Generalized Newton-Raphson Method Research Articles

Numerical simulation for two species time fractional weakly singular model arising in population dynamics

ABSTRACT In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order . The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme.

NAO robots as context to teach numerical methods

This article examines the contextual role that NAO humanoid robots play in the learning environment of a numerical methods course through a design-based research approach. Under the design of intentional and reflective planning that emanates from the socio-critical paradigm in education, the feasibility of using this type of robot in the classroom as a motivation element to engage students in understanding numerical analysis issues is addressed. In particular, the active learning of engineering students is achieved by studying such robots' forward and inverse kinematics. The didactical sequence to teach the generalized Newton–Raphson method for solving non-linear equation systems is validated as part of a Hypothetical Learning Trajectory proposed. Besides, this research provides a brief theoretical discussion on reflection-in-action of the teacher of Higher Education level.

Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation

In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a space-time finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the so-called streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each space-time slab are solved by the generalized Newton-Raphson method. Exploiting the band-structured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each time-step a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.

Microwave dielectric property based classification of renal calculi: Application of a kNN algorithm

The proper management of renal lithiasis presents a challenge, with the recurrence rate of the disease being as high as 46%. To prevent recurrence, the first step is the accurate categorization of the discarded renal calculi. Currently, the discarded renal calculi type is determined with the X-ray powder diffraction method which requires a cumbersome sample preparation. This work presents a new approach that can enable fast and accurate classification of discarded renal calculi with minimal sample preparation requirements. To do so, first, the measurements of the dielectric properties of naturally formed renal calculi are collected with the open-ended contact probe technique between 500 MHz and 6 GHz with 100 MHz intervals. Cole–Cole parameters are fitted to the measured dielectric properties with the generalized Newton–Raphson method. The renal calculi types are classified based on their Cole–Cole parameters as calcium oxalate, cystine, or struvite. The classification is performed using k-nearest neighbors (kNN) machine learning algorithm with the 10 nearest neighbors, where accuracy as high as 98.17% is achieved.

Hybrid Approach by Use of Linear Analogs for Gas-Network Simulation With Multiple Components

Summary The generalized Newton-Raphson method is routinely deployed in industrial and academic applications in order to solve complex systems of highly nonlinear equations. Prime candidates for this solution methodology are complex natural-gas transportation networks, of which the nonlinear governing equations can be written in terms of nodal, loop, or nodal/loop formulations to solve for all network pressures and flows. A well-known issue of the Newton-Raphson iterative methodology is its hapless divergence characteristics when poorly initialized or when flow loops are poorly defined in loop or nodal/loop formulations. In this study, a method of linear analogs is discussed, which eliminates the need for user-prescribed flow or pressure initializations or loop definitions in the solution of the highly nonlinear gas-network governing equations. A comprehensive solution strategy based on analog transformations is presented for the analysis of a gas-pipeline network system comprised of not only pipes, but also common nonpipe network elements such as compressors and pressure-dependent gas supplies (e.g., wellheads). The proposed approach retains advantages of Newton-nodal formulations, while removing the need for initial guesses, Jacobian formulations, and calculation of derivatives. Case studies are presented to showcase the straightforward and reliable nature of the methodology when applied to the solution of a steady-state gas-network system analysis with pipeline, compressor and well-head components.

Thermal design of an orthotropic flat fin in fin-and-tube heat exchangers operating in dry and wet environments

A semi-analytical method was described to determine the thermal performance of fin-and-tube heat exchangers with orthotropic fin materials and wet surfaces. The effect of the orthotropic thermal conductivity was investigated over a range of design parameters for both wet and dry fins, and design optimization was carried out using a generalized Newton–Raphson method. The optimized designs were analyzed to gain insight into the relative importance of the different parameters under varying conditions, and a set of curves was determined to provide design optimization guidelines.

Soil Parameter Identification and Driving Force Prediction for Wheel-Terrain Interaction

This paper considers wheeled vehicles traversing unknown terrain, and proposes an approach for identifying the unknown soil parameters required for vehicle driving force prediction (drawbar pull prediction). The predicted drawbar pull can potentially be employed for traversability prediction, traction control, and trajectory following which, in turn, improve overall performance of off-road wheeled vehicles. The proposed algorithm uses an approximated form of the wheel-terrain interaction model and the Generalized Newton Raphson method to identify terrain parameters in real-time. With few measurements of wheel slip, i, vehicle sinkage, z, and drawbar pull, DP, samples, the algorithm is capable of identifying all the soil parameters required to predict vehicle driving forces over an entire range of wheel slip. The algorithm is validated with experimental data from a wheel-terrain interaction test rig. The identified soil parameters are used to predict the drawbar pull with good accuracy. The technique presented in this paper can be applied to any vehicle with rigid wheels or deformable wheels with relatively high inflation pressure, to predict driving forces in unknown environments.

Variable dimension Newton-Raphson method

The classical Newton-Raphson method is generalized to solve nonsquare and nonlinear problems of size m/spl times/n with m/spl les/n. Using this generalized Newton-Raphson method as a core, a new variable dimension Newton-Raphson (VDNR) method is developed. The VDNR method is verified to have a better convergence property than the classical Newton-Raphson method by benchmark testing.

A generalized Newton–Raphson method using curvature

AbstractA numerical method for finding the roots of any function is developed. This method considers a circle using the concept of curvature instead of the tangential line in the Newton‐Raphson method. The compared results between the proposed method and the Newton‐Raphson method are listed. The proposed method has a wider convergent region of initial points and finds more proper solutions than the Newton‐Raphson method. In particular, the paper proposes that the curvature method is replaced by the modified Newton method discussed by Ralston in dealing with multiple roots.

Minimization of Energy Consumption for a Manipulator with Nonlinear Friction in PTP Motion.

Robot engineering is developed mainly in the field of intelligibility such as in a manipulation. Considering the wide use of robots in the future, the robots should be studied from a viewpoint of saving energy because a robot is a kind of machine with an energy conversion.This paper deals with minimizing the energy consumption of a manipulator which is driven in a point-to-point control method. When the links of a manipulator are decelerated for positioning, the motors at the joints generate electric power. Since this energy can be regenerated to the source by using a chopper, the energy consumption of a manipulator is only by the heat loss of the electric and the frictional resistances. The minimization of the sum of these losses is reduced to a two-point boundary-value-problem of a nonlinear differential equation. The solutions of the equations are obtained by the generalized Newton-Raphson method. In this case the starting function for an optimal solution is selected from a view point of saving energy. It is seen from simulations that the second link of a two-link manipulator should be rotated backward in a certain condition.

A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation

This paper is concerned with the problem of convexity-preservng (orc-preserving) interpolation by using Exponential Splines in Tension (or EST's). For this purpose the notion of ac-preserving interpolant, which is usually employed in spline-in-tension interpolation, is refined and the existence ofc-preserving EST's is established for the so-calledc-admissible data sets. The problem of constructing ac-preserving and visually pleasing EST is then treated by combining a generalized Newton-Raphson method, due to Ben-Israel, with a step-length technique which serves the need for “visual pleasantness”. The numerical performance of the so formed iterative scheme is discussed for several examples.

A New Algorithm for Limit Cycle Analysis of Nonlinear Control Systems

A numerical method is presented for the limit cycle analysis of multiloop nonlinear control systems with multiple nonlinearities. Describing functions are used to model the first harmonic gains of the nonlinearities. Existence of a limit cycle is sought by driving the least damped eigenvalues to the imaginary axis. The evolution of the limit cycle is studied next as a function of a critical system-parameter. It is shown that by defining a suitable error function it is possible to use both eigenvalue as well as the eigenvector sensitivities to formulate a generalized Newton-Raphson method to solve simultaneously for the updates of state variable amplitudes in a minimum norm sense. Several case studies have been presented and the development of a numerical procedure to test the stability of the limit cycle has also been reported.

Numerical computation of solutions to the KdV equation in double Chebyshev series

The Korteweg-de Vries equation is studied by a computational technique using double Chebyshev series expansions. The non-linear determining equations for the unknown coefficients are solved by the generalized Newton-Raphson method. The solitary wave solution is chosen as a test case to illustrate the efficiency of the method.

Phosphonate complexes. Part 5. An improved multiparametric refinement program, MUCOMP, for complex formation studies from potentiometric data

A new version of the refinement program MUCOMP is described. The stability constants of the complex species as well as the parameters liable to comport systematic errors (total concentration of main components and of impurities, and electrodes characteristics) are adjusted so as to minimize the weighted error-square sum of the added volume, the partial derivatives with respect to the unknown parameters being evaluated analytically. A generalized Newton–Raphson method is employed for the simultaneous calculation of the free concentrations and of the added volume. To deal with carbonate-contaminated reagents and the subsequent carbon dioxide evolution which occurs during titrations under nitrogen flow, an appropriate function for carbon dioxide departure is introduced in the proton balance and the corresponding equilibrium defect is taken into account in the weighting procedure.

Application of quasilinearization technique to the vertical channel flow and heat convection

The quasilinearization technique, also known as the generalized Newton-Raphson method, is employed to solve the problem of fluid injection through one side of a long vertical channel. With very approximate starting values, only a few iterations are needed to obtain a result with high accuracy. The results are compared with those obtained by the perturbation technique. Although the perturbation technique is perhaps the simplest and the most straight-forward method to achieve solutions to non-linear boundary value problems, the limitation on the magnitude of perturbed parameters hinders its applicability.

An empirical interpretation for the time evolution of the Caii K line

An empirical approach to interpret the time evolution of the high spatial resolution Ca ii K line is presented. We specify the physical parameters, such as electron temperature, hydrogen density, and velocity (microturbulent and systematic) as functions of height. The electron density is obtained from scaled non-LTE solutions for hydrogen ionization. The population indices, and thus the Ca ii source functions, for a 5-level Ca ii atom are computed by using the generalized Newton-Raphson method. K line profiles are then synthesized for different evolutionary stages and are compared with the observed ones. The explanation of the ‘peculiar’ type profile is also attempted.

The time-optimal design of linear dynamic systems

The following general time-optimal design problem is studied: determine a real constant square matrix, A , subject to specified constraints, to minimize the transit time between specified endpoints while satisfying the vector differential equation \dot{x}(t) = Ax(t) . Two specific kinds of constraints on A are considered: 1) where the individual elements of A are free but the matrix as a whole must satisfy Q(A) \leq \hat{Q} where Q(A) is a specified homogeneous function of the elements of A and \hat{Q} is a given upper bound and 2) where the elements of A are individually bounded. Theoretical results show that both problems can be solved by first solving a related minimum cost fixed time problem. The latter problem is solved iteratively by using the generalized Newton-Raphson method for two point boundary value problems to provide a set of linear equations at each iteration. The cost function Q is then minimized subject to these equations using appropriate optimization techniques.

Quasilinearization technique for solution of unsteady state diffusion problems

A new iterative method is given for the numerical solution of the diffusion equation ∂C/∂t = ∂/∂x(D∂C/∂x) in a semi-infinite medium. The nonlinear partial differential equations for unsteady state diffusion problems with variable diffusion coefficients are transformed into nonlinear ordinary differential boundary value problems. It is shown that the quasilinearization technique (also known as the generalized Newton-Raphson method) is an effective tool for solving these non-linear ordinary differential boundary value problems. The versatility and convergence properties of this method are demonstrated by the solution of several representative cases. The following cases discussed herein for which the diffusion coefficients are functions of concentration possess solutions with differing stabilities and rates of convergence: D = D 0(1 + αC C 0 ), D 0(1 + αC C 0 + βC 2 C 0 2 ) , D 0(1 + αC C 0 + βC 2 C 0 2) , D 0 (1 − αC C 0 ), D 0 (1 − αC C 0) 2 , D 0e αC C 0 and D 0(α + β log (γ + δC C 0 )).

Applying quasilinearization to the nonlinear analysis of elastic bars

The quasilinearization technique, also known as the generalized Newton-Raphson method, is used to solve the nonlinear differential equations with a set of integral form of boundary conditions representing the deflection of an elastic bar with immovable ends. With very approximate starting values, only two to five iterations are needed to obtain a result with very high accuracy. The results are compared with those obtained by the perturbation technique.

On the application of the generalized Newton-Raphson method in radiative transfer problems

The simulataneous excitation and transfer of radiation in a plasma gives rise to non-linear operator equations for the atomic occupation numbers. An iterative method for solving such equations, based on a generalized Newton-Raphson scheme, is described and applied to the problem of the formation (under the condition of complete redistribution) of a resonance emission line interacting with two associated subsidiary lines (the three level atom or (Lα, Lβ, Hα) problem).