Higher-order Differential Equations Research Articles

Mixed convective flow of engine oil-based non-Newtonian tri-hybrid nanofluid across a porous rotating disk

Purpose Flow induced by rotating disks is of great practical importance in several engineering applications such as rotating heat exchangers, turbine disks, pumps and many more. The present research has been freshly displayed regarding the implementation of an engine oil-based Casson tri-hybrid nanofluid across a rotating disk in mass and heat transferal developments. The purpose of this study is to contemplate the attributes of the flowing tri-hybrid nanofluid by incorporating porosity effects and magnetization and velocity slip effects, viscous dissipation, radiating flux, temperature slip, chemical reaction and activation energy. Design/methodology/approach The articulated fluid flow is described by a set of partial differential equations which are converted into one set of higher-order ordinary differential equations (ODEs) by using convenient conversions. The numerical solution of this transformed set of ODEs has been spearheaded by using the effectual bvp4c scheme. Findings The acquired results show that the heat transmission rate for the Casson tri-hybrid nanofluid is intensified by, respectively, 9.54% and 11.93% when compared to the Casson hybrid nanofluid and Casson nanofluid. Also, the mass transmission rate for the Casson tri-hybrid nanofluid is augmented by 1.09% and 2.14%, respectively, when compared to the Casson hybrid nanofluid and Casson nanofluid. Originality/value The current investigation presents an educative response on how the flow profiles vary with changes in the inevitable flow parameters. As per authors’ knowledge, no such scrutinization has been carried out previously; therefore, our results are novel and unique.

A numerical study of MHD nonlinear mixed convection flow over a nonlinear vertical stretching sheet with the buoyancy and suction/injection effects

The current research has a primary focus on characterizing buoyancy-driven nonlinear mixed convection. This phenomenon is studied within the context of an MHD Newtonian flow over a vertically stretching sheet with nonlinear characteristics. Flow patterns resembling those studied in this context have practical applications in various industrial sectors. Such flow patterns have practical applications in several industrial sectors, including the article and pulp industry, polymer manufacturing, electronic device cooling, solar energy collection, gas turbine plants, and nuclear power generation. The governing partial differential equations (PDEs) for nonlinear mixed convection are transformed into higher-order nonlinear ordinary differential equations (ODEs) through appropriate transformations, and these transformed ODEs are subsequently solved numerically. The software MATHEMATICA is used to solve the boundary layer flow equations, employing the semi-analytical Galerkin weighted residual method (GWRM). This approach yielded simultaneous solutions for both cases suction and injection. For both the suction and injection situations, graphical results, such as velocity, concentration, and temperature profiles are included. An augmentation in the nonlinear stretching parameter results in a reduction in velocity near the surface of the sheet. Conversely, away from the sheet, velocity increases. Moreover, an increase in nonlinear stretching parameters corresponds to higher values for both temperature and concentration. An increase in the suction/injection parameter causes a decrease in the velocity, temperature, and concentration fields. In the scenario of assisting flow, an increase in the thermal buoyancy parameter leads to higher velocity but lower temperature. Conversely, in the case of opposing flow, an increase in the thermal buoyancy parameter results in higher values for both velocity and temperature. Furthermore, a numerical examination is conducted to analyze the values of skin friction, Sherwood number, and Nusselt number.

Characteristic Mode Solution of Complex-Coefficient Complex-Solution Differential Equations

Computation of complex-coefficient complex-solution differential equations is a problem that arises in various domains of science and engineering. This paper aims at applying the Theory of Characteristic Modes (TCM) approach along with the Method of Moments (MoM) in solving these problems with emphasis on procedures for higher differential equations. Several available methods, known in literatures, are available for solving the problem. The complexity of the available methods differs based on the accuracy of the solution. In this paper, the general method is first presented and then a simplified version of it is proposed to solve high order differential equations. Two examples are illustrated, a third and a fourth order complex-coefficients complex-solution differential equations, to show the simplicity of the proposed method. The proposed approach can be also introduced along with other methods to solve these special occurrences differential equations and other boundary value problems.

Analytical Study of Nonlinear Systems of Higher-Order Difference Equations: Solutions, Stability, and Numerical Simulations

This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the local stability of the unique positive equilibrium point of the proposed systems. The theoretical findings are validated through numerical examples using MATLAB, facilitating graphical illustrations of the results.

Higher-order derivative of uncertain process and higher-order uncertain differential equation

Higher-order derivative of uncertain process and higher-order uncertain differential equation

Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations

BackgroundNumerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping.ResultsThe results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s.ConclusionsThe solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.

Property (A) and Oscillation of Higher-Order Trinomial Differential Equations with Retarded and Advanced Arguments

In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations y(n)(t)+p(t)y(τ(t))+q(t)y(σ(t))=0 is established. We offer new criteria for so-called property (A) and oscillation of the considered equation. Examples are provided to illustrate the importance of our results.

An Efficient New Scheme for Direct Simulation of Higher Order Third and Fourth Oscillatory Differential Equations

In this research, we have examined the general block approach for solving higher-order oscillatory differential equations using the linear block approach (LBA). The basic properties of the new method, such as order, error constant, zero-stability, consistency, convergence, linear stability, and region of absolute stability, were also analyzed and satisfied. Some distinct fourth-order oscillatory problems were directly applied to the new method in order to overcome the setbacks of the reduction method. The results obtained were compared with those in the literature, and the new method takes away the burden of solving fourth-order oscillatory differential equations. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantages of the new method is that it does not require much computational burden and is also self-starting. Convergence, Consistency, Direct simulation, Linear stability, Oscillatory differential equations

Dynamical Behavior and Expressions of Solutions of a Class of Higher-Order Nonlinear Difference Equations

The aim of this article is to get the forms of the solutions of the following nonlinear higher-order difference equations χs+1=χs−3k+1/±1±Πl=1kχs−3l+1,s=0,1,2,…, where the initial conditions χ−j,j=0,1,2,…,3k−1 and k∈1,2,… are arbitrary real numbers. Also, we examine stability, boundedness, oscillation, and the periodic nature of these solutions.

New Analytic Free Vibration Solutions of L-Shaped Moderately Thick Plates by Symplectic Superposition

L-shaped moderately thick plates have widespread applications in diverse engineering structures. Exploring the benchmark analytic solutions for the free vibration of L-shaped moderately thick plates is important in order to accurately analyze and efficiently design structures. Nevertheless, analytical solutions, which serve as the benchmarks, have been rarely documented in previous literature due to the challenge of finding suitable solutions that satisfy both the governing higher-order partial differential equations (PDEs) and the boundary conditions of the plates. The symplectic superposition method was employed in our recently published study to present the benchmark vibration solutions of clamped rectangular moderately thick plates. In this study, we expand upon this method to solve the free vibration problem of L-shaped moderately thick plates. By employing domain decomposition, we construct an irregular domain by combining multiple rectangular domains. First, the construction of the superposition system is carried out, followed by the import of the sub-problems into the Hamiltonian system, utilizing the fundamental governing equations of the plate. Then, the sub-problems are resolved through the application of the symplectic geometry methodology in an analytical manner. Ultimately, the analytical solutions for frequencies and mode shapes are derived through ensuring the equivalence between the initial problem and the combination of sub-problems. The finite element method is used to validate and present a comprehensive analysis of the natural frequencies and mode shapes obtained from this method. This method possesses the benefits of rapid convergence and accurate precision, rendering it well suited for the analytic modeling of a broader range of plate-related problems.

The Impact of Thermal Radiation on Mixed Convective Unsteady Nanofluid Flow in a Revolving Vertical Cone

This study investigates the effects of an unsteady mixed convection nanofluid flow in a rotating vertical cone submerged in spinning nanofluid. Our analysis considered the impacts of heat flux, chemical reactions, and thermal radiation, with the thermal and concentration Biot numbers serving as constraints at the boundary. The governing unsteady and coupled partial differential equations were solved through appropriate similarity transformations, addressing the nonlinear terms inherent in these equations. The spectral quasi-linearisation method (SQLM) was employed to solve the higher-order nonlinear differential equations. This study elucidates and assesses the impact of diverse physical constraints and parameters through the use of graphical representations. Notably, the temperature distribution of the liquefied substance was intensified as the thermal and solutal Biot numbers increased.

Heat and mass transfer phenomena for fractal interfacial layer model on metallic-nanomaterials nanofluids flow with chemical reaction effect

This study presents a comprehensive investigation into the intricacies of single-phase simulation concerning the behavior of fractal interfacial magnetized metallic chemical-reactive nanoparticles within fluid systems. The research focuses on unraveling the multifaceted phenomena associated with heat and mass transfer in complex nanoscale environments. To achieve this, we employ numerical techniques to address the intricate fluid flow problems that arise due to the presence of these nanoparticles. Fractal complexities are linked to critical phenomena, such as mixing and mass transport, at the turbulent/non-turbulent interface of shear layers and boundary layers. This study primarily concentrates on the phenomena of heat and mass transfer, with a focus on estimating thermal conductivity models that have significant impacts on the flow of metallic nanofluids under convective conditions. The combination of a high thermal conductivity model with a novel interfacial fractal theory reveals a substantial effect on heat and mass transfer enhancement. Furthermore, this research explores hybrid types of porous channel geometries under the combined influence of magnetohydrodynamics and chemical reaction mechanisms in the presence of metallic nanoparticles. A comprehensive numerical analysis of higher-order nonlinear differential equations is conducted, which exposes the flow field’s behavior concerning momentum, energy, and mass transfer. These insights are envisioned through single-phase simulations of nanofluids. The influence of the nanoparticles’ radius and diameter is found to play a significant role in thermal performance, with nanoparticles ranging from 1% to 8% exhibiting a notable impact on thermal conductivity. Highlights A new mathematical relation of mass, as well as heat transfer for metallic-nanomaterial being affected by thermal conductivity (T.C.), has been established. The important role of the physical number is observed under the effect of T.C. and heat and mass transfer through orthogonal porous surfaces. An injection/suction, as well as an expansion/contraction phenomenon related to the porous surface is observed.

Romanovski-Jacobi spectral schemes for high-order differential equations

We develop direct solution techniques for solving high-order differential equations with constant coefficients using the spectral tau method. The spatial approximation is based on Romanovski-Jacobi polynomials {Rnα,β(x)}n=0N with α>−1, β<−2N−α−1, x∈(0,∞) and n is the polynomial degree. Then, a hybrid approach combining the Romanovski-Jacobi tau method with the Romanovski-Jacobi collocation technique is introduced for the numerical solution of high-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces the problems to those of solving systems of algebraic equations, which greatly simplifies the problem. Finally, the Romanovski-Jacobi collocation method is presented for solving nonlinear high-order initial value problems. Numerical examples are included to demonstrate the validity and applicability of the techniques, and comparisons are made with the existing results. The method is easy to implement and yields very accurate results.

Hamiltonian system-based analytic thermal buckling solutions of orthotropic rectangular plates

New analytic solutions for the thermal buckling of orthotropic rectangular plates are presented using the symplectic superposition method (SSM). The SSM deals with higher-order partial differential equations within the Hamiltonian framework, distinguishing it from conventional methods in the Lagrangian framework. The solution procedure involves dividing the original problem into two subproblems, utilizing the techniques in the symplectic space, like separating variables and symplectic eigen expansion, to achieve analytic solutions, and using superposition to obtain the final solutions. One notable advantage of the SSM is the absence of predetermined assumptions, which is a departure from conventional semi-inverse methods. The new analytic solutions under challenging non-Lévy-type boundary conditions (BCs) were rarely addressed in prior research due to their complex mathematical characteristics, which reveals the main novelty. Comprehensive thermal buckling results are provided for isotropic/orthotropic plates under various BCs, validated by those from the finite element method with a maximum difference being 2.7374% and literature with a maximum difference being 5.9201%. Parameter analyses quantify the effects of BCs, aspect ratio, and thickness-to-length ratio on critical buckling temperatures and mode shapes.

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

&lt;abstract&gt;&lt;p&gt;In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.&lt;/p&gt;&lt;/abstract&gt;

The existence of solutions to higher-order differential equations with nonhomogeneous conditions

The existence of solutions to higher-order differential equations with nonhomogeneous conditions

Construction of modular differential equations by using Rankin-Cohen brackets

In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic or skew-holomorphic Jacobi forms paired with elliptic modular forms. They provide a unified method of construction of higher-order modular differential equations for the modular forms above. In particular, with these extended Rankin-Cohen brackets, we can reproduce the modular differential equations of previous studies.

Conditions for the Oscillation of Solutions to Neutral Differential Equations of Higher Order

In this research, we applied three techniques—the comparison technique, the Riccati technique, and the integral averages technique to analyze and establish various conditions and properties associated with the oscillatory behavior of even-order neutral differential equations. These findings contribute to a better understanding of the dynamics of such equations. To demonstrate the efficacy of these new conditions and properties, we present illustrative examples. This study offers valuable insights into the behavior of neutral differential equations, advancing our knowledge in this field.

Nonlinear thermo-mechanical stress-driven modeling of nano arches augmented by higher order double-scaled kernel

Modeling the nonlocal behavior of small-scale structures is almost controversial. The recently developed approach, known as the stress-driven nonlocal model, in which the stress is the input of nonlocal integral, has been proved to pose as a remedy for the ill-posing behavior of strain-driven solutions. Though it is still in its early stages, stress-driven modeling of the nonlinear behavior of nano-scaled problems needs further thought. The coupled displacement field, in particular, makes nano-arches more challenging. The stress-driven formulation, enhanced by constitutive boundary conditions, reveals the thermo-mechanical buckling and post-buckling features of shallow nano-arches for the first time in this research. Furthermore, the single-scaled Helmholtz kernel is replaced with the more appropriate double-scaled kernel known as the bi-Helmholtz averaging nonlocal kernel. Higher-order continuity of the bi-Helmholtz kernel and, hence, constitutive boundary conditions and higher-order differential equations corresponding to the stress-driven scheme hold the key to efficiency. The arch is subjected to uniform temperature and radial pressure at the same time. To demonstrate the response of the nano-arch during buckling and post-buckling, the nonlinear equilibrium path, whether for mechanical or thermal loading, is given for the first time by prevailing on the challenges of the higher-order problems. Moreover, it would be beneficial if one could have the chance to recognize how and when the arch buckles. These valuable data are available in this study to give the limiting parameters at the moment of buckling, including limiting temperature, nonlocal parameter, and geometry.

MHD flow of a rotating micropolar fluid between two plates by using DTM

The primary purpose of this work is to investigate the influence of a magnetic field on a micropolar fluid, which is rotating between two horizontally arranged parallel plates. A uniform magnetic field acts perpendicularly to the porous plate, which consumes fluid with a steady suction velocity. The obtained higher-order nonlinear partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by using similarity transformation. A semi-analytical methodology has been applied to obtain the result to the problem. The influence of various flow parameters of the flow field like coupling parameter, viscosity parameter, Reynolds number, porosity parameter, rotation parameter and magnetic field parameter has been examined and explained by graphs. A resemblance between the results obtained by solving by the differential transform method (DTM) and the outcomes of the former research works has also been displayed which shows the correctness and efficiency of the used analytical method. The numerical values of the coefficient of skin friction at the lower plate are also displayed in tabular form and compared with the literature.