Solution Of Transcendental Equation Research Articles

Method of the Nonlinear Monotonic Tangent in the Solution of Transcendental Equations

Method of the Nonlinear Monotonic Tangent in the Solution of Transcendental Equations

Interface coupling model of Stefan phase change during thermal dissociation of natural gas hydrate

The thermal dissociation process that occurs in a natural gas hydrate is a Stefan phase-change problem with moving boundaries. Based on the integral form of a heat conduction model, for a continuous single-phase hydrate control volume, the energy conservation condition for interfacial coupling during the Stefan phase-change was built for the control volume of thermal dissociation of a natural gas hydrate with a sharp moving boundary. A Neumann solution of the Stefan phase-change model was obtained for the thermal dissociation in a semi-infinite natural gas hydrate reservoir, using Boltzmann similar variables. A unique solution for the Stefan model was found using a monotonic of the transcendental equation. An example analysis was used to verify, the monotonicity of the transcendental equation, and uniqueness of the solution of the Stefan model. The MATLAB program was used to study, the laws of temperature distribution, and dissociation frontal brim during the thermal dissociation process in a hydrate reservoir. Sensitivity fitting studies showed that, the solution of transcendental equation and interface position gradually increased, whereas, the depth penetrated and time penetrated gradually decreases with increasing temperature.

General Dynamic Collapse Criterion for Elastic–Plastic Structures Under Double Impulse as Substitute of Near-Fault Ground Motion

A dynamic collapse criterion for elastic-plastic structures under near-fault ground motions is derived analytically by approximately transforming near-fault ground motions into double impulse and using an energy balance law. A negative post-yield stiffness is introduced to treat the P-delta effect in the single-degree-of-freedom (SDOF) model. The principal part of fling-step near-fault ground motions is modeled by a one-cycle sine wave and then a double impulse. The double impulse enables the efficient use of the energy approach in the derivation of compact expressions of complicated elastic-plastic responses of structures with the negative post-yield stiffness. In contrast to the previous work (Kojima and Takewaki 2016b) for the resonant critical case, a general collapse criterion is provided for the velocity amplitude and the frequency of the double impulse. It is significant that no iteration is needed in the derivation of the dynamic collapse criterion except the solution of transcendental equations. It is shown that discussions on several patterns of dynamic collapse behaviors introduced in the previous critical case are useful for deriving a boundary between the collapse and the non-collapse in the plane of the input velocity and the input frequency. The most important point to be remarked is that the critical state (Kojima and Takewaki 2016b) corresponding to the nonlinear resonance does not necessarily provide the minimum input velocity level with respect to arbitrary impulse timing. The validity of the proposed dynamic collapse criterion is examined by the numerical response analysis for structures under double impulses with collapse or non-collapse parameters.

Method of Measuring Effective Dielectric Permittivity of Partially Filled Waveguides Using a Mismatched T-Bridge

Introduction. Waveguides, partially filled with dielectric material (partially filled waveguides) are widely used in the super high frequency equipment. They have certain advantages over hollow waveguides, including the possibility of reducing sizes of a cross-section, increasing the power of radiation and suppressing undesired types of waves. Following the production of diverse new dielectric materials intended for use in super high frequency range devices, there is a need to continuously develop methods of calculation and measuring characteristics of partially filled waveguides.Statement of the problem. The theory of completely filled waveguides and waveguides with dielectric filling along narrow walls is developed quite thoroughly. However, its application to various waveguide devices requires the solution of transcendental equations, which is possible only using numerical methods. This makes it difficult to obtain information about any characteristics of a device. There is also an effect of a large number of factors influencing the characteristics of partially filled waveguides (degree of filling, position of a waveguide plate, magnitude of the dielectric permittivity, etc.). For the study of partially filled waveguides with different sample height, among others, this paper presents an approach, which is based on the representation of the relative permittivity of the medium in the form of two real functions, each of which depends on the cross-section from one coordinate. This is an approximate method for determining the proper scalar and vector functions of partially filled waveguides. The solution of the electrodynamic problem for partially filled waveguides is reduced to determining the propagation constant in a waveguide using exact or approximate methods. This paper presents the method which allows calculating the propagation constant at any frequency, by measuring the value of effective dielectric permittivity.Results and discussion. According to the results of the analysis, it is shown that known methods of measuring effective dielectric permittivity (using a measuring line, a panoramic indicator, and a bridge meter) have certain shortcomings in relation to the modification of partially filled waveguides, bandlimitedness, and a significant relative error of measurement with increasing effective dielectric permittivity. In particular, the lack of bridge meters is a very narrow frequency range in which the bridge remains matched. It is almost necessary to match the bridge at each frequency, reaching the absence of a signal in the arm E for identical loads that are connected to the side arm. For this, bridges have tuning elements in the form of pins, diaphragms, etc. The method for measuring effective dielectric permittivity of partially filled waveguides using an unmatched T-bridge, which does not have these deficiencies, is introduced.Conclusions. The scientific novelty of the proposed method for measuring effective permittivity of partially filled waveguides using an unmatched T-bridge is the possibility of providing broadbandness, increasing the accuracy of measurements, and the universality through the use of a panoramic indicator of the standing wave ratio using the voltage and electron-computer. One more point is the ability to measure effective dielectric permittivity when other measurement methods are not suitable. The results obtained should be used when designing new antenna systems, which include partly filled waveguides, as well as part of teaching and learning activities for creating new workplaces or improving existing ones aimed at laboratory and practical training using the above method of measurement.

Solution of Transcendental Equation Using Clamped Cubic Spline

Bisection and regular false-position methods are widely used to find roots of a transcendental function f(x) in a certain interval [a,b] satisfying f(a). f(b) < 0. The paper develops a new algorithm to find roots of transcendental functions based on false position method. We use two end points of the interval to interpolate f(x) by an equivalent cubic polynomial using clamped cubic spline formula. We consider one of the roots of the interpolated function to define a new interval and to approximate the root of f(x). Dhaka Univ. J. Sci. 61(1): 47-52, 2013 (January) DOI: http://dx.doi.org/10.3329/dujs.v61i1.15095

Analysis of free oscillations of thin-walled cylindrical shells containing a compressible liquid

Free oscillations of a thin-walled cylindrical cell containing a compressible liquid were investigated. At some values of the parameters of the system, its eigenfrequencies were determined, and the influence of the geomet- ric and physical parameters of the cylindrical shell-liquid system on the free oscillation of cylinder was in- vestigated. Introduction. Shells as elements of machines and constructions are widely used in aircraft industry, shipbuild- ing, and so on. Therefore, in recent years the problems associated with the dynamic behavior of thin-walled structures that are in contact with environment under operating conditions attract considerable interest of researchers. The frequencies and forms of free oscillations of spherical and cylindrical shells coming in contact with elas- tic and liquid media were investigated in (1, 2). Approximate simple formulas were obtained by asymptotic methods for calculating the frequency and determining the form of oscillations of the system considered; this limits the use of the results obtained, because in a number of cases the possibility of carrying out a qualitative analysis of investigated processes is eliminated. Statement of the Problem. In this work we consider free asymmetric oscillations of an infinite thin-walled cylindrical shell containing a compressible liquid. Since finding the eigenfrequencies of the cylindrical shell-liquid sys- tem is associated with the solution of transcendental equations, the frequency of oscillations of a shell without a liquid is expressed in terms of the frequency of oscillations of the system in an explicit form; this allows one to investigate the spectra of the system's frequencies both analytically and graphically. The equations of radials oscillations of cylindrical shells have the form (3) ∂ 2 u

On the Propagation of Second-Sound in Nonlinear Media: Shock, Acceleration and Traveling Wave Results

An exhaustive study of thermal shock, acceleration (or temperature-rate) and traveling waves in media with temperature-dependent thermal conductivity, and within which the flux is described by the Maxwell–Catteneo law, is presented. The resulting quasilinear, hyperbolic system of equations predicts a variety of interesting phenomena in such media including, but not limited to, dynamic thermal shock waves with jump-dependent wave speed, finite-time temperature-rate wave blow-up, and temperature-rate wave formation as traveling waves reach a critical speed. Both analytical (integral transforms, singular surface theory, solution of transcendental equations using special functions) and numerical approaches are used; the former are benchmarked against the latter where appropriate. Parallels are drawn between the nonlinear heat waves discussed in the present work and nonlinear wave phenomena in related “classical” continuum theories such as acoustics.

Synthesis of PID controllers for a class of time delay systems

In this paper we use the Hermite–Biehler theorem to establish results on the design of proportional plus integral plus derivative (PID) controllers for a class of time delay systems. Using the property of interlacing at high frequencies of the class of systems considered and linear programming we obtain the set of all stabilizing PID controllers. As far as we know, previous results on the synthesis of PID controllers rely on the solution of transcendental equations. This paper also extends previous results on the synthesis of proportional controllers for a class of delay systems of retarded type to a larger class of delay systems.

Optimal control for a class of systems subjected to disturbances

A special class of linear dynamical systems acted upon by controls and bounded disturbances is considered. It is assumed that the disturbances are due to an error in the control implementation; there are no disturbances in the case of the zero control, but the range of the disturbances increases with the control intensity. The problem of constructing a control which provides a minimax to a specified optimality criterion for an arbitrary permissible form of the disturbances is formulated. The solution of the minimax problem is reduced to the solution of transcendental equations. Under certain conditions the solution is obtained in explicit form. An example is considered in which the optimal control is constructed both in the form of a program and in the form of a synthesis.

On the problem of negative ions photodetachment in intense circularly polarized laser field

We present the theoretical calculation of photodetachment rate of loosely bound atomic states in an intense circularly polarized monochromatic laser field. Different calculation methods are used: the perturbation theory with respect to laser field intensity, the semiclassical approximation and the exact numerical solution of transcendental equation for complex quasienergy of an atomic level. In the last case we use the regularization method for divergent integrals with Gamow wave function proposed by Zel'dovich. We present the comparison of semiclassical and exact results and discuss the accuracy of semiclassical approximation in the theory of ionization. Results are applied for the calculation of photodetachment rate for negative ions of hydrogen and alkali metals.

On the analytic inversion of functions, solution of transcendental equations and infinite self-mappings

The solution of seemingly simple transcendental equations is in effect equivalent to the general problem of analytical inversion of functions. Within a powerful and systematic method, based on the solution of an associated Riemann–Hilbert boundary value problem, beautiful explicit results for various inverse functions of physical importance have been found which inevitably take on the guise of integral representations of these functions. In an attempt to reduce one such solution to a standard-function expression which would then be easy to evaluate, we recognize an infinite ladder self-mapping solution. This new perspective, born out of complicated complex analysis, is straightforwardly and uniquely related to the systematic generation of fast converging expansions within the corresponding regions of single-valuedness of the inverse function.

A new quadrature method for the computation of critical loads in problems of elastic buckling

The computation of critical loads in problems of elastic buckling of bars and frames is generally reduced to the numerical solution of a transcendental equation. In this paper, we apply a new method, based on Gaussian quadrature rules, for the solution of transcendental equations to the determination of critical loads in buckling problems. Two such problems of practical interest are completely solved for the illustration of the method and numerical results are presented. The main advantages of the present method are its simplicity and the availability of closed-form formulae for the critical loads, from which the numerical results are obtained by using Gaussian quadrature rules. A comparison between the present method and the method of Burniston and Siewert is also made.

The use of a pocket computer in gas dynamics

A pocket computer (Casio model FX-702P) has been utilized to solve some problems arising in gas dynamics. The programs reported cover algebraic calculations, solution of transcendental equations and finite difference solutions of coupled, nonlinear ordinary differential equations. In gas dynamics terms, the algebraic computations pertain to properties variation in isentropic flow, Rayleigh flow and Fanno flow. The transcendental equations solved are those arising in Prandtl-Meyer flow and oblique shock analysis. Problems of pipe flow with simultaneous friction and heat transfer, unsteady discharge of a variable-volume tank through a convergent nozzle, Taylor-Maccoll flow over a cone and flow in a rocket nozzle with secondary injection of oxidant in the divergent section, are used as illustrative examples of coupled, nonlinear ordinary differential equations.

Solution of transcendental equations by series reversion

Solution of transcendental equations by series reversion

Tables of Bounds for Distributions with Monotone Hazard Rate

Abstract This paper presents in a form convenient for applications a number of bounds for distributions with monotone hazard rate. These bounds, together with their proofs and various generalizations have been obtained by Barlow and Marshall (1964). However, many of them can be characterized only through solutions of transcendental equations, and machine calculation has been necessary to make them accessible. Section 2 gives a listing of the results, and those without explicit forms are tabulated. Section 3 lists various related bounds which are of interest for purposes of comparison, and for use when the hypothesis of monotone hazard rate is not satisfied. Applications of the bounds are discussed in Section 4, together with some numerical examples.

Absolute iteration and the solution of transcendental equations

A simple, general method of solving transcendental equations is presented. Quantitative relations concerning the rate of convergence of the process are developed, and several examples are worked.