Third-order Ordinary Differential Equation Research Articles

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Solution of Third-Order Ordinary Linear Differential Equations in Discriminant Domains Implementing the Variational Iteration Method

Stochastic neuro-swarming intelligence paradigm for the analysis of magneto-hydrodynamic Prandtl–Eyring fluid flow with diffusive magnetic layers effect over an elongated surface

In recent years, the integration of stochastic techniques, especially those based on artificial neural networks, has emerged as a pivotal advancement in the field of computational fluid dynamics. These techniques offer a powerful framework for the analysis of complex fluid flow phenomena and address the uncertainties inherent in fluid dynamics systems. Following this trend, the current investigation portrays the design and construction of an important technique named swarming optimized neuro-heuristic intelligence with the competency of artificial neural networks to analyze nonlinear viscoelastic magneto-hydrodynamic Prandtl–Eyring fluid flow model, with diffusive magnetic layers effect along an extended sheet. The currently designed computational technique is established using inverse multiquadric radial basis activation function through the hybridization of a well-known global searching technique of particle swarm optimization and sequential quadratic programming, a technique capable of rapid convergence locally. The most appropriate scaling group involved transformations that are implemented on governing equations of the suggested fluidic model to convert it from a system of nonlinear partial differential equations into a dimensionless form of a third-order nonlinear ordinary differential equation. The transformed/reduced fluid flow model is solved for sundry variations of physical quantities using the designed scheme and outcomes are matched consistently with Adam's numerical technique with negligible magnitude of absolute errors and mean square errors. Moreover, it is revealed that the velocity of the fluid depreciates in the presence of a strong magnetic field effect. The efficacy of the designed solver is depicted evidently through rigorous statistical observations via exhaustive numerical experimentation of the fluidic problem.

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

Advanced color image encryption using third-order differential equations and three-dimensional logistic map

Image encryption stands out as a crucial technique employed to securely transmit images across the Internet. In this paper, we introduce a novel algorithm for encrypting color images. The algorithm is built upon the principles of differential equations, XOR operations, and chaotic maps. First, the plain image is three-dimensional pixel shuffled via a logistic map. Afterward, the differential equations are used as a mathematical tool for encrypting images. The third-order ordinary differential equations are used to encrypt the shuffled images. The color values of the plain image are considered coefficients for the independent variable X. Subsequently, an alternate matrix of the same size is generated using a three-dimensional logistic map, taking into account its color values as the exponents linked to the independent variable X. A set of third-order differential equations emerged, containing an equivalent number of elements as the color values present in the plain image. This set of differential equations is addressed in the following manner: combining XOR and integration three times with respect to the independent variable X for each set of obtained differential equations while treating the integration constant as zero. Ultimately, a set of ordinary equations involving the independent variable X is derived, where the coefficients of X represent color values for the cipher image. The results from experiments and the security analysis affirm the resilience of the proposed algorithm against established security attacks. It exhibits a substantial key space, heightened key sensitivity, and a strong encryption effect.

Hemodynamics of ventricular-arterial coupling under enhanced external counterpulsation: An optimized dual-source lumped parameter model

Background and objectiveEnhanced external counterpulsation (EECP) is a mechanically assisted circulation technique widely used in the rehabilitation and management of ischemic cardiovascular diseases. It contributes to cardiovascular functions by regulating the afterload of ventricle to improve hemodynamic effects, including increased diastolic blood pressure at aortic root, increased cardiac output and enhanced blood perfusion to multiple organs including coronary circulation. However, the effects of EECP on the coupling of the ventricle and the arterial system, termed ventricular-arterial coupling (VAC), remain elusive. We aimed to investigate the acute effect of EECP on the dynamic interaction between the left ventricle and its afterload of the arterial system from the perspective of ventricular output work. MethodsA neural network assisted optimization algorithm was proposed to identify the ordinary differential equation (ODE) relation between aortic root blood pressure and flow rate. Based on the optimized order of ODE, a lumped parameter model (LPM) under EECP was developed taking into consideration of the simultaneous action of cardiac and EECP pressure sources. The ventricular output work, in terms of aortic pressure and flow rate cooperated with the LPM, was used to characterize the VAC of ventricle and its afterload. The VAC subjected to the principle of minimal ventricular output work was validated by solving the Euler-Poisson equation of cost function, ultimately determining the waveforms of aortic pressure and flow rate. ResultsA third-order ODE can precisely describe the hemodynamic relationship between aortic pressure and flow rate. An optimized dual-source LPM with three energy-storage elements has been constructed, showing the potential in probing VAC under EECP. The LPM simulation results demonstrated that the VAC in terms of aortic pressure and flow rate yielded to the minimal ventricular output work under different EECP pressures. ConclusionsThe ventricular-arterial coupling under EECP is subjected to the minimal ventricular output work, which can serve as a criterion for determining aortic pressure and flow rate. This study provides insight for the understanding of VAC and has the potential in characterizing the performance of the ventricular and arterial system under EECP.

MHD boundary layer micropolar fluid flow over a stretching wedge surface: Thermophoresis and brownian motion effect

To investigate the consequence of thermophoresis and Brownian diffusion on convective boundary layer micropolar fluid flow over a stretching wedge-shaped surface. The effects of non-dimensional parameters namely coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5), vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7) on the steady MHD heat and mass transfer is investigated in the present study. The coupled non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations employing similarity transformation. Furthermore, by using the Runge-Kutta method followed by the shooting technique, the transformed equations are solved. The main goal of this study is to investigate the numerical analysis of nanofluid flow within the boundary layer region with the effects of the microrotation parameter and velocity ratio parameter. The novelty of this paper is to propose a numerical method for solving third-order ordinary differential equations that include both linear and nonlinear terms. To understand the physical significance of this work, numerical analyses and tabular displays of the skin friction coefficient, Nusselt number, and Sherwood number are shown. The new approach of the present study contributes significantly to the understanding of numerical solutions to non-linear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing.

An Algorithm for Approximating Third-Order Ordinary Differential Equations with Applications to Thin Film Flow and Nonlinear Genesio Problems

Third-order problems have been found to model real-life phenomena such as thick film fluid flow, boundary layer problems, and nonlinear Genesio problems, to name a few. This research focuses on the development of an algorithm using a basis function that combines an exponential function with a power series. This algorithm, called the Exponential Function Induced Algorithm (EFIA), was formulated using the collocation and interpolation techniques. The theoretical analysis of the algorithm was also carried out in this research. The outcome of the analysis showed that the algorithm is consistent, convergent, and zero-stable. The EFIA was applied to solving some real-life third-order problems like thin film flow and nonlinear Genesio problems. The numerical and graphical results obtained showed that the EFIA is accurate, has high precision, and is easy to implement. Algorithm, Exponentially-fitted function, Hybrid block method, Nonlinear Genesio equations, Thin film flow problem, Third-order

Matrix factorizations for the generalized Charlier and Meixner orthogonal polynomials

The Cholesky factorization of the moment matrix is considered for the generalized Charlier and generalized Meixner discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre–Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient γn, that happen to be forms of the Painlevé deg-PV in disguise. New Laguerre–Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche.

Solution of Third Order Initial value Problems of Ordinary Differential Equations using the Block Backwards Method

This paper introduces a highly effective method for solving third-order ordinary differential equations. The method employs a block backward formula to implement the primary scheme for numerical solutions. The numerical examples provided demonstrate its exceptional applicability and efficacy, outperforming existing methods and providing solutions that are nearly identical to exact solutions. This new method is unequivocally superior in solving third-order ordinary differential equations.

Artificial neural network with dynamic synapse model

The purpose of this study is to develop and investigate a new short-term memory model based on an artificial neural network without short-term memory effect and a dynamic short-term memory model with astrocytic modulation. Methods. The artificial neural network is represented by a classical convolutional neural network that does not have short-term memory. Short-term memory is modeled in our hybrid model using the Tsodyks-Markram model, which is a system of third-order ordinary differential equations. Astrocyte dynamics is modeled by a mean field model of gliotransmitter concentration. Results. A new hybrid short-term memory model was developed and investigated using a convolutional neural network and a dynamic synapse model for an image recognition problem. Graphs of dependence of accuracy and error on the number of epochs for the presented model are given. The sensitivity metric of image recognition d-prime has been introduced. The developed model was compared with the recurrent neural network and the configuration of the new model without taking into account astrocytic modulation. A comparative table has been constructed showing the best recognition accuracy for the introduced model. Conclusion. As a result of the study, the possibility of combining an artificial neural network and a dynamic model that expands its functionality is shown. Comparison of the proposed model with short-term memory using a convolutional neural network and a dynamic synapse model with astrocytic modulation with a recurrent network showed the effectiveness of the proposed approach in simulating short-term memory.

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

ON THE ASYMPTOTICS OF SOLUTIONS OF NON-AUTONOMOUS DIFFERENTIAL EQUATIONS ASYMPTOTICALLY CLOSE TO LINEAR

The necessary and sufficient conditions are obtained for the existence of all possible types of Pω(Y0, λ0)-solutions of non-autonomous second and third-order ordinary differential equations that are asymptotically close to linear equations. The asymptotic representation of each possible type of Pω(Y0, λ0)-solutions has been determined, and the question of their quantity has been clarified. Consequences have also been derived from the obtained theorems for cases when the equation is a linear differential equation.

Turing-Hopf bifurcation and bi-stable spatiotemporal periodic orbits in a delayed predator-prey model with predator-taxis

When Turing-Hopf bifurcation occurs, it is naturally worth discussing the types of spatiotemporal patterns and their stabilities. In this paper, for partial functional differential equations with nonlinear diffusion, we establish the explicit formulae for the coefficients of normal form associated with Turing-Hopf bifurcation. This provides an explicit connection between the original system and the third-order ordinary differential equations (i.e. normal form) restricted on center manifold. Subsequently, with the aid of extended formulae, we discuss a delayed predator-prey model with predator-taxis and find that Turing-Hopf bifurcation leads to the emergence of a pair of stable single-peak spatially inhomogeneous steady states, a pair of stable multi-peak spatially inhomogeneous periodic orbits and so on. Also, the parameter regions in which these phenomena arise are given quantitatively. It is further shown that the pair of bistable spatially inhomogeneous periodic orbits arise due to the interaction of predator-taxis and time delay, which won't emerge in the same system without predator-taxis.

A class of single-step hybrid block methods with equally spaced points for general third-order ordinary differential equations

This study presents a class of single-step, self-starting hybrid block methods for directly solving general third-order ordinary differential equations (ODEs) without reduction to first order equations. The methods are developed through interpolation and collocation at systematically selected evenly spaced nodes with the aim of boosting the accuracy of the methods. The zero stability, consistency and convergence of the algorithms are established. Scalar and systems of linear and nonlinear ODEs are approximated to test the effectiveness of the schemes, and the results obtained are compared against other methods from the literature. Significantly, the study shows that an increase in the number of intra-step points improves the accuracy of the solutions obtained using the proposed methods.

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.

On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

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.

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.

An exact finite difference scheme for a three-layer beam model

ABSTRACT In this work, we propose an efficient computational technique for a recognized three-layer beam model which is described by a third-order linear ordinary differential equation. By using the Mickens' methodology, an exact finite difference (EFD) scheme for this model is constructed. The EFD scheme provides an explicit recurrence formula to compute the solution of the beam model, but it is more efficient than the formula of the exact solution. Moreover, the EFD scheme is also more efficient than well-known standard finite difference schemes in both computational time and accuracy. Finally, two numerical experiments are performed to illustrate the theoretical results as well as to show advantages of the EFD scheme.

ELASTIC TRANSFORMATION AND ITS INVERSE TRANSFORMATION FOR SOLVING FIRST- AND THIRD-ORDER NONLINEAR VARIABLE COEFFICIENT ORDINARY DIFFERENTIAL EQUATIONS

Here elastic transformation and its inverse transformation are used to solve two kinds of first- and third-order nonlinear ordinary differential equations (ODEs) with variable coefficients. First, elastic transformations are used to upgrade and reduce the first- and third-order nonlinear ODEs to a class of second-order linear homogeneous ODEs. Then the general solutions of the first- and third-order ODEs are obtained separately by using the general solution of the obtained second-order linear homogeneous ODEs, elastic transformation, inverse transformation of elastic transformation and related operation rules. Second, the steps for solving these two kinds of ODEs are summarized. Finally, examples and parametric analysis are given to prove that it is effective, simple and feasible to solve two kinds of first- and third-order nonlinear ODEs by elastic transformation and its inverse transformation. Elastic transformation and its inverse transformation provide a new method for solving ODEs. The research in this paper expands the solvable classes of ODEs.