Third-order Differential Equation Research Articles

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation.

Existence of Infinite, Everywhere Discontinuous Spectra of Upper Exponents of Oscillation of Signs, Zeros, and Roots of Third-Order Differential Equations

Existence of Infinite, Everywhere Discontinuous Spectra of Upper Exponents of Oscillation of Signs, Zeros, and Roots of Third-Order Differential Equations

Scaled boundary perfectly matched layer for wave propagation in a three-dimensional poroelastic medium

In this paper, we introduce a novel direct time-domain artificial boundary condition, called the scaled boundary perfectly matched layer, for wave problems in 3D unbounded fluid-saturated poroelastic medium. The poroelastic medium is conceptualized as a two-phase medium composed of a solid skeleton and an interstitial fluid based on the u-p formulation of Biot's theory. The scaled boundary perfectly matched layer proposed in this work originates from an extension of a recently developed perfectly matched layer for wave problems in a single-phase solid medium. The dependence of the conventional perfectly matched layer on the global coordinate system is overcome in the present method. As a result, this method permits the utilization of an artificial boundary with general geometry (not necessarily convex) and can consider planar physical surfaces and interfaces extending to infinity. The proposed method is formulated by a mixed unsplit-field form, which includes both the mixed displacement-stress field and pore pressure-relative displacement field. The resulting scaled boundary perfectly matched layer formulation can be directly described by a system of third-order ordinary differential equations in time, which can be easily integrated into the standard finite-element framework of poroelastic theory. Finally, numerical benchmark examples are presented to demonstrate the accuracy and robustness of the proposed scaled boundary perfectly matched layer, as well as the versatility in practical engineering problems.

Method of Variation of Parameters for the Third-Order Linear Proportional Dynamic Equations

The method of variation of parameters for the third-order linear nonhomogeneous differential equations is given in this article, utilizing the proportional derivative and its characteristics on the time scale. Then, an example is provided to illustrate how to apply the provided approach.

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Stability analysis and numerical implementation of the third-order fractional partial differential equation based on the Caputo fractional derivative

Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems

In this study, we developed a new modification of the well-known differential transform method (DTM) that differs from the classical one by the algorithm for calculating the coefficients of an approximate solution given as a series. The proposed new algorithm we will call α-parameterized DTM (α-p DTM,for short). By using the proposed α-p DTM, we solve the boundary value transmission problem for the third-order differential equation. For the same problem, an approximate solution is also found using also the classical DTM. The solutions obtained were compared graphically.

On the Size of the Safety Area around the Launch Trajectory of a Rocket

The safety zone around the flight path of a rocket is determined by the fall of debris in the case of an accidental explosion or commanded termination. The trajectory of a tumbling body in a vertical plane is determined by specifying the velocity, flight path angle and angle of attack as functions of time. This involves the lift, drag and pitching moment coefficients as functions of the angle of attack over a full circle—0 to 360 degrees—to account for the tumbling motion. The problem is reduced to a third-order non-linear differential equation for the angle of attack by using the approximation of free fall coordinates. The analytical and numerical solutions show that two types of tumbling fall are possible, one with rotation and the other with oscillation. The tumbling trajectories are plotted and discussed for a variety of initial conditions, mass and aerodynamic properties of the tumbling body.

Solvability of an infinite system of three point BVP in c0 and ℓ1 spaces using measure of noncompactness

Various existing works in the literature have addressed the solvability of different infinite systems of boundary value problems in different types of Banach spaces. In most of these works, the authors have restricted themselves in systems of differential equations with boundary conditions defined at two points. In this work, we have addressed the solvability of an infinite system of third-order differential equations with boundary conditions defined at three points in the sequence spaces [Formula: see text] and [Formula: see text]. We have also illustrated the results for both sequence spaces with suitable examples.

Non-Newtonian Casson Fluid Behaviour in the Presence of Nanofluid Particles During MHD Flow Through a Vertical Cone Filled With Porous Material

The flow of nanofluids around a vertical cone with porous media and Casson fluid characteristics is being looked at in this study. Thermophoresis, Brownian motion, and chemical reactions are also looked at. There are some ways to change the connected partial differential equations into a set of third-order ordinary differential equation with variable coefficients. This is called a similarity transformation. The Runge-Kutta method is used to solve third-order boundary layer equations. Physical processes, such as Epidermis slippage, velocity, temperature, but instead fluid density, mass transfer, heat transference coefficients, besides rate of heat handover coefficients, may be studied in this research. These processes may be looked at in this study. There are graphs that show a lot of different physical processes. Current numerical results are compared to results that have been published in the past to make sure computer programmes work. The resultant velocity profiles are decreasing utilising an increasing trendy captivating field as a result of Lorentz potency. Species concentration of Casson-When the oxidizing agent factor is increased, the microspheres decrease. Temperature profile areas a result of the rise in Thermo Scattering movements but instead heat conduction and Brownian motion parameters. Also, roles about increasing values of Biot number and this same criterion of radiant heat would be to surge the room’s temperature hybrid Nanofluid flow as well as rate of heat flows so at exterior. Concentration profiles remain rising with increasing the morals of Thermo migration limitation and contrary effect occurs as a consequence of Brownian motion parameter.

Inverse Spectral Problem for the Third-Order Differential Equation

Inverse Spectral Problem for the Third-Order Differential Equation

Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

INTEGRATION OF A THIRD-ORDER ODE VIA ANALYTICAL AND GEOMETRICAL METHODS

Analytical and geometrical methods are applied to integrate an ordinary differential equation of third order. The main objective is to compare both approaches and show the possibilities that each one of them offers in the integration process of the considered equation, specially when not only Lie point symmetries but also generalized symmetries are involved. The analytical method of order reduction by using a generalized symmetry provides the general solution of the equation but in terms of a primitive that cannot be explicitly evaluated. On the other hand, the application of geometrical tools previously reported in the recent literature leads to two functionally independent first integrals of the equation without any kind of integration. In order to complete the integration of the given third-order equation, a third independent first integral arises by quadrature as the primitive of a closed differential one-form. From these first integrals, the expression of the general solution of the equation can be expressed in parametric form and in terms of elementary functions.

Some Characteristic Properties of Non-Null Curves in Minkowski 3-Space 𝔼13

This paper gives new characteristic properties of non-null spherical and rectifying curves in Minkowski 3-space E13. In the light of the causal characteristics, we give some representations of rectifying non-null curves. Additionally, we proved that the tangential function of every non-null curve fulfills a third-order differential equation. Then, a number of well-known characteristic properties of rectifying, Lorentzian, and hyperbolic spherical curves are consequences of this differential equation.

Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

We consider a certain class of third order nonlinear delay differential equations in this work. The results that we obtained are an improvement and extension of some results mentioned in previous literature, as the criteria we obtained are less restrictive compared to the previous results reported in literature. An example is provided to illustrate new results.

Controllability of consensus heterogeneous multi-agent networks over continuous time scale

Abstract The research, presented in this paper, concernes the controllability of a multi-agent network with a directed, unweighted, cooperative, and time-invariant communication topology. The network’s agents follow linear and heterogeneous dynamics, encompassing first-order, second-order, and third-order differential equations over continuous time. Two classes of neighbour-based linear distributed control protocols are considered: the first one utilises average feedback from relative velocities/relative accelerations, and the second one utilises feedback from absolute velocities/absolute accelerations. Under both protocols, the network’s agents achieve consensus in their states asymptotically. We observe that both of the considered dynamical rules exploit the random-walk normalised Laplacian matrix of the network’s graph. By categorising the agents of the network into leaders and followers, with leaders serving as exogenous control inputs, we analyse the controllability of followers within their state space through the influence of leaders. Specifically, matrix-rank conditions are established to evaluate the leader–follower controllability of the network under both control protocols. These matrix-rank conditions are further refined in terms of the system matrices’ eigenvalues and eigenvectors. The inference diagrams presented in this work provide deeper insights into how leader–follower interactions impact the network controllability. The efficacy of the theoretical findings is validated through numerical examples.

Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator

ABSTRACT The aim of the paper is to study the oscillation behavior for third-order differential equations with several delays and with p-Laplacian operator. Our technique bases essentially on the presentation of sharper estimates of positive solutions of the equation studied using an iterative method. What is more, the iterative nature of the new criteria allows us to check for oscillation of all solutions, even if the known findings relevant to this fail to apply.

On the Asymptotic Behavior of Class of Third-Order Neutral Differential Equations with Symmetrical and Advanced Argument

In this paper, we aimed to study some asymptotic properties of a class of third-order neutral differential equations with advanced argument in canonical form. We provide new and simplified oscillation criteria that improve and complement a number of existing results. We also show some examples to illustrate the importance of our results.

ADEQUATE SOLUTIONS OF JERK OSCILLATORS CONTAINING VELOCITY TIMES ACCELERATION-SQUARED: HAQUE’S APPROACH WITH MICKENS’ ITERATION METHOD

Haque’s Approach with Mickens’ Iteration Method is used to find the exact analytic solution of the nonlinear equation involving velocity times acceleration squared. A truncated Fourier series is used in different rhythms with different repetition steps. Our results are very close to the exact results and our results are comparatively closer to the exact results than others. Our solution method is obtained around the second-order angular frequency using Newton’s method. For some third-order (jerk) differential equations with cubic nonlinearities and nonlinear second-order differential equations; Mickens’ iteration method is used to determine the exact analytical approximate periodic solution. A numerical experiment of general differential equations with third-order, one-dimensional, autonomous, quadratic, and cubic nonlinearity has uncovered several algebraically simple equations involving the shaking of time-dependent acceleration that contain chaotic solutions.

General solutions’ laws of linear partial differential equations II

This paper uses Z transformations to obtain the general solutions of a large number of second-order, third-order and fourth-order linear partial differential equations for the first time, including one-dimensional inhomogeneous wave equation. Using general solutions, we have obtained plenty of exact solutions of the typical definite solution problems, which proves the important value of general solutions. We propose the Z4 transformation for the first time and initially use it to solve a specific case. We successfully obtain the Fourier series solution using the series general solution of the one-dimensional homogeneous wave equation, which successfully solves a famous unresolved debate in the history of mathematics.

Exact traveling wave solutions for (2+1)-dimensional Konopelchenko-Dubrovsky equation by using the hyperbolic trigonometric functions methods

Abstract In this research, the extended rational sinh-cosh method and the modified extended tanh-function method for mathematically constructing traveling wave solutions to the (2+1)-dimensional integro-differential Konopelchenko-Dubrovsky evolution equation are successfully employed to obtain specific appropriate solutions for the first time. A traveling wave transformation was utilized to turn the provided model into a third-order nonlinear ordinary differential equation. Solitary and periodic wave solutions for the model under investigation are obtained in terms of various complex hyperbolic trigonometric and rational functions. Several of the aforementioned solutions have been represented by two- and three-dimensional graphics with appropriate arbitrary parameters to highlight their physical implications. Two-dimensional graphs have presented the influence of time evolution on the solution’s structures.