acoustic wave equation is proposed and designed by using data parallel approach and message passing schemes. The algorithm is implemented on shared memory parallel system using MATLAB parallel computing mode. The parallel performance of the designed algorithm is analyzed on different mesh sizes and time steps. It is revealed that the proposed algorithm may reduce computational time up to 3 times as compared to sequential solution algorithm. The proposed parallel algorithm remains more efficient on workers while for the efficiency of the algorithm drops because of the high communication time. The results of the proposed research could be utilized in the large-scale simulations of 3D acoustic wave equations and may enable to simulate the acoustic wave pressure at more refined meshes on high performance computing systems.

On k-fuzzy metric spaces with applications

With application point of view, Gopal et al. [D. Gopal, W. Sintunavarat, A.S. Ranadive, S. Shukla, The investigation of k-fuzzy metric spaces with the first contraction principle in such spaces, Soft Comput., 27:11081–11089, 2023] generalized the conceptions of a fuzzy metric space and introduced the definition of k-fuzzy metric space. Here a fuzzy set defined in k-fuzzy metric space is a membership function FY : X × X × (0, +∞)k -> [0; 1], that is, the fuzzy distance between two points of the set depends on more than one parameter, and then also introduced first contraction principle in this space. In this sequel, we extend the work on k-fuzzy metric spaces by generalizing Banach contraction principle by introducing various type of inequalities. Here we introduce Tirado-type k-fuzzy contraction condition and prove fixed point theorem for Tirado-type contractive mapping. We also discuss the k-fuzzy ψ-contractive mapping, where ψ ∈ Ψ, and Ψ is a class of mappings defined from ψ : [0; 1] -> [0; 1] that has certain properties, and also obtained fixed point for such class of mappings. Later, we define Ćirić-type contraction inequalities to prove fixed point results by restricting ourselves on l-natural property of the fuzzy space to ensure the existence of fixed point. Between all results, a set of supportive examples are also produced to validate the results. In application section, we discuss the solutions of Volterra-type integral equations and second-order nonlinear ordinary differential equation.

Some new oscillation results for second-order differential equations with neutral term

<p>In this paper, we study the oscillatory behavior of second-order differential equations. Using the comparison method, we obtain new oscillation criteria that improve the relevant results in the literature. Additionally, an example is given to illustrate the importance of the obtained oscillation criteria.</p>

A mathematical foundation for QUMOND

We link the QUMOND theory with the Helmholtz–Weyl decomposition and introduce a new formula for the gradient of the Mondian potential using singular integral operators. This approach allows us to demonstrate that, under very general assumptions on the mass distribution, the Mondian potential is well-defined, once weakly differentiable, with its gradient given through the Helmholtz–Weyl decomposition. Furthermore, we establish that the gradient of the Mondian potential is an Lp vector field. These findings lay the foundation for a rigorous mathematical analysis of various issues within the realm of QUMOND. Further, we prove that the once weakly differentiable Mondian potential solves a second-order partial differential equation in distribution sense. Thus, the question arises whether the potential has second-order derivatives. We affirmatively answer this question in the situation of spherical symmetry, although our investigation reveals that the regularity of the second derivatives is weaker than anticipated. We doubt that a similarly general regularity result can be proven without symmetry assumptions. In conclusion, we explore the implications of our results for numerous problems within the domain of QUMOND.

Study of the Impact of Loads on the Deformation of Building Structures using Differential Equations

The paper investigates the influence of different types of loads on the deformation of building structures, particularly beams, using second-order differential equations that allow modeling deformation processes. For this purpose, we considered two beam deflection equations under concentrated and distributed loads. Several experiments were conducted under different geometrical conditions (variable parameters of the beam cross-section, length, and loads). The solutions were studied using graph-analytical methods: constructing a graph of solutions reflecting the system's behaviour, a hodograph allowing tracking of additional nuances of the studied process, and amplitude and phase-frequency characteristics. The influence of conditions on the nature of system behavior is determined. The study's results can be used to optimise design solutions in construction.

Application of Laplace Transform Method to Solve Neutral-Type Delay Differential Equation

This study investigates the use of the Laplace transform method to resolve a particular class of second-order linear initial-value neutral delay differential equations, which are prevalent in fields such as physics and engineering. By employing this analytical approach, we derive exact solutions that facilitate a deeper awareness of the dynamics of the system and its stability characteristics. A thorough illustrative example is provided to illustrate the Laplace transform method's efficacy, showcasing its practical utility in addressing real-world problems. This study not only highlights the advantages of the Laplace transform in providing clear insights into complex delay systems but also emphasizes its relevance in both theoretical and applied contexts.

Analysis of solution methods for high order ordinary differential equations used in electrical circuits

This article presents a detailed methodology for applying the most common methods in solving high-order ordinary differential equations in the simulation of electrical circuits. The methods analyzed include the Laplace transform, the direct method in the time domain, the z-transform, the finite difference method, and methods based on difference equations. A detailed development of each of these methods is provided, along with practical examples that demonstrate their application in specific electrical circuits. The examples include: a circuit modeled with a second-order ordinary differential equation, a circuit modeled with a third-order ordinary differential equation, and an electrical network whose modeling results in an eighth-order ordinary differential equation. The article compares the results obtained with each method, using the Laplace transform solution as a reference. A deep analysis of the deviations between the methods is conducted, considering different time steps and parameters, allowing conclusions to be drawn about the effectiveness and accuracy of each approach.

Variant of the Mathematical Theory of Non-Thin Multilayer Nonlinearly Elastic Plates of Symmetric Structures

A new version of the mathematical theory of non-thin multilayer nonlinearly elastic Kauderer plates of symmetric structures has been constructed. The components of the stress-strain state (SSS) and boundary conditions on the side surface of the plate are considered functions of three spatial coordinates. The variant is based on the development of components of the SSS of the plate along the transverse coordinate with the help of combinations of Legendre polynomials within each layer. The three-dimensional physically nonlinear boundary value problem for a multilayer non-thin plate is reduced to a two-dimensional one using the three-dimensional equations of the theory of elasticity and Reissner's variational principle. The main dependencies are obtained. Derived systems of high-order differential equations of equilibrium with partial derivatives and boundary conditions containing nonlinear terms from the SSS. The conjugation conditions at the boundary of adjacent layers (rigid connection) are exactly fulfilled for displacements, transverse tangential and normal stresses. The boundary conditions for stresses on the horizontal faces of the plate are also exactly fulfilled. The system of high-order differential equations of equilibrium is transformed into two systems that independently describe skew-symmetric and symmetric deformation relative to the median plane of the plate. In turn, each of these systems is transformed into differential equations that describe vortex boundary effects and equations that describe an internal SSS with a potential boundary layer-type boundary effect. The solution of systems of high-order equations by the developed method is reduced to the solution of second-order differential equations (Poisson and Helmholtz equations). General solutions of the system of differential equilibrium equations are obtained. The validity and accuracy of the variant of the mathematical theory is confirmed by comparisons with exact solutions of bending problems for linearly elastic plates with different mechanical and geometrical parameters. The newly constructed variant of the theory and the developed method provide a real opportunity to solve boundary problems in a spatial formulation with high accuracy for multilayer platens under various loads and boundary conditions on the lateral surface.

Modeling of shear wave propagation in piezo-viscoelastic microbeam over quadratic heterogeneous viscoelastic plate with sliding contact

Abstract This study analyzes the phase and attenuation dynamic behavior of piezo-viscoelastic microbeam overlying quadratic heterogeneous viscoelastic plate under sliding contact. Using the Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Maxwell’s relations are used to incorporate the electric potential function. The solutions for both media are derived separately by solving the second-order hyperbolic differential equation using the method of separation of variables and asymptotic expansions of Bessel functions. The system of linear homogeneous equations is obtained by applying admissible boundary conditions to determine fundamental physical quantities. The key contribution of the current work is demonstrating the influence of dissipation factors, sliding contact, micro-length, heterogeneity, and thickness ratio parameters on shear wave propagation. The micro-length effect is found to suppress the attenuation of shear waves through the analysis and discussion of the dispersion and attenuation curves.

Derivation of Four-Point Integrator for the Solutions of Second-order Oscillatory Problems

The research introduces a new four-point integrator (FPI) for solving second-order differential equations characterized by oscillatory behavior. The proposed FPI is developed through a continuous scheme within a linear multistep framework, utilizing two off-step points to enhance efficiency. Unlike conventional methods that reduce higher-order equations to first-order systems, often resulting in loss of inherent characteristics, the FPI maintains essential equation properties. Through this approach, the FPI enables a self-starting, block-by-block algorithmic implementation that does not require predictor values, simplifying computation. The integrator’s properties are rigorously analyzed, confirming its consistency, zero-stability, and convergence, all of which contribute to its reliability in handling oscillatory problems. Additionally, stability analysis, including the region of absolute stability, demonstrates the A-stability of the integrator. Numerical experiments conducted on some second-order problems reveal the FPI’s computational accuracy, effectively comparing with established solvers such as ode45, ode15s and other methods. These results highlight the potential of the FPI as a viable alternative for directly solving oscillatory differential equations with damping characteristics, providing computational efficiency and accuracy.

Applying Theorems on b-Metric Spaces to Differential and Integral Equations Through Connected-Image Contractions

This paper introduces a new concept of a connected-image set for a mapping, which extends the notion of edge-preserving properties with respect to mapping. We also present novel definitions of connected-image contractions, with a focus on fixed-point theorems involving auxiliary functions in b-metric spaces. The relationships between these mathematical concepts are explored, along with their applications to solving differential and integral equations. In particular, we discuss existence results for solving integral equations and second-order ordinary differential equations with inhomogeneous Dirichlet boundary conditions, as well as theorems related to contractions of the integral type.

SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUILIBRIUM EQUATIONS OF NON-FLAT TIMOSHENKO TYPE SHELLS OF NON-ZERO GAUSSIAN CURVATURE IN ISOMETRIC COORDINATES

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates.The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in Sobolev space, the solvability of which is established using the principle of compressed mappings.

ON THE EXISTENCE OF PERIODIC SOLUTIONS OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH QUASI-HOMOGENEOUS NON-LINEARITY

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.

Dynamic Behaviours of Non-Uniform Rayleigh Beam under Variable-Magnitude Accelerating Masses and Resting on Non-Uniform Bi-parametric Foundation with General Boundary Conditions

This paper investigates the dynamic behaviours of non-uniform Rayleigh beam under variable-magnitude accelerating masses and resting on non-uniform bi-parametric foundation with general boundary conditions. The problem is governed by a fourth-order partia l differential equation, generalised Galerkin’s method use to reduces the fourth-order partial differential equation to sequence second-order ordinary differential equations. Two cases are examined: moving force (neglecting inertia term) and moving mass (considering inertia). In order to solve the moving force problem, the transverse displacement response is obtained by variation of parameters. In order to solve the moving mass problem, the Runge-Kutta of fourth order is used to obtain the approximate solution because the commonly used Struble asymptotic method was unable to simplify the coupled second order ordinary differential equation due to the variability of the load magnitude. Analytical and numerical solutions (Runge-Kutta) are compared for validation of accuracy of the Runge-kutta scheme and found compared favourably. The results are presented in plotted curves, illustrating the effects of shear modulus, rotatory inertia correction factor results show increased shear moduli and rotatory inertia decrease response amplitudes, and critical speed for moving mass is lower than for moving force, leading to earlier resonance. Resonance conditions for the dynamical system are also established.