Higher-order Nonlinear Ordinary Differential Equations Research Articles

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.

Numerical analysis of mixed convection and Thomson-Troian slip effects on ternary nanofluid flow and heat transfer over a stretching sheet with porous media: Darcy-Forchheimer model

The present study unveils research that examines the laminar motion of water-infused nanofluid comprised of nanoparticles of TiO2 (titanium dioxide), Ag (silver), and Cu (copper) over a stretching sheet. The flow undergoes a novel slip condition based on Thomson and Troian to model complex fluid behavior near solid walls and incorporate the Darcy-Forchheimer model to comprehensively analyze the influence of a porous medium on flow behavior. Moreover, a heat source/sink and radiation are included to ensure the outcome of the energy equation will resemble most actual-world applications. Partial differential equations (PDEs) of higher order, originally used to describe the system are then reformulated into higher-order non-linear ordinary differential equations (ODEs) by taking symmetry variables that are chosen thoughtfully. The resulting boundary layer equations are then turned into a set of ODEs by implementing relevant similarity transformations, which can be solved using the M ATLAB function bvp4c. Through graphs, the variations in the velocity, Nusselt number, temperature, and skin friction coefficient were displayed. The outcome showed that the incorporation of silver nanoparticles (0.01–0.03) enhanced the skin friction coefficient by ≈ 1.68 % and the Nusselt number reduced up to ≈ 5 %. The Forchheimer number also reported a 6.5% enhancement and 1.5% reduction in the skin friction coefficient and the Nusselt number, respectively. The velocity slip parameter γ1 correlates with an upswing in temperature and skin friction coefficient for the ternary nanofluid while observing a decline in the velocity and Nusselt number.

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.

Solution of Second- and Higher-Order Nonlinear Two-Point Boundary-Value Problems Using Double Decomposition Method

The present paper makes use of the efficient double decomposition method to propose a method for solving two-point boundary-value problems, featuring second- and higher-order nonlinear ordinary differential equations. The efficacy of the proposed method is demonstrated on numerous test problems. In the end, a high level of exactitude between the obtained approximate solution and the available exact solution is achieved.

On time dependent MHD nanofluid dynamics due to enlarging sheet with bioconvection and two thermal boundary conditions

The current study pertains to heat and mass transportation of magnetic fluid flow having dilute diffusion of nanoparticles and motile microorganisms over a permeable stretched sheet to examine the influence of thermal radiation and activation energy. Similarity functions are utilized to convert the highly mixed non-linear partial differential equations into higher-order non-linear ordinary differential equations. Five coupled equations are derived to be resolved numerically by employing a computing function Bvp4c, built-in Matlab. Two sets of thermal boundaries prescribed surface temperature (PSF) and prescribed heat flux (PHF) are considered. Basic physical quantities, temperature distribution, concentration, velocity field, and motile micro-organism profiles are observed as influenced by emerging parameters. The microorganisms distribution undergoes decreasing behavior against growing values of bio-convection Lewis number and Peclet number. These results are highly useful in the application of heat-transmitting devices and microbial fuel cells. It is seen that decreasing trend is observed in velocity profile when parameters Nr and Nc are uplifted. Also, the motility of the nanofluid decreases when the Lb parameter is raised. On the other hand, an increase in Peclet number Pe showed a rising trend in motility profile. Additionally, the implications of Brownian motion, Rayleigh number, Bioconvection Lewis number thermophoresis parameter, Peclet number, and buoyancy ratio parameter are discussed. Moreover, the obtained outcomes are validated as compared to the existing ones as limiting cases. Representative findings for microorganism concentration, skin friction coefficient, temperature gradient, local Sherwood number and density number of motile microorganisms, velocity field, temperature, the volumetric concentration of nanoparticles, are discussed in tabulated and graphical form.

Bioconvection Cross Diffusion Effects on MHD Flow of Nanofluids over Three Different Geometries with Melting

Currently, nanofluid is a hot area of interest for researchers. The nanofluid with bioconvection phenomenon attracted the researchers owing to its numerous applications in the field of nanotechnology, microbiology, nuclear science, heat storage devices, biosensors, biotechnology, hydrogen bomb, engine of motors, cancer treatment, the atomic reactor, cooling of devices, and in many more. This article presents the bioconvection cross-diffusion effects on the magnetohydrodynamic flow of nanofluids on three different geometries (cone, wedge, and plate) with mixed convection. The temperature-dependent thermal conductivity, thermal diffusivity, and Arrhenius activation energy applications are considered on the fluid flow with melting phenomenon. The flow is analyzed under thermal and solutal Robin’s conditions. The problem is formulated in the mathematical formulation of partial differential equations (PDEs). The similarity transformations are applied to diminish the governing nonlinear coupled boundary value problems into higher-order non-linear ordinary differential equations (ODEs). The resulting expressions/equation numerically tackled utilizing the famous bvp4c package by MATLAB for various interesting parameters. The results were physically and numerically calculated through graphics and tables for the velocity field, energy distribution, nanoparticles concentration, and microorganisms profile for numerous parameters. From the obtained results, we discern that the transfer of heat and mass coefficient is high over a plate and cone in the flow, respectively. The velocity profile is reduced via a larger magnetic parameter. Temperaturedependent thermal conductivity enhances the thermal field. Larger thermophoresis enhanced the concentration of nanoparticles. The microorganisms’ Biot number boosts the microorganism’s profile.

Existence Theory for Higher-Order Nonlinear Ordinary Differential Equations with Nonlocal Stieltjes Boundary Conditions

‎In this paper‎, ‎we develop the existence theory for some boundary‎ value problems of nonlinear $nth$-order ordinary differential‎ ‎equations supplemented with nonlocal Stieltjes boundary‎ conditions‎. ‎Our results are based on some standard theorems of‎ ‎fixed point theory and are well illustrated with the aid of‎ ‎examples‎.

Magnetohydrodynamic and heat transfer effects on the stagnation-point flow of an electrically conducting nanofluid past a porous vertical shrinking/stretching sheet in the presence of variable stream conditions

A steady two-dimensional magnetohydrodynamic stagnation-point flow of an electrically conducting fluid and heat transfer with thermal radiation of a nanofluid past a shrinking and stretching sheet is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis. A similarity transformation is used to convert the governing nonlinear boundary-layer equations into coupled higher-order nonlinear ordinary differential equations. The result shows that the velocity, temperature, and concentration profiles are significantly influenced by the Brownian motion, heat radiation, and thermophoresis particle deposition.

MHD BOUNDARY-LAYER FLOW AND HEAT TRANSFER OVER PERMEABLE PLATE WITH CONVECTIVE SURFACE BOUNDARY CONDITION

A numerical analysis has been carried out to investigate the problem of magnetohydrodynamic (MHD) boundary-layer flow and heat transfer of a viscous incompressible fluid over a fixed plate. Convective surface boundary condition is taken into account for thermal boundary condition. A problem formulation is developed in the presence of thermal radiation, magnetic field and heat source/sink parameters. A similarity transformation is used to reduce the governing boundary-layer equations to couple higher-order nonlinear ordinary differential equations. These equations are numerically solved using Keller–Box method. The effect of the governing parameters such as radiation, Prandtl number, Hartman number, heat source/sink parameter on velocity and temperature profile is discussed and shown by plotting graphs. It is found that the temperature is an increasing function of convective parameter A, radiation and heat source parameters. Besides, the numerical results for the local skin friction coefficient and local Nusselt number are computed and presented in tabular form. Finally a comparison with a previously published results on a special case of the problem has done and shows excellent agreement.

Existence of positive solutions for a system of higher-order multi-point boundary value problems

We investigate the existence of positive solutions of a system of higher-order nonlinear ordinary differential equations, subject to multi-point boundary conditions.

Positive solutions for a system of higher-order multi-point boundary value problems

We present some results for positive solutions of a system of higher-order nonlinear ordinary differential equations, subject to multi-point boundary conditions.

A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions

This paper investigates some new existence results for an nth-order nonlinear differential equation with four-point nonlocal integral boundary conditions (strip/slit like conditions). Our results are based on some standard fixed point theorems and Leray-Schauder degree theory.

Hamiltonization of Higher-Order Nonlinear Ordinary Differential Equations and the Jacobi Last Multiplier

It is known that Jacobi's last multiplier is directly connected to the deduction of a Lagrangian via Rao's formula (Madhava Rao in Proc. Benaras Math. Soc. (N.S.) 2:53---59, 1940). In this paper we explicitly demonstrate that it also plays an important role in Hamiltonian theory. In particular, we apply the recent results obtained by Torres del Castillo (J. Phys. A Math. Theor. 43:265202, 2009) and deduce the Hamiltonian of a second-order ODE of the Lienard type, namely, $\ddot{x}+f(x){\dot{x}}^{2}+g(x)=0$ . In addition, we consider cases where the coefficient functions may also depend on the independent variable t. We illustrate our construction with various examples taken from astrophysics, cosmology and the Painleve-Gambier class of differential equations. Finally we discuss the Hamiltonization of third-order equations using Nambu-Hamiltonian mechanics.

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.

A boundary value problem on the half-line for higher-order nonlinear differential equations

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.

Necessary Conditions for the Existence of Global Solutions of Systems of Higher-Order Nonlinear Ordinary Differential Equations and Inequalities

Necessary Conditions for the Existence of Global Solutions of Systems of Higher-Order Nonlinear Ordinary Differential Equations and Inequalities