The shooting method for the solution of ordinary differential equations: A control-theoretical perspective

The velocity and temperature distributions in a long dead-end channel, where the convective flow is driven by the buoyancy flux across the free surface, were investigated. The longitudinal variations of the temperature and velocity were determined using the similarity profiles for the vertical distribution of the temperature and velocity. The governing partial differential equations were reduced to ordinary differential equations by substituting the expressions for the longitudinal variations of the temperature and velocity. The resulting ordinary differential equations were solved numerically by employing a shooting method and a fourth order Runge-Kutta technique. The sensitivity to the temperature and velocity distributions of the various parameters were studied.

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A note on the non-inertial similarity solution for von Kármán swirling flow

Modern Physics Letters B ◽

10.1142/s0217984921500305 ◽

2020 ◽

pp. 2150030

Author(s):

Madeleine L. Combrinck

Keyword(s):

Boundary Layer ◽

Partial Differential Equations ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Similarity Solution ◽

Shooting Method ◽

Swirling Flow ◽

Boundary Layer Equations ◽

Von Karman ◽

Von Kármán

This note proposes a non-inertial similarity solution for the classic von Kármán swirling flow as perceived from the rotational frame. The solution is obtained by implementing non-inertial similarity parameters in the non-inertial boundary layer equations. This reduces the partial differential equations to a set of ordinary differential equations that is solved through an integration routine and shooting method.

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Non-Linear Static Analysis of 2D Steep Wave Riser Under Current Load

Volume 5: Pipelines, Risers, and Subsea Systems ◽

10.1115/omae2016-54460 ◽

2016 ◽

Cited By ~ 1

Author(s):

Hongdong Qiao ◽

Weidong Ruan ◽

Zhaohui Shang ◽

Yong Bai

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Difference Method ◽

Shooting Method ◽

Non Linear ◽

Newton Raphson ◽

Raphson Method ◽

Steep Wave

A new solution combining finite difference method and shooting method is developed to analyze the behavior of steep wave riser subjected to current loading. Based on the large deformation beam theory and mechanics equilibrium principle, a set of non-linear ordinary differential equations describing the motion of the steep wave riser are obtained. Then, finite difference method and shooting method are adopted and combined to solve the ordinary differential equations with zero moment boundary conditions at both the seabed end and surface end of the steep wave riser. The resulting non-linear finite difference formulations can be solved effectively by Newton-Raphson method. To improve iterative efficiency, shooting method is also employed to obtain the initial value for Newton-Raphson method. Results are compared with that of FEM by OrcaFlex, to verify the accuracy and reliability of the numerical method. Finally, a series of sensitivity analyses are also performed to highlight the influencing parameters in the steep wave riser.

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Combined Effects of Variable Viscosity and Thermal Conductivity on Dissipative Flow of Oil-Based Nanofluid over a Permeable Vertical Surface

Diffusion Foundations ◽

10.4028/www.scientific.net/df.16.158 ◽

2018 ◽

Vol 16 ◽

pp. 158-176 ◽

Cited By ~ 2

Author(s):

Christian John Etwire ◽

Ibrahim Yakubu Seini ◽

Rabiu Musah ◽

Oluwole Daniel Makinde

Keyword(s):

Thermal Conductivity ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Shooting Method ◽

Variable Viscosity ◽

Vertical Surface ◽

Skin Friction Coefficient ◽

Third Order ◽

Suction Parameter ◽

Dissipative Flow

Abstract: The combined effect of variable viscosity and thermal conductivity on dissipative flow of oil-based nanofluid over a permeable vertical plate with suction has been studied. The governing partial differential equations have been transformed into a coupled third-order ordinary differential equations using similarity techniques. The resulting third-order ordinary differential equations were then reduced into a system of first-order ordinary differential equations and solved numerically using the fourth-order Runge-Kutta algorithm with a shooting method. The results revealed that both viscosity and thermal conductivities of CuO oil-based nanofluid enhances the intensity of the skin friction coefficient and the rate of heat transfer at the surface of the plate. Furthermore, the thermal boundary layer thickness is weakened by the viscosity of CuO oil-based nanofluid, the Prandtl number, the suction parameter, the permeability of the medium and the thermal Grashof number

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A note on some solutions of micropolar fluid in a channel with permeable walls

Multidiscipline Modeling in Materials and Structures ◽

10.1108/mmms-06-2017-0053 ◽

2018 ◽

Vol 14 (1) ◽

pp. 91-101 ◽

Cited By ~ 1

Author(s):

Jawad Raza ◽

Azizah M. Rohni ◽

Zurni Omar

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Micropolar Fluid ◽

Design Methodology ◽

Shooting Method ◽

Similarity Transformation ◽

Physical Parameters ◽

Content Type ◽

Single Solution ◽

Permeable Walls

Purpose The purpose of this paper is to investigate different branches of the solution of micropolar fluid in a channel with permeable walls. Moreover, the intention of the study is to examine the effect of different physical parameters on fluid flow. Design/methodology/approach The mathematical modeling is performed on the basis of law of conservation of mass, momentum and angular momentum. The governing partial differential equations were transformed into ordinary differential equations by applying suitable similarity transformation. Afterwards, the set of nonlinear ordinary differential equations was solved numerically by a shooting method. Findings The study reveals that various branches of the solution of the proposed problem exist only in the case of strong suction. Originality/value The investigation of new branches of the solution of non-Newtonian micropolar fluid is relatively difficult as far as the single solution is concern. This study explores the new branches of the solution of a micropolar fluid in a channel with suction/injection. Simultaneous effect of suction Reynolds number and vortex viscosity parameter on velocity and micro-rotation profile is examined for different branches of solution in order to make the analysis more interesting.

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