Higher-order Differential Equations Research Articles

Two-dimensional (p,q)-heat polynomials of Gould–Hopper type

We introduce a new class of holomorphic polynomials extending the classical Gould–Hopper to two complex variables. The considered polynomials include the 1-D and 2-D holomorphic and polyanalytic Itô–Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication, Runge addition and Nielson types formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.

On a family of higher order recurrence relations: symmetries, formula solutions, periodicity and stability analysis

In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry analysis as part of our approach together with some number theoretic functions. Our findings generalize certain results in the literature.

Stability of forced higher-order continuous-time Lur'e systems: a behavioural input-output perspective

We consider a class of forced continuous-time Lur'e systems obtained by applying nonlinear feedback to a higher-order linear differential equation which defines an input-output system in the sense of behavioural systems theory. This linear system directly relates the input and output signals and does not involve any internal, latent or state variables. A stability theory subsuming results of circle criterion type is developed, including criteria for input-to-output stability and strong integral input-to-output stability, concepts which are very much reminiscent of input-to-state stability and strong integral input-to-state stability, respectively. The methods used in the paper combine ideas from the behavioural approach to systems and control, absolute stability theory and input-to-state stability theory.

WKB analysis of the linear problem for modified affine Toda field equations

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the {D}_r^{(1)} - and {D}_{r+1}^{(2)} -type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Using $G_{\alpha}$-transform to Study Higher-order Differential Equations with Polynomial Coefficients

In this study, solutions of higher-order differential equations with polynomial coefficients(HODEPCs) were obtained by applying the $G_{\alpha}$-transform. Based on some characterizations, the solutions of HODEPCs were investigated. With the general solution of the HODEPCs, the curves of the general solution can be shown in several examples.

Mathematical analysis of MHD hybrid nanofluid flow with variable viscosity and slip conditions over a stretching surface

There are number of real-world uses for hybrid nanofluids. Studies have shown that hybrid nanofluid is better at transferring heat than nanofluid with only one type of nanoparticle. It can be used in new ways, like in microfluidics, dynamic sealing, heat dissipation, and so on. The use of hybrid nanofluids in MHD flow over a stretching surface with varying viscosity can enhance heat transfer rates. This has various applications in industries such as, cooling systems in electronics, thermal management in aerospace, and energy system. But the properties of boundary layer flow and rate of heat transfer for hybrid nanofluids could be studied more in a space with more dimensions and with other physical assumptions around the flow of fluids. The current study's major purpose is to investigate the flow of a three-dimensional hybrid nanofluid over a stretched sheet as its viscosity changes. Furthermore, the effects of the Smoluchowski temperature and Maxwell velocity slip boundary conditions are considered. Hybrid nanofluid was created by dispersing copper (Cu) and alumina (Al2O3) nanoparticles in a base fluid (H2O). Similarity variables are employed to transform a set of nonlinear partial differential equations (PDEs) that govern fluid flow and heat transfer into a system of higher-order ordinary differential equations (ODEs). The RKF method in Mathematica solves the higher-order nonlinear ODEs. Graphical analyses and in-depth discussions examine how leading variables affect velocity and temperature profiles. The expressions for skin friction and local Nusselt number are explicitly formulated, and graphs illustrate the variations of these two quantities across different parameter values. When velocity slip parameter goes up, velocity profile f′ηgoes down, and the temperature profile θ(η) also goes down when the temperature slip parameter goes up. As the velocity slip parameter goes up, skin friction goes down, while the Nusselt number goes up as the temperature slip parameter goes up. When results of this study are compared to some of results that have already been published, they are found to be very interesting as heat transfer increases due to the impacts of physical parameters.

On a hybrid updating method for modeling vibroacoustic behaviors of composite panels

Composite panels, by virtue of their outstanding mechanical properties, have found various applications in different sectors like aerospace and transportation. Under certain assumptions like Kirchhoff–Love and Mead–Markus hypotheses, certain high-order differential equations can be used for vibroacoustic modeling of composite panels. However, for accurate modeling, updating parameters is an essential stage. Herein, we aim to theoretically and experimentally review this stage. For this purpose, a hybrid updating method is proposed, incorporating hierarchical functions, inhomogeneous wave correlation approach, and least squares optimizations. Then various laboratory measurements, including Laser Doppler Vibrometry measurements as well as sound pressure levels, are analyzed. The measurements were performed for a thick composite (sandwich) panel, and a thin composite (laminate) one, along with two isotropic (steel and aluminum) plates for additional validation. The experiments indicate the ability of the hybrid approach to adjust parameters and precisely model vibroacoustic behaviors of the panels. Furthermore, the proposed hybrid method can be used in studies whose goal is accurate vibroacoustic modeling for psychoacoustic issues and perceptual validations, which is also one of the future targets of this research.

Asymptotic Properties of the Solutions of Higher-Order Differential Equations on Generalized Hölder Classes

Asymptotic Properties of the Solutions of Higher-Order Differential Equations on Generalized Hölder Classes

Non-Newtonian Slippery Nanofluid Flow Due to a Stretching Sheet Through a Porous Medium with Heat Generation and Thermal Slip

The domains of engineering, electrical, and medicine all have a significant demand for nanofluids. Applications for nanofluid flow include electronic device storage, industrial cooling and heating frameworks, and associated medicinal management information systems. Nanofluids are utilized generally as coolants in heat exchangers such as thermostats, electronic cooling systems, and radiators due to their enhanced thermal characteristics. This study aims to explain the mixed convection phenomenon’s applications on the thermal impact of Maxwell nanofluid. The mass diffusivity is supposed to be a function of concentration, whereas the thermal conductivity and viscosity of Maxwell nanofluid are assumed to be functions of temperature. It is recommended to consider the additional thermal effects of thermal slip, magnetic fields, and heat generation phenomena. The fluid flow motion was caused by the vertically stretched sheet. The dimensionless formulation of the suggested physical model is shown by the suitable variables interacting. The shooting approach is used in the numerical simulations, and it is based on lowering higher-order nonlinear differential equations to first-order. The slip velocity and the magnetic parameters have a direct impact on the local skin friction coefficient and velocity, as indicated by the research findings. Also, the increase in values of the Maxwell parameter, porous parameter, and viscosity parameter leads to the enhancement of temperature distribution, while the decline in velocity distribution can be attributed to the same factors. A comparison is also made with the results described in the literature that is currently available, and a superb agreement is discovered.

Intersection numbers from higher-order partial differential equations

We propose a new method for the evaluation of intersection numbers for twisted meromorphic n-forms, through Stokes’ theorem in n dimensions. It is based on the solution of an n-th order partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivariate residue. We illustrate our approach with a number of simple examples from mathematics and physics.

Applications of the Tarig Transform and Hyers–Ulam Stability to Linear Differential Equations

In this manuscript, we discuss the Tarig transform for homogeneous and non-homogeneous linear differential equations. Using this Tarig integral transform, we resolve higher-order linear differential equations, and we produce the conditions required for Hyers–Ulam stability. This is the first attempt to use the Tarig transform to show that linear and nonlinear differential equations are stable. This study also demonstrates that the Tarig transform method is more effective for analyzing the stability issue for differential equations with constant coefficients. A discussion of applications follows, to illustrate our approach. This research also presents a novel approach to studying the stability of differential equations. Furthermore, this study demonstrates that Tarig transform analysis is more practical for examining stability issues in linear differential equations with constant coefficients. In addition, we examine some applications of linear, nonlinear, and fractional differential equations, by using the Tarig integral transform.

Numerical investigation of the magnetohydrodynamic hybrid nanofluid flow over a stretching surface with mixed convection: A case of strong suction

The present work explores the physical aspects of the alumina and silver nanoparticles on the magnetohydrodynamic (MHD) flow of mixed convection micropolar hybrid nanofluid with ethylene glycol + water [Formula: see text] base fluid via stretching surface embedded in a porous medium. A strong magnetic field is employed normally in the flow direction. The behavior of the suction on the presented flow analysis is discussed strongly. Heat transport phenomena are analyzed. The current model’s mathematical modeling is based on higher-order nonlinear partial differential equations, which are then translated into higher-order nonlinear ordinary differential equations using appropriate similarity transformations. The modeled higher-order nonlinear ordinary differential equations are solved using NDSolve technique. The physical significance of the different flow parameters on the velocity, microrotation, and temperature profiles of the hybrid nanofluid are described in a graphical form. In a tabular form, the skin friction coefficients for nanofluid and hybrid nanofluid against various flow parameters are calculated. Some important results from this investigation are demonstrated that the velocity of the hybrid nanofluid is higher for the stretching ratio parameter and it is detected that the suction parameter enhanced the microrotation profile of the hybrid nanofluid. From the comparison, it is noted that the velocity, microrotation, and temperature of the hybrid nanofluid are higher as compared to the velocity, microrotation, and temperature of the alumina nanofluid and silver-nanofluid.

Analysis of Kerosene oil conveying silver and Manganese zinc ferrite nanoparticles with hybrid Nanofluid: Effects of increasing the Lorentz Force, Suction, and volume fraction

The current study aims to explore the magnetic field on a spinning disk with the hybrid nanofluid flow and Cattaneo-Christov heat theory in the existence of nonlinear thermal radiation incorporating Ag and MnZnFe2O4 nanoparticles. Because silver may increase the thermal characteristics of the base material, it has a wide variety of industrial, pharmaceutical, power generation, and heating and cooling applications. The thermal properties of a hybrid nanofluid were to be found by exploring the aspect of nanomaterials on heat transfer and fluid flow. The principal partial differential equations are converted into ordinary differential equations using appropriate similarity treatments. With the aid of the shooting strategy, higher-order ordinary differential equations are converted to first-order ordinary differential equations. To present the numerical data and graphical results of the flow parameters, the built-in solver Bvp4c in the computational tool MATLAB is used. Several plots are also used to investigate the upshot of physical parameters on the graphically depicted profiles, such as the suction and injection parameter, magnetic parameter, Prandtl number, temperature ratio parameter, thermal radiation parameter, thermal relaxation parameter, and volume friction nanoparticles. For the magnetic parameter, both velocity profiles dropped, but the heat profile increased. When the temperature ratio and heat source-sink parameters were increased while the Prandtl number was decreased, the heat field increased.

Photo-thermoelastic wave propagation with changing thermal conductivity on excited pulsed laser nanoscale microelongated semiconductor material

Indroduction: The novel model of a non-local elastic semiconductor material that is microelongated is created. The process of photothermal transfer is responsible for the stimulation of the material. Photo-thermoelastic theories are used when the thermal conductivity is changed for the non-local medium. Under the influence of laser pulses, the effective framework describes the nanoscale microelongation instance as well as the interference between the photo-thermoelastic propagation waves in the non-local medium. It is possible to think of thermal conductivity as a linear function of temperature when electronic and thermoelastic deformation mechanisms are described. Two-dimensional deformation (2D) is used to extract the main fields, which obtained in non-dimension.Methods: Harmonic wave analysis, which is described by the normal mode, has been used to convert the basic equations into non-homogeneous higher-order ordinary differential equations. Applying a small subset of the possible non-local semiconductor surface conditions leads to exhaustive solutions.Result and Discussion: The outcomes of numerical simulations for silicon (Si) are graphically shown. There are comparisons made and explanations given for the investigated physical factors like their thermal conductivity, laser pulses and microelongation parameters.

Vibration Analysis of a Unimorph Nanobeam with a Dielectric Layer of Both Flexoelectricity and Piezoelectricity.

In this study, for the first time, free and forced vibrational responses of a unimorph nanobeam consisting of a functionally graded base, along with a dielectric layer of both piezoelectricity and flexoelectricity, is investigated based on paradox-free local/nonlocal elasticity. The formulation and boundary conditions are attained by utilizing the energy method Hamilton's principle. In order to set a comparison, the formulation of a model in the framework of differential nonlocal is first presented. An effective implementation of the generalized differential quadrature method (GDQM) is then utilized to solve higher-order partial differential equations. This method can be utilized to solve the complex equations whose analytic results are quite difficult to obtain. Lastly, the impact of various parameters is studied to characterize the vibrational behavior of the system. Additionally, the major impact of flexoelectricity compared to piezoelectricity on a small scale is exhibited. The results show that small-scale flexoelectricity, rather than piezoelectricity, is dominant in electromechanical coupling. One of the results that can be mentioned is that the beams with higher nonlocality have the higher voltage and displacement under the same excitation amplitude. The findings can be helpful for further theoretical as well as experimental studies in which dielectric material is used in smart structures.

A higher-order extension of Atangana–Baleanu fractional operators with respect to another function and a Gronwall-type inequality

This paper aims to extend the Caputo–Atangana–Baleanu (ABC) and Riemann–Atangana–Baleanu (ABR) fractional derivatives with respect to another function, from fractional order mu in (0,1] to an arbitrary order mu in (n,n+1], n=0,1,2,dots . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an ABC operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.

High-Order Nonlinear Functional Differential Equations: New Monotonic Properties and Their Applications

We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation test requires three independent conditions, but we provide criteria with two-conditions without checking the additional conditions. Lastly, we give examples to highlight the significance of the findings.

Mechanical characteristics of MHD of the non-Newtonian magnetohydrodynamic Maxwell fluid flow past a bi-directional convectively heated surface with mass flux conditions

In engineering and manufacturing industries, stretching flow phenomena have numerous real-world implementations. Real-world applications related to stretched flow models are metalworking, crystal growth processes, cooling of fibers, and plastics sheets. Therefore, in this work, the mechanical characteristics of the magnetohydrodynamics of the non-Newtonian Maxwell nanofluid flow through a bi-directional linearly stretching surface are explored. Brownian motion, thermophoresis, and chemical reaction impacts are considered in this analysis. Additionally, thermal convective and mass flux conditions are taken into consideration. The mathematical framework of the existing problem is constructed on highly non-linear partial differential equations (PDEs). Suitable similarity transformations are used for the conversion of partial differential equations into ordinary differential equations (ODEs). The flow problem is tackled with the homotopy analysis method, which is capable of solving higher-order non-linear differential equations. Different flow profiles against various flow parameters are discussed physically. Heat and mass transference mechanisms for distinct flow factors are analyzed in a tabular form. The outcomes showed that both primary and secondary velocities are the declining functions of magnetic and Maxwell fluid parameters. The heat transfer rate rises with the cumulative values of the Brownian motion and thermal Biot number. In addition, the mass transfer rate decreases with the rising Schmidt number, Brownian motion parameter, and chemical reaction parameter, while it increases with the augmenting thermophoresis parameter. It has been highlighted that streamlines in the current work for Maxwell and Newtonian models are in fact different from one another.

Kramer-type sampling theorems associated with higher-order differential equations

For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.