Method Of Undetermined Coefficients Research Articles

A sub-equation method for solving the cubic–quartic NLSE with the Kerr law nonlinearity

In this paper, we study a class of cubic–quartic NLSE with the Kerr law nonlinearity via a new sub-equation method. Considered method of undetermined coefficients is applied to obtain the introduced solutions. The outcomes are useful in describing the diffusion of optical solitons. The performance of the approach is reliable, useful and gives more new general exact solutions than the other methods.

Fully developed magnetohydrodynamics natural convection flow in a vertical micro-porous-channel with Hall effects

An exact solution is presented for the steady hydromagnetic fully developed natural convection flow in a vertical microporous channel due to asymmetric heating. The governing momentum and energy equations are presented in dimensionless form and solved analytically using the method of undetermined coefficient. The effects of Hall current and suction/injection parameters on the primary and secondary velocity, volume flow rates, and skin frictions are discussed with the help of line graphs and tables. It is observed that injection accelerates the flow, whereas suction retards the flow in both the primary and secondary flow directions.

Plane surface milling force prediction with fillet end milling cutter under pre-determined inclination angle

Fillet end milling is widely used in mould, aerospace, shipping manufacture and other fields. As one of the important parameter in milling machining, milling force is related to machining efficiency and quality, as well as tool wear, but it is difficult to predict. Aiming at fillet end milling under a pre-determined inclination angle, considering cutter run-out, this paper presents a cutting force coefficients identification model related to chip thickness and axial position angle, and then achieve milling force prediction. A parametric relationship among feed direction, cutter axis and to-be-machined surface are established, and then, the analytic expression of previous swept surface, to-be-machined surface and cutter envelope surface is derived. Boundary curves of cutter workpiece engagement (CWE) are established based on spatial surface intersection analytically. The criterion of the cutter edge element involved in cutting is proposed. Judging element along the cutter edge one by one, whereafter the algorithm of in-cut-cutting edge (ICCE) is proposed. Combining mechanical force model with chip thickness considering feed direction and cutter run-out, a milling force prediction model is established. Based on the method of undetermined coefficients, combining average instantaneous measured force, coefficients corresponding to axial position angle and chip thickness are solved. Further instantaneous peak measured force is used to obtain cutter run-out parameters. Simulation and experiments show that CWE agrees well with that of experiment and ICCE is consistent well with that of simulation. Milling force experiments of different machining parameters show the milling force coefficients and run-out parameters can be applied to fillet end milling under pre-determined inclination angle well.

Effect of electro-thermal conductivity on MHD free convection flow in an inclined porous channel

The twin effect of thermal conductivity and electroconductivity on the flow of fluid in an inclined porous channel was carried out. A combination of Laplace transform and undetermined coefficients method were used to determine the solution of the governing equations of motion and energy. Analysis of the solution using graphs showed that decrease in temperature was observed as a result of increase in radiation, prandtl number, thermal conductivity and free stream frequency while an increase in temperature brings about a corresponding increase in heat generation term. For decrease in velocity profile of the fluid, increase in magnetic field, angle of inclination and electroconductivity are the causes while increase in buoyancy force, Reynolds’ number, porosity and stream frequency, increases the velocity profile of the fluid.

The existence of homoclinic orbits in the Lorenz system via the undetermined coefficient method

In this paper, we are concerned about homoclinic orbits of the Lorenz system{x˙=−σx+σy,y˙=rx−y−xz,z˙=−bz+xy,where σ, b, r > 0 are three parameters. Firstly, by Fredholm alternative theorem and Taylor expansions, one can figure out the local invariant manifolds of the trivial equilibrium E0=(0,0,0), including the stable and unstable ones. Secondly, the undetermined coefficient method is used to find out the negative part of the homoclinic orbit, which is supposed to have the form of exponential type series. Such negative semiorbit together with the positive semiorbit that starts from the locally stable manifold consists of a homoclinic loop. As these two semiorbits should be continuous at the intersection point, and the corresponding series should be convergent, one can establish the conditions for the existence of homoclinic orbits. Finally, as an application, we will exhibit the existence theorem of homoclinic orbits for the Lorenz system with the parameters σ=10,b=8/3,r=13.926.

Optical solitons in fiber Bragg gratings with dispersive reflectivity for parabolic law nonlinearity using undetermined coefficients

This paper reveals bright, dark and singular solitons in fiber Bragg gratings with dispersive reflectivity having parabolic form of nonlinearity. The method of undetermined coefficients yielded such soliton solutions. The existence criteria of such solitons are also enumerated that are listed as constraint conditions.

Regime switching in the present value models: A backward-solving method

We develop a new method for incorporating regime shifts in the present value models. Existing studies solve the present value problem forward by projecting the expectations of investors and regimes into the infinite future. Representing the problem in a recursive form, we propose a backward-solving approach in which the initially guessed formulation of the present value is verified by the undetermined coefficients method. Unlike the forward-solving methods, ours does not resort to unnecessary approximations or overly restricted forms of regime switching. Applied to the Korean housing market, our regime-switching present value model sharply detects two distinctive regimes in the behaviour of the price-rent ratio.

Heron triangles with figurate number sides

By the theory of Pellian equation and the method of undetermined coefficients, we show that there exist infinitely many isosceles Heron triangles whose sides are polygonal numbers and binomial coefficients.

Direct similarity reductions and new exact solutions of the short pulse equation

In this paper, we present some similarity reductions of the short pulse equation(SPE) based on the direct similarity reduction method proposed by Clarkson and Kruskal. These similarity reductions have a more general form than those obtained by using the Lie group method. Especially, we obtain one new similarity reduction which can not be obtained by Lie group method. Furthermore, we derive one new exact analytic solutions by the method of undetermined coefficients.

The Method of Determined Coefficients

The method of undetermined coefficients is a well-established procedure for finding a particular solution to any inhomogeneous linear differential equation with constant coefficients, in which the ...

Analytical modelling of driven pendulum with small angular displacement using Undetermined Coefficients Method

Many researchers have worked on dynamical systems in recent times because of its relevance in Engineering and Applied Physics. This paper attempts to model a damped driven simple pendulum as dynamical system and solve the model using Undetermined Coefficients Method. The model is a second order differential equation. The results obtained show that the motion of the pendulum is affected, significantly, by the driving force and the torque of the pendulum. Specifically, as driving force increases the response maximum amplitude of the pendulum decreases.

Solitons in optical fiber Bragg gratings with dispersive reflectivity

This paper retrieves bright, dark and singular solitons of both types for Bragg gratings with dispersive reflectivity. The method of undetermined coefficients is the integration algorithm adopted in this paper. Several constraints guarantee the existence of such solitons.

Levy solution for bending response of FG carbon nanotube reinforced plates under uniform, linear, sinusoidal and exponential distributed loadings

A new analytical approach for bending response of functionally graded single-walled carbon nanotube (FGCNT) reinforced plate resting on double-layered elastic foundations in thermal environments is presented based on Levy and Navier types solutions. The distribution of the carbon nanotubes is varied through the plate thickness in accordance with a modified power law. Four types of carbon nanotube distributions are considered. The four edges of the plate are simply supported for Navier method, whereas for Levy method, two opposite only of them are simply supported and the other ones are arbitrary. The present FGCNT plate is subjected to uniform, linear, sinusoidal or exponential distributed loadings. A refined shear deformation plate theory with four unknown functions is employed to obtain the closed form solution. The four coupled governing partial differential equations are derived utilizing Hamilton’s principle. Applying Levy solution and then the state space concept to the governing equations, a nonhomogeneous first-order linear ordinary differential system with constant coefficients is obtained. The solution of the homogeneous system (homogeneous solution) is obtained by using the matrix method. While, the method of undetermined coefficients is applied to find the particular solution of the nonhomogeneous linear system. The results obtained by Navier and Levy methods are compared with available results in the literature. Several examples are discussed for various values of the foundation stiffness, CNT volume fraction, various types of plate geometries, CNT distributions, external applied loads and boundary conditions.

Dynamic Behaviors and the Equivalent Realization of a Novel Fractional-Order Memristor-Based Chaotic Circuit

This paper proposed a novel fractional-order memristor-based chaotic circuit. A memristive diode bridge cascaded with a fractional-order RL filter constitutes the generalized fractional-order memristor. The mathematical model of the proposed fractional-order chaotic circuit is established by extending the nonlinear capacitor and inductor in the memristive chaotic circuit to the fractional order. Detailed theoretical analysis and numerical simulations are carried out on the dynamic behavior of the proposed circuit by investigating the stability of equilibrium points and the influence of circuit parameters on bifurcations. The results show that the order of the fractional-order circuit has a great influence on the dynamical behavior of the system. The system may exhibit complicated nonlinear dynamic behavior such as bifurcation and chaos with the change of the order. The equivalent circuits of the fractional-order inductor and capacitor are also given in the paper, and the parameters of the equivalent circuits are solved by an undetermined coefficient method. Circuit simulations of the equivalent fractional-order memristive chaotic circuit are carried out in order to validate the correctness of numerical simulations and the practicability of using the integer-order equivalent circuit to substitute the fractional-order element.

Advanced Family of Newton-Cotes Formulas

In this paper, we introduced an advanced family of numerical composite integration formulas of closed Newton–Cotes-type that uses the function values on uniformly spaced intervals only without any derivative values. To increase the accuracy, we divide the given interval into a number of equal subintervals and integrating on each interval by using integration rules with abscissas outside integration interval. Since there are more unknowns when using including function values outside integration interval in addition to function values of on interval, the order of accuracy of these numerical integration formulas is higher than the standard closed Newton-Cotes formulae. These new formulae are obtained using the method of undetermined coefficients which are based on the concept of the precision of the quadrature formula. The error terms are found using the concept of precision. Also, we compared the errors in an advanced family of numerical composite integration formulas with the errors in composite closed Newton–Cotes-type. Finally, we have presented some examples and then mentioned the related MATLAB codes.

An identification model of cutting force coefficients for five-axis ball-end milling

Five-axis ball-end milling is widely used in energy, ship, aerospace, and other fields. As one of the difficulties in milling research, milling force is often critical to the processing efficiency and quality of part. For predicting the milling force of the five-axis ball-end milling better, this paper presents a cutting force coefficients identification model which is related to the instantaneous chip layer thickness and axial position angle considering the cutter run-out. For five-axis ball-end milling of the oblique plane, the relationship of feed direction, cutter axis vector, and machined surface is parameterized. The boundary curves of cutter workpiece engagement (CWE) are determined by intersecting spatial surfaces. The boundary curves are discretized, and an algorithm of in-cut cutting edge (ICCE) is proposed by defining the distance between discrete points and the cutting edge. Combining the instantaneous chip thickness considering arbitrary feed direction and cutter run-out, the five-axis milling force model of ball-end mill is established. Based on the undetermined coefficient method and the instantaneous average milling force of teeth, the cutting force coefficients identification model corresponding to the instantaneous chip layer thickness and the axial position angle is established. Furthermore, cutter run-out parameters are obtained combined with instantaneous measured milling force of single tooth. The experimental and simulation examples demonstrate that the CWE determined by the spatial surfaces is consistent with the experimental results. The ICCE is in good agreement with the simulation results based on the solid modeling method. A large number of milling experiments under different processing parameters show that the cutting force coefficients and cutter run-out parameters identification model can be effectively applied to five-axis ball-end milling.

Finite-Order VAR Representation of Linear Rational Expectations Models: With Some Lessons for Monetary Policy

This paper considers the solution of a large class of linear rational expectations (LRE) models and its characterization via finite-order VARs. The solution of the canonical LRE model can be cast in state-space form and solved for by the method of undetermined coefficients. In this paper I propose an approach that simplifies the systematic characterization of the solution into finite-order VAR form and checks existence and uniqueness based on the solution of a companion Sylvester equation. Solving LRE models with a finite-order VAR representation via the Sylvester equation is straightforward to implement, efficient, and can be handled easily with standard matrix algebra. An application to the workhorse New Keynesian model with accompanying Matlab codes is provided to illustrate the implementation of the procedure in practice.

The Fracture Analysis of Collinear Periodic Cracks in an Infinite Piezoelectric Plate

The fracture problem of collinear periodic cracks in an infinite transversely isotropic piezoelectric plate subjected to the anti-plane shear stress and the in-plane electric load at infinity is studied. Using the complex function method, the mechanical problem is turned into the boundary value problem of partial differential equations. The solutions of the boundary value problem of partial differential equation are obtained by undetermined coefficients method. Then, considering the periodicity of cracks, the stress intensity factors and the electric displacement intensity factors for mode III near the right tip of every crack are defined, the expressions of the stress fields, electric displacement fields, displacement fields, electric potential fields and the mechanical strain energy release rate around the crack tip are obtained with the assumption that the surface of the crack is electrically impermeable. Finally, interference effect and scale effect of collinear periodic cracks and the mechanical strain energy release rate are discussed by analysis of examples. It can be seen interference effect of collinear periodic cracks is strong when 1 < b / a < 2. The scale effect of the singularity of the stress intensity factors and electric displacement intensity factors in crack tip is obvious. Stress always promotes extension of the cracks, the mechanical strain energy release rate is related to the size and direction of the electric field, the positive electric field can promote the expansion of the cracks, the negative electric field can inhibit the extension of cracks.

Fully developed mixed convection flow in a vertical microtube with time periodic heating boundary condition

Purpose The purpose of this paper is to investigate fully developed mixed convection flow in the steady-periodic regime for a Newtonian fluid in a vertical microtube in the presence of velocity slip and temperature jump, which has not been accounted for in the literature. Design/methodology/approach To achieve this objective, the governing equations for the problem are separated into steady and oscillatory components using separation of variable method; this gives a pair of independent boundary value problems. This is then solved along with its boundary conditions and constraint equations using the method of undetermined coefficient. The exact solutions of momentum and energy equations are obtained under the velocity slip and temperature jump conditions. Findings The significant result from the study is that increase in rarefaction parameter as well as fluid–wall interaction parameter decreases the oscillation amplitude of the dimensionless velocity. Furthermore, it was found that the product of dimensionless frequency and Prandtl number initiate a strong convection current inside the microtube. Practical implications Such type of study may be used on the determination of the thermal and tangential momentum accommodation coefficients and be applicable to the designs and fabrications of microheat exchanger. Moreover, it provides the possibility to get a bench mark for numerical solvers with reference to basic flow configuration. Originality/value These solutions generally deserve great attention, since the application of a magnetic field has been found to be effective tool in controlling the convection current. The current work is aimed as an extension of the previous analytical studies to prove some insight into a number of industrial applications, which use similar configurations.

Optical solitons with polarization mode dispersion for Lakshmanan–Porsezian–Daniel model by the method of undetermined coefficients

This paper obtains bright, dark and singular optical soliton solutions to the Lakshmanan–Porsezian–Daniel model that describes soliton propagation through polarization-mode dispersive fibers, without the effect of four-wave mixing. The method of undetermined coefficients is employed to retrieve these soliton solutions. The existence criteria for these solitons are also presented.