Papers by Keyword: Shooting Method

Abstract: This paper examines the problem of nonlinear heat transfer in a cylindrical solid of combustible materials with two-step exothermic kinetics and radiative heat loss to the ambient surrounding. The reactant diffusion and temperature dependent pre-exponential factors with respect to sensitized, Arrhenius, and bimolecular kinetics are taken into account in the model energy balanced equation. Both regular perturbation method and numerical shooting technique coupled with Runge-Kutta-Fehlberg iteration scheme are employed to tackle the nonlinear model problem. The effects of various thermophysical parameters on the reactive cylindrical solid temperature, Nusselt number and thermal stability are discussed quantitatively with the help of computational illustrations. It is found that radiative heat loss enhances thermal stability of the material while the two-step exothermic kinetics promotes the onset of thermal instability.

Abstract: This article explores the problem of Blasius flow of water based hybrid nanofluid containing Al₂O₃ and Cu as nanoparticles over a convectively heated surface. Five different geometries of nanoparticles shape viz spherical, bricks, cylindrical, platelets and blades are considered in our analysis. The nonlinear model equations are obtained and tackled numerically using shooting method coupled with Runge-Kutta Fehlberg numerical scheme. The effects of nanoparticle shapes and other relevant thermophysical parameters on fluid velocity, temperature, skin friction and Nusselt number are discussed with the help of computational illustrations. The result for skin friction coefficient is compared with already existing results in the literature and excellent agreement was obtained. It is found that the heat transfer rate of hybrid nanofluid (Cu-Al₂O₃/Water) is higher than that of nanofluid (Al₂O₃/Water) and the Nusselt number increment for blade shaped nanoparticles is the highest as compared to that of platelet, cylindrical, brick and spherical shaped nanoparticles.

Abstract: The paper deals with eigenvalues excited-state energy eigenvalues and wave-function of a particle under harmonics oscillator asymmetric potential using numerical shooting method. The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schrodinger equation directly, though in the numerical sense. If the value of parameter μ is small the energy eigenvalues of single particle will large and the parameter μ large the energy eigenvalues of single particle will small.

Abstract: This paper treats an iterative shooting method based on sensitivity functions for solving non–linear two–point boundary value problems (BVPs), in the form of a fourth–order differential equation and more than four boundary conditions. The solution of this problem is possible only when the equation includes the required number of unknown parameters. In order to use this method, it is necessary to convert the BVP to an appropriate initial value problem (IVP). The presented method has been illustrated with a numerical example.

Abstract: This paper treats an iterative shooting method based on sensitivity functions for solving non–linear two–point boundary value problems (BVPs), in the form of a second–order differential equation and four boundary conditions. The solution of this BVP constitutes an in–run profile of a ski jumping hill. It is characterized by reduced a normal reaction force, which has impact on ski jumper’s legs during sliding downhill. In order to use this method, it is necessary to convert the BVP to an appropriate initial value problem (IVP). Consequently, in each iteration, we must solve a system of six first–order differential equations.

Abstract: In the present study, nonlinear bending problem of functionally graded material (FGM) cantilever beams resting on a Winkler elastic foundation under distributed load are discussed. Based on the large deformation theory and considering the axial extension of the beam, the equilibrium equations with geometric nonlinearity of FGM beams subjected to distributed load are established. In the analysis, it is assumed that the material properties of the beam vary continuously as a power function of the thickness. By using shooting method, the nonlinear boundary-value problem is numerically solved as well as the non-linear bending characteristic curves of the deformed beam versus the load are presented. The effects of material gradient property and foundation stiffness parameter on the bending deformation of the beam are discussed in detail.

Abstract: Due to the capability of high strain, dielectric elastomers are promising for applications as transducers in cameras, robots, valves, pumps, energy harvesters and so on. This paper focuses on the large deformation analysis of a dielectric elastomer membrane.The membrane is initially flat and attached to a disk in the inner circle and to a rigid ring in the outer circle, then a weight is applied to the disk and the membrane deforms into an axisymmetric shape, undergoing large out-of-plane deformation. The membrane is assumed to behave elastically in accordance with the ogden law. The governing equations are derived by combining kinematics and thermodynamics and a set of ordinary differential equations (ODEs) are obtained finally. The ODEs are solved by using shooting method. The obtained results show that the deformation field in the membrane is very inhomogeneous.

Abstract: This paper proposes a corrected shooting method for a general non-linear system with impacts. We define the global Poincaré mapping for period orbits by the discontinuous mapping. It is suitable to construct the strategy of shooting method. As an illustrated example, we investigate the stability of period orbits in a Duffing system with impacts. In Addition, coexistence of attractors and bifurcations for period orbits are considered.

Abstract: With the remarkable development of parallel computers, computing scale is becoming larger. Recently, it is an important trend to use iterative approaches to solve differential equations. Among them, the waveform relaxation method as a dynamic iterative method has good parallelism. This method will be compared with direct integration and shooting method using A Mathematical Programming Language (AMPL) in nonlinear circuit to verify the efficiency of waveform relaxation approach.

Abstract: Based on the accurate geometrical theory for the extensible elastic beams, an exact mathematical model of post-buckling transverse free vibration of Euler beams subjected to a distributed tangential follower force along the central axis are established. By using shooting method, pre-buckling free vibrations of both simply supported and fixed Euler beam are solved and the responses of small amplitude vibration are obtained. The numerical results show that all the frequencies of unbuckled beams decrease continuously with the increment of the load parameters.