Analytical Solutions Of Nonlinear Equations Research Articles

Reactive Power Compensation for Preventing Voltage Instabilities in Distribution Lines Enriched with PV Power Plants

The efficient voltage regulation method by capacitive reactive power usage is represented in the article. The stochastic electricity generation of multiple PV stations placed in local distribution lines induces voltage instabilities. These voltage fluctuations may overcome the allowable limits and cause technical problems and economic losses. The method of applying reactive power generated by a capacitor bank at the load is proposed. Primarily, a strict analytical solution of nonlinear equations describing voltage deviations in the distribution line connected with photovoltaic systems was obtained. The control system can fast select optimal capacitors based on the analytical solution and then connect them smoothly to a load. The developed experimental setup verified and approved the correctness and accuracy of theoretical analysis. Experimental measurements confirmed the proposed method’s ability to improve voltage stability in distribution grids. The proposed method is supposed to be used together with the transformer tap-changers in electrical substations. The capacitive power is applied in a time of inappropriate voltage deviations only and is disconnected as the tapchanger corrected the transformation ratio.

PMSM Torque Ripple Minimization Based on Novel Low Carrier Ratio PWM Technique

Harmonic injection is commonly used in permanent magnet synchronous motor torque ripple minimization (TRM) of electric vehicle. Traditional inverter switching frequency limits TRM application at high speed due to harmonic modulation accuracy and inverter loss. To achieve TRM in the whole speed range and reduce inverter loss, a novel online calculated low carrier ratio pulsewidth modulation (LCRPWM) is proposed in this article. The carrier ratio determining regular according to the injected harmonics is established first. The relationship between injected harmonics and LCRPWM switching angle is revealed and a set of nonlinear equations is established. Thus, the problem is transformed from solving nonlinear equations into optimization by establishing a cost function. The analytical solutions of nonlinear equations are obtained, which avoids iterative operation and plunging into local minimum. Therefore, the switch angle solution method is suitable for online calculation. The LCRPWM is verified in simulation compared with the traditional regularly sampled space vector PWM, which achieves better results in voltage harmonic injection and flux tracking at a much lower switching frequency. At last, the LCRPWM is used in a test bench experiment that shows good TRM effects. The influence of sideband harmonics on torque ripple is analyzed.

Capacitive reactive power compensation to prevent voltage instabilities in distribution lines

This article presents an efficient voltage regulation method using capacitive reactive power. Simultaneous operation of photovoltaic power systems with the local grids induces voltage instabilities in the distribution lines. These voltage fluctuations cross the allowable limits on several occasions and cause economic losses. In the proposed method, the reactive power is applied at the load and generated using a capacitor bank. The capacitors are arranged in a binary order of capacitances to enable the 2n equally dispersed combinations. Initially, a strict analytical solution is developed to study the outcome of capacitance connection at load. The capacitance is selected based on the analytical solution of non-linear equations describing voltage deviations in the distribution line connected with photovoltaic systems. The control system is developed for fast and appropriate capacitance selection and a smooth transient process. The experimental setup is developed to verify the theoretical analysis. The measured data shows good agreement with the calculated one, verifying the correctness and accuracy of the proposed method. It is recommended that the reactive power compensation can be applied for a shorter time because the source current enhances substantially as the capacitance is connected to the load. The proposed method can be applied together with the tap-changers functionality, and the capacitance should be disconnected as the tap-changer corrects the transformation ratio.

A problem-based learning proposal to teach numerical and analytical nonlinear root searching methods

Nonlinear equations are usually solved numerically. However, approximated analytical solutions of nonlinear equations are useful as an initial iteration point for numerical methods. Furthermore, these analytical approximations are sometimes quite close to the actual root. In order to teach at undergraduate level how to perform this kind of approximation, we propose problem-based learning with some nonlinear equations extracted from Applied Sciences. We also present a simple method to analytically solve nonlinear equations which are not solvable by standard computer algebra solvers.

Sinusoidal motion of small particles through a Darcy-Brinkman-Forchheimer microchannel filled with non-Newtonian fluid under electro-osmotic forces

This article deals with the electro-osmotic flow of non-Newtonian fluid in the presence of small particles moving in the sinusoidal form in a Darcy-Brinkman-Forchheimer medium. A transverse magnetic field is applied by ignoring the effects of an induced magnetic field with a small magnetic Reynolds number. To examine the non-Newtonian effects, Jeffrey fluid model is considered. The proposed mathematical modeling is based on the Poisson–Boltzmann equation, Debye length approximation, and ionic Nernst Planck equation. The modeled problem is formulated in a fixed frame, which is further transformed into a wave frame. Lubrication theory is also taken into account. Analytical solutions of nonlinear equations are obtained by using perturbation method. The impact of all the parameters is discussed for both fluid- and particulate-phases. The present effects are beneficial in designing and manufacturing microfluidics devices to determine the transport mechanism.

Exact analytic solutions of nonlinear equations

The improved sub-equation method is used to obtain the exact analytical solution of (2 + 1) - dimensional BBM Equation. Some traveling wave solutions such as soliton solution and periodic wave solution of trigonometric function are obtained. Practice shows that the sub-equation method is widely used in nonlinear evolution equations.

New Exact Solutions of a Perturbed Nonlinear Schrodingers in Nonlinear Optics

Here, new exact travelling wave solutions of a perturbed nonlinear Schrodingers equation arising in nonlinear optics are successfully obtained via the generalized Kudryashov method. The proposed method has been successfully implemented to seek exact solutions for the modeles coming for describing nonlinear optics. The results obtained by the generalized Kudryashov method is straightforward and concise mathematical tool to establish the exact analytical solutions of nonlinear equations. The solutions obtained here are new and have not been reported in former literature.

Determination of the Correct Range of Physical Parameters in the Approximate Analytical Solutions of Nonlinear Equations Using the Adomian Decomposition Method

Physical parameters in dimensionless form in the governing equations of real-life phenomena naturally occur. How to control them by determining their range of validity is in general a big issue. In this paper, a mathematical approach is presented to identify the correct range of physical parameters adopting the recently popular analytic approximate Adomian decomposition method (ADM). Having found the approximate analytical Adomian series solution up to a specified truncation order, the squared residual error formula is employed to work out the threshold and the existence domain of certain physical parameters satisfying a preassigned tolerance. If the current procedure is not closely pursued, the presented results with the ADM may not be up to the desired level of accuracy (the worst is the divergent physically meaningless solutions), or much more ADM series terms need to be computed to satisfy certain accuracy. Examples reveal the necessity of the present approach to make sure that the results embark the correct range of physical parameters in the study of a physical problem containing several dominating parameters.

The Approximate Analytical Solution of Non-Linear Equation for Simultaneous Internal Mass and Heat Diffusion Effects

For the first time a mathematical modelling of porous catalyst particles subject to both internal mass concentration gradients as well as temperature gradients, in endothermic or exothermic reactions has been reported. This model contains a non-linear mass balance equation which is related to rate expression. This paper presents an approximate analytical method (Modified Adomian decomposition method) to solve the non-linear differential equations for chemical kinetics with diffusion effects. A simple and closed form of expressions pertaining to substrate concentration and utilization factor is presented for all value of diffusion parameters. These analytical results are compared with numerical results and found to be in good agreement.

Approximate Analytical Solution of Non-linear Equation in One Dimensional Instability Phenomenon in Homogeneous Porous Media in Horizontal Direction by Variational Iteration Method

The present paper analytically discusses Instability phenomenon in double phase flow through homogeneous porous media in horizontal direction. This phenomenon is very important in oil recovery. Mathematical model for Instability phenomenon in homogeneous porous media in horizontal direction without source or sink is derived. Governing nonlinear partial differential equation has been solved using Variational iteration method with the help of specific initial and boundary conditions.

Approximate Analytical Solution of Non-linear Equation in one Dimensional Imbibitions Phenomenon in Homogeneous Porous Media by Homotopy Perturbation Method

Imbibitions process is very important process in oil recovery. In this paper mathematical model for imbibition phenomenon in one dimensional homogeneous porous media is derived. Governing nonlinear partial differential equation is solved by using homotopy perturbation method. Homotopy perturbation method is a new perturbation method is proposed. The initial approximation is chosen using initial and boundary conditions in the form of quadratic equation.

Analytical Solutions of Non-Linear Equations of Power-Law Fluids of Second Grade over an Infinite Porous Plate

The flow of an incompressible fluid of modified second grade past an infinite porous plate subject to either suction or blowing at the plate is studied. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. Equations of motion in dimensionless form are derived. Analytical solutions of the outcoming non-linear differential equations are found by using the homotopy analysis method (HAM), which is a powerful semi-analytical method. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

Analytical Investigation of Jeffery-Hamel Flow with High Magnetic Field and Nano Particle by RVIM

Many researchers have been interested in application of mathematical methods to find analytical solutions of nonlinear equations and for this purpose, new methods have been developed. One of the newest analytical methods to solve nonlinear equations is Reconstructio n of variational Iteration Method (RVIM) which is an accurate and a rapid convergence method in finding the approximate solution for nonlinear equations. By applying Laplace Transform, RVIM overcomes the difficulty of the perturbation techniques and other variational methods in case of using small parameters and Lagrange multipliers, respectively. In this study RVIM is applied for the effects of magnetic field and nano particle on the Jeffery-hamel flow. The traditional Navier-Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. Finally the effect of nano particle volume fraction in the absence of magnetic field is investigated, too. The validity of RVIM method is ascertained by comparing our results with numerical (Runge Kutta method) results.

Simulating fractional order systems using multistage Adomian decomposition method

AbstractNonlinear phenomena play a crucial role in applied mathematics and physics. Analytic solutions of nonlinear equations are of fundamental importance and various methods for obtaining analytic solutions have been proposed. In this paper an application of the Adomian decomposition method is introduced for solving nonlinear fractional equations. The results reveal that the proposed method is very effective and simple and leads to accurate, approximately convergent solutions of nonlinear equations.

An approximation of the analytical solution of the Jeffery–Hamel flow by decomposition method

Many researchers have been interested in application of mathematical methods to find analytical solutions of nonlinear equations and for this purpose, new methods have been developed. Since most of fluid mechanics problems due to boundary layer are strongly nonlinear, so analytical solution of them is confronted with some difficulty. In this Letter, the Jeffery–Hamel flow—a nonlinear equation of 3rd order—is studied by Adomian decomposition method. After introducing Adomian decomposition method and the way of obtaining Adomian's polynomial, we solved the problem for divergent and convergent channels. Finally, velocity distribution and shear stress constant is depicted at various Reynolds numbers and comparing our results with some earlier works illustrated their excellent accuracy.

Homotopy perturbation method for the solution of the electrostatic potential differential equation

This paper obtains an explicit analytical solution for nonlinear Poisson‐Boltzmann equation by the homotopy perturbation method, which does not require a small parameter in the equation under study, so it can be applied to both the weakly and strongly nonlinear problems. The obtained results show the evidence of the usefulness of the homotopy perturbation method for obtaining approximate analytical solutions for nonlinear equations.

Dynamo and Electrical Jet in Hall Plasmas, Application to Astrophysics

The magnetic field in Hall plasmas is frozen in the electron component and is advected not only with the plasma motion but also with the electrical current flow. Its coupling with the plasma may be not as strong as characteristic of the MHD approximation. The rotation and slipping of the magnetic field result in a very different and less efficient magnetic field generation by dynamo-rotating plasma disk in magnetic field. We found some exact analytical solutions of nonlinear equations describing the dynamo. In particular, the dynamo may not dissipate the energy in the steady state limit. The 3-component magnetic field and magnetic energy are generated and accumulated during the transition time only. An electrical jet is another unusual phenomenon for MHD. It was investigated theoretically and experimentally for laboratory plasmas during the last 15 years and now is used for fast current switching. We found periodical or shock-like nonlinear wave traveling along a Hall plasma column and not associated with plasma motion. A nonlinear equation describing a possible steady state magnetic field distribution and current flow in a Hall plasma conductor is derived, which differs from the Grad-Shafranov equation. In conclusion we discuss how transfer our results onto astrophysical plasmas. This physics could be important for understanding the evolution of dusty plasma disks and jets around new stars.

Iteration Perturbation Method for Strongly Nonlinear Oscillations

In this paper, the author proposes a new perturbation technique coupling with iteration method, yielding a powerful mathematical tool for an analytical solution of nonlinear equations. The obtained results are valid not only for weakly nonlinear problems but also for strongly nonlinear ones. Furthermore, the approximate solutions are valid for the whole solution domain, and even the first-step iteration leads to high accuracy Some examples are given to illustrate its effectiveness.

Spectrum and wave shape of intense laser pulse propagating through Raman active medium

Results of the analytical solution of non-linear equations describing the propagation of an ultra-short intense laser pulse through a Raman-active medium are presented. The pulse passing through the medium generates a large number of high-order harmonics if the pulse intensity is close to some thresholds. The medium is described by the density matrix for two-level systems. The laser pulse duration was much shorter than both the period of oscillation and the relaxation times of an oscillator. Using this solution, the spectrum of such a pulse was expressed in terms of integrals of the initial field. The solution allows the harmonic spectrum dependence on the pulse intensity and effects of propagation to be investigated. Estimations of the conversion efficiency of the initial pumping frequency into high-order harmonics are given. The evolution of the pulse shape as a function of the distance passed by the pulse through the media was investigated.

Analytic solutions for nonlinear equations

Solutions obtained by the decomposition method for physical problems modeled by nonlinear differential (and partial differential) equations without linearization or pertubation are not only continuous but analytic as well. Thus the solution and all derivatives are obtained.