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This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved. A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams. The proposed algorithm, named as two-parameter multiple shooting method, is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions. The impact of the axial force and additional point masses is also taken into account. Due to the fact that the method is based directly on the fourth-order ordinary differential equation, the structures do not have to be divided into many small elements to obtain an accurate enough solution, even though the geometry is very complex. To verify the proposed method, three different examples are considered, i.e., a three-segment non-prismatic beam, a prismatic column subject to non-uniformly distributed compressive loads, and a two-segment beam with an additional point mass. Numerical analyses are carried out with the software MATHEMATICA. The results are compared with the solutions computed by the commercial finite element program SOFiSTiK. Good agreement is achieved, which confirms the correctness and high effectiveness of the formulated algorithm. [ABSTRACT FROM AUTHOR]

In most cases, the research on the buckling of a helical spring is based on the column, the spring is equivalent to the column, and the torsion around the axial line is ignored. A three-dimensional (3D) helical spring model is considered in this paper. The equilibrium equations are established by introducing two coordinate systems, the Frenet and the principal axis coordinate systems, to describe the spatial deformation of the center line and the torsion of the cross section of the spring, respectively. By using a small deformation assumption, the variables of the deflection can be expanded into Taylor's series, and the terms of high orders are ignored. Hence, the equations can be simplified to the functions of the twist angle and the arc length, which can be solved by a numerical method. The reaction loads of the spring caused by the axial load subjected to the center point are also discussed, giving the boundary conditions for the solution to the equilibrium equations. The present work is useful to the research on the behavior of the post-buckling of the compressed helical spring. [ABSTRACT FROM AUTHOR]

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lubrication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter. [ABSTRACT FROM AUTHOR]

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method. [ABSTRACT FROM AUTHOR]

A class of periodic initial value problems for two-dimensional Newton-Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved. [ABSTRACT FROM AUTHOR]

This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has been shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new alternating group explicit-implicit (nAGEI) scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment. [ABSTRACT FROM AUTHOR]