Solution Of Nonlinear Problem Research Articles

ПЕТРОВСКИЙ БОЙЫНША ЭЛЛИПСТІК ЖҮЙЕ ҮШІН СЫЗЫҚТЫҚ ЕМЕС БІР ЕСЕПТІҢ ШЕШІЛУІ ЖАЙЛЫ

The article presents the results of study the existence of the solution of nonlinear problem for elliptic by Petrovsky system in unbounded domain. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Applying the fixed point and embedding theorems the solvability of nonlinear problem for elliptic by Petrovsky system in weighted Sobolev space are established.

Chemical Reactors Catalyst bed and Analytical Solution of Complicated Partial Nonlinear Differential Equations by new Approach AKLM

In this paper, for the first time, we investigate and solve a complicated highly partial nonlinear differential equations of chemical reactions in the cylindrical and spherical coordinates on the catalyst bed for the design of chemical reactors. We challenge and prove the power of this method that it can easily analyze the most difficult nonlinear problems in a completely analytical way, which we named it AKLM (Akbari Kalantari Leila’s Method). Certainly we know that the chemical reaction process on the solid catalysts is very complex, and the governing differential equations governing them are nonlinearity complex. In this paper we present a new analytical solution which can easily analyze all such problems and make a great evolution in the nonlinear industries in the reactors design in chemical engineering. Finally, this scientific approach can create a great phenomenon in the analytical solution of nonlinear problems in engineering sciences, especially in the chemical and mechanical engineering.

Stable Generalized/eXtended Finite Element Method with global–local enrichment for material nonlinear analysis

This paper presents a novel methodology for the solution of nonlinear problems using the Stable Generalized/eXtended Finite Element Method (SGFEM) in two-scale approach. The nonlinear analysis is performed in a coarse mesh and the solution of a local problem, represented by a fine mesh, is used to enrich the partition of unity of the coarse mesh in the so-called global–local strategy. The SGFEM is adopted to deal with the problem of ill-conditioning of the stiffness matrix triggered by the enrichment strategy. Here, this problem is investigated for the first time for global–local analysis of physically nonlinear problem. In addition to the use of SGFEM, it is noteworthy that the relationship between the incremental-iterative process and changes in the constitutive variables are simultaneously updated on both scales. Differently from original GFEM global–local strategies, the evolution of the state variables is conduced only in the global scale. The local problem heritages the material degradation of the global scale, by using a mapping technique, and provides the numerically obtained functions that enriches the global approximation. Different constitutive models and material laws are adopted to incorporate the material degradation. Numerical examples are presented for validating this approach and to show its generality and efficiency.

Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Based on a new kind of analytical approach, namely the Optimal Auxiliary Functions Method (OAFM), a new analytical procedure is proposed to solve the problem of the annular axisymmetric stagnation flow and heat transfer on a moving cylinder with finite radius. As a novelty, explicit analytical solutions were obtained for the considered complex problem. First, the Navier–Stokes equations were simplified by means of similarity transformations that depended on different parameters and some combinations of these parameters, and the problem under study was reduced to six nonlinear ordinary differential equations with six unknowns. The OAFM proves to be a powerful tool for finding an accurate analytical solution for nonlinear problems, ensuring a fast convergence after the first iteration, even if the small or large parameters are absent, since the determination of the convergence-control parameters is independent of the magnitude of the coefficients that appear in the nonlinear differential equations. Concerning the main novelties of the proposed approach, it is worth mentioning the presence of some auxiliary functions, the involvement of the convergence-control parameters, the construction of the first iteration and much freedom to select the procedure for determining the optimal values of the convergence-control parameters.

Why heat transfer coefficients are unnecessary and undesirable, and how heat transfer problems are solved without them.

For 200 years, convective heat flux q has been calculated by multiplying heat transfer coefficient h times boundary layer temperature difference T. Since h times T equals q, h must be a symbol for (q/T) because (q/T) times T equals q. h (ie q/T) is generally calculated from correlations derived from experiments in which q data and T data are used to obtain (q/T){T} correlations-ie h{T} correlations. (It is not possible to obtain h data because h is not a parameter. h is the ratio of two parameters). Heat transfer coefficients are unnecessary and undesirable. It is self-evident that any problem that can be solved using q, q/T (ie h), and T can also be solved using only q and T. Therefore h (ie q/T) is unnecessary. h (ie q/T) is undesirable because, when q is a nonlinear function of T (as in free convection, condensation, and boiling), h (ie q/T) is an extraneous variable, and it greatly complicates problem solutions. When h has been abandoned, convective heat flux is determined from q{T} correlations that result from q data and T data, or from the transformation of h{T} correlations. (Transformation from h{T} correlations to q{T} correlations requires that h be replaced by q/T, and that q and T be separated.). The text includes example problems that validate the conclusion that h (ie q/T) is unnecessary and undesirable, and demonstrate that the solution of nonlinear problems is much simpler if h is abandoned.

Numerical Verification of Solutions for Nonlinear Parabolic Problems

In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakao’s projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.

Multiscale model reduction for the Allen–Cahn problem in perforated domains

In this paper, we consider a class of multiscale methods for the solution of nonlinear problem in perforated domains. These problems are of multiscale nature and their discretizations lead to large nonlinear systems. To discretize these problems, we construct a fine grid approximation using the finite element method with implicit time approximation and the Newton’s method. In order to solve these large systems efficiently, we will develop a model reduction procedure. To perform the model reduction, we construct a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM). The GMsFEM consists of an offline and online stages. In the offline stage, we construct multiscale basis functions based on the solution of some local spectral problems defined in the snapshot space. Then, we enrich the offline multiscale space by additional multiscale basis that handle non-homogeneous boundary condition. For the accurate solution of the nonlinear problem, we use two techniques on the online stage: (1) residual based multiscale basis functions and (2) residual based local correction. We will present numerical results for two-dimensional Allen–Cahn problems in perforated domains.

Neuro-optimized numerical solution of non-linear problem based on Flierl–Petviashivili equation

The article presents, a reliable numerical framework supported by feed-forward Artificial Neural Network (ANN) optimized with hybrid swarm intelligence technique for non-linear system of Flierl–Petviashivili (FP) problem. The universal approximation capabilities of ANN are exploited for mathematical approximation of the system in an unsupervised way based upon various performance metrics like fitness value, absolute error and execution time. The optimization of the cost function is subject to finding the appropriate weights which are highly stochastic in nature for the problem as well as its initial and boundary conditions. Therefore, hybrid approach based on Particle Swarm Optimization (PSO) and Interior Point Algorithm (IPA) is exploited for tuning of the adaptive weights in such a way that exploration is performed by PSO while the exploitation is done using IPA algorithm. The designed scheme is evaluated for standard FP problem along with its variants supported on various scenarios. The reliability, accuracy and robustness of the solvers are validated through a statistical analysis applied on two hundred independent runs.

Method of application of analytical solution of the piecewise linear approximation of magnetization curve on the example of calculation the peak-transformer parameters

The purpose. Paper purpose is working out of a technique of the solution of nonlinear problems in an analytic form, namely a piecewise linear approximation of magnetization curve of a magnetic circuit by discontinuous functions, and the solution of the nonlinear differential equation for an electric circuit which contains this type of device. For an example of use of a technique the approximate calculation of the peak-transformer working in a regime of saturation of a magnetic circuit is resulted. Methodology. By working out of a technique of piecewise linear approximation in the capacity of the basic mathematical apparatus individual discontinuous functions were used. Calculation of a nonlinear problem for a current finding in the transformer was analogous to a nonlinear problem for magnetic field strength finding in a ferromagnetic. Results. The design procedure of currents and voltage of the transformer working in a regime of saturation, on the basis of analytical record of a piecewise linear approximation of magnetization curve of a magnetic circuit is developed. Originality. For the first time with reference to the characteristic of magnetization of the peak-transformer the method of piecewise linear approximation is applied. Practical value. The offered technique can be directly used for electromagnetic calculation of peak - transformer as a part of the general (operated) electrotechnical device at a design stage and optimization.

The Convergence Estimation of the Parallel Algorithm of the Linear Cauchy Problem Solution for Large Systems of First-Order Ordinary Differential Equations Using the Solution as Expansion over Orthogonal Polynomials

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process.

Solution of nonlinear problems by a new analytical technique using Daftardar-Gejji and Jafari polynomials

This article shows the solution of nonlinear differential equations by a new analytical technique called modified optimal homotopy perturbation method. Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the equation. Four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders are solved by modified optimal homotopy perturbation method as well as optimal homotopy perturbation method. The achieved consequences are authenticated by comparison with the results gained by the existing method—optimal homotopy perturbation method. The method consists of few steps and gives better results. The easy applicability and fast convergence are goals of the applied technique. The applied technique has fewer limitations and can be used for the phenomena containing ordinary differential equation, partial differential equation, integro-differential equation, and their systems.

Influence of Dynamics Viscosity on the Water Base Graphene Oxide–Ethylene Glycol/Graphene Oxide–Water Nanofluid Flow Over a Stretching Cylinder

This research paper presents the impact of dynamics viscosity of water base GO–EG (graphene oxide–ethylene glycol)/GO–W (graphene oxide–water) nanofluid over a stretching cylinder with non-porous medium. The impact of different parameter for both velocity and temperature profile are displayed and discussed graphically. The similarity transformation are used to convert the partial differential equation to nonlinear ordinary differential equation. The solution of the problem is obtained using the optimal homotopy analysis method (OHAM). (Liao, S. J., 2010. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15, pp.2003–2016) used this method for the solution of nonlinear problem and show that this method is quickly convergent to the approximate solution. This method gives us series solution in the form of function, and all the physical parameters of the problem involved in this method. The stability of the problems is also obtained up to the 30th order approximation using the BVPh 2.0 package. The outputs are displayed graphically and discussed. The effects of various parameters on the skin friction coefficient and Nusselt number coefficient of base GO–EG/GO–Ware displayed and discussed.

An Approximate Solution of Fractional Kolmogorov-Petrovskii-Piskunov Equations

An approximate-analytical method known as Reduced Differential Transform Method (RDTM) is proposed to approximate the Fractional Kolmogorov- Petrovskii –Piskunov Equations. It is a powerful and convenient approximation analytical tool used for linear and nonlinear equations related to various science, engineering and industrial applications. Some illustrative examples are given to exemplify the competence of the proposed scheme and provide precise solutions for nonlinear problems. In addition, a comparison with the classical equation and Homotopy perturbation method illustrates the competency of the technique. The results show that the proposed technique requires minimum computational cost at rapid convergence. This method can be applied to other partial differential equations of fractional order for the analysis of their solutions.

The impact of the magnetic field and viscous dissipation on the thin film unsteady flow of GO-EG/GO-W nanofluids

The unsteady flow of nanoliquid film over a flexible surface has been inspected. The water and ethylene glycol are used as the base liquids for the graphene oxide platelets. The comparison of the two sorts of the nanoliquids has been used for the heat transfer enhancement applications. The thickness of the nanoliquid film kept variable under the influence of applied magnetic field and viscous dissipation. The governing equations for the flow problem have been altered into the set of nonlinear differential equations. The BVP 2.0 package has been used for the solution of the problem. The sum of the square residual error has been calculated up to the 10th order approximations. It has been observed that the graphene oxide ethylene glycol based nanofluid (GO-EG) has more efficient for the heat transfer enhancement as compared to the graphene oxide water based nanofluid (GO-W). The impact of the physical parameters has been plotted and discussed. The problem has been solved through homotopy analysis method (HAM) as described by Liao et al. [26]. This method is frequently used for the solution of nonlinear problems and show that this method is quickly convergent to the approximate solution. This method gives us series solution in the form of function and all the physical parameter of the problem involved in this method.

Code verification of soil dynamics simulations: A case study using the method of numerically manufactured solutions

Through quantitative measurement of the accuracy and the convergence of error, code verification can objectively assess the quality of a simulation code. A lack of analytical benchmark solution for nonlinear problems makes it difficult to verify soil dynamics simulations in geotechnical engineering. To overcome such difficulty, this study makes use of a novel code verification technique, the method of numerically manufactured solutions (MNMS). By utilizing MNMS, the accuracy of a simulation code for nonlinear soil dynamics problems was measured. The orders of the convergence rate of the errors, with respect to the spatial and to the temporal discretization, were found to be in good agreement with the theoretical orders of the numerical methods. By demonstrating the applicability of MNMS for verifying nonlinear soil dynamics simulations, it is hoped that the concept and the necessity of code verifications can be appreciated for simulations of geotechnical engineering problems.

Economic Load Dispatch in Power System by Hybrid Swarm Intelligence

Conserving energy and efficient power systems are very important for us to reduce pollution levels and also reduces wasted fuel resources which are already depleting on the planet. In power operation system, the potential of energy conservation and also much less emission of greenhouse gas because of the wise usage of cleaner non-renewable fuels burned in combined heat and power (CHP) models like the natural gas which provide them benefit from the usual electric power systems. Mixed generator systems have been widely employed by the industry. The industry which requires both power and heat can supply the demands with cogeneration of heat-power systems. Co-generation (CHP) systems could be constructed in cities and used in the form of distributed electricity sources. To get the optimal usage of CHP devices, economic load dispatch (ED) should be requested more for the process of energy conservation. Economic load dispatch plays a vital role and a large number of different approaches and methods have been used in solving such kind of problems. The methods like lambda-iteration and Gradient are used for finding out the optimized solution of non-linear problem. The purpose of this thesis is to utilize the algorithmic optimization approaches like particle swarm optimization (PSO), genetic algorithm (GA) and PSO-GA. In this work, the method of PSO-GA Optimization is used to find out the minimized cost at four of the generating units of heat and power. The base of the work is already published where in the loss coefficients are also presented with max-min cost function and power limit. This work is implemented in the MATLAB simulation environment. The work starts by initializing the load/power and then generators load power flow. The generators are allocated to initialize the cost and this cost is optimized by PSO with GA. At the end the experimental results of GA ,PSO and PSO-GA algorithm is equated with each other and it seems better convergence is achieved by PSO-GA Algorithm.

Using Anisotropic Magnetic Permeability for Solution of Nonlinear Problems of Magnetostatics for Structures with Lamination Steel

This article discusses the solution of nonlinear problems of magnetostatics for structures with lamination steel by specifying the magnetic permeability coefficient. It has been demonstrated that the solution with a specified steel layer is reduced to the solution of anisotropic problem with increased number of layers.

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (J Sci Comput 77:579–596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics equations to several standard test problems with a variety of boundary conditions.

Non-invasive implementation of nonlinear isogeometric analysis in an industrial FE software

PurposeThe purpose of this paper is to further simplify the use of NURBS in industrial environnements. Although isogeometric analysis (IGA) has been the object of intensive studies over the past decade, its massive deployment in industrial analysis still appears quite marginal. This is partly due to its implementation, which is not straightforward with respect to the elementary structure of finite element (FE) codes. This often discourages industrial engineers from adopting isogeometric capabilities in their well-established simulation environment.Design/methodology/approachBased on the concept of Bézier and Lagrange extractions, a novel method is proposed to implement IGA from an existing industrial FE code with the aim of bringing human implementation effort to the minimal possible level (only using standard input-output of finite element analysis (FEA) codes, avoid code-dependent subroutines implementation). An approximate global link to go from Lagrange polynomials to non-uniform-rational-B-splines functions is formulated, which enables the whole FE routines to be untouched during the implementation.FindingsAs a result, only the linear system resolution step is bypassed: the resolution is performed in an external script after projecting the FE system onto the reduced, more regular and isogeometric basis. The novel procedure is successfully validated through different numerical experiments involving linear and nonlinear isogeometric analyses using the standard input/output of the industrial FE software Code_Aster.Originality/valueA non-invasive implementation of IGA into FEA software is proposed. The whole FE routines are untouched during the novel implementation procedure; a focus is made on the IGA solution of nonlinear problems from existing FEA software; technical details on the approach are provided by means of illustrative examples and step-by-step implementation; the methodology is evaluated on a range of two- and three-dimensional elasticity and elastoplasticity benchmarks solved using the commercial software Code_Aster.

Existence and stability of heat and fluid flow in the presence of nanoparticles along a curved surface by mean of dual nature solution

This study examines the possible existence and stability of the heat transfer solutions due to unsteady nanofluid flow over a curved surface. Alumina oxide (Al2O3) is considered as one of the important nanoparticles due to its potential use in various fields like purification of water, ceramics and insulation products. It is considered that the small quantity of nanoparticles is incorporated in the Ethylene Glycol to form the nanofluid. Mathematical form of the problem is obtained by using the fundamental laws of motion and heat transfer. These equations are modified for the presence of solid concentrations in an ordinary liquid. The complicated coupled unsteady system is transformed into non-dimensional form by the use of relevant transformations. The solution of nonlinear problem is produced by the use of numerical scheme, available in the form of BVP4C package in MATLAB. In the case of surface shrinking towards the surface, a reverse flow situation is also developed, and requires careful selection of solution by examining the stability of solution. Detailed stability analysis is carried, and critical values are determined for the possible existence of dual solutions. Moreover, various parameters variation is also analyzed by plotting graphs and tables. The numerical values are also calculated for the reduced Nusselt number and skin friction values caused due to variation in various flow parameters. Results have shown that for the curved shrinking surfaces, one should expect multiple solutions for a set of parameter values like mass suction, curvature, nanoparticles volume fraction and unsteadiness. The isotherms suggest linear decay of heat away from the surface.