Method For Approximate Solution Research Articles

Axial compressive behavior of laminated bamboo lumber columns with a chamfered section

The axial compressive behavior of laminated bamboo lumber (LBL) columns with a chamfered section was investigated using 15 specimens with lengths varying from 600 mm to 3000 mm; all considered samples had same square cross section of 100 mm × 100 mm with 10 mm chamfers at each corner. Axial compression tests were carried out to analyze the impact of slenderness ratios on failure modes, strain and ultimate bearing capacity of LBL columns. Two failure modes were observed i.e. compression failure and buckling failure. Obtained results on the ultimate vertical strain and ultimate load showed a downward trend with the increase in slenderness ratios; the correlation between the ultimate strain, ultimate load and the slenderness ratios were fitted. The approximate solution method was adopted to reveal the lateral deflections of buckling columns by a quartic functional model. Based on the ideal elastic–plastic constitutive model and Hill failure criterion, FEM was carried out to simulate the axial compression tests. The simulation results agreed well with both test and theoretical results, which verified the feasibility of proposed methods used under similar working conditions in this paper.

Numerical examination of the effect of different boundary conditions on the method of approximate particular solutions for scalar and vector problems

This work presents a numerical study of the influence of different boundary conditions (BC) on the precision of the method of approximate particular solutions (MAPS) with radial basis functions in two dimensions. The state of the art merely mentions that the MAPS accuracy decreases near the boundaries. Therefore, the influence that the choice of boundary condition has on this method should be studied with more detail. The present analysis with numerical experiments was applied to the solution of homogeneous Stokes flow equations, backward facing step flow, Slip flow, convective flow between parallel vertical plates and to elliptic differential equations with variable coefficients. The results show that care must be taken in choosing the boundary conditions for this type of solution methods because this selection affects the precision of the results. It is shown that the error associated with Neumann boundary condition, can be reduced by using a refined nodal distribution towards the inlet and outlet zones. Conversely, mixed boundary conditions require careful combination of traction and velocity to achieve better accuracy with MAPS for some flow problems.

On a method for approximate solution of a mixed boundary value problem for an elliptic equation

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.

A Comparison of Genetic Representations and Initialisation Methods for the Multi-objective Shortest Path Problem on Multigraphs

This paper compares different solution approaches for the multi-objective shortest path problem (MSPP) on multigraphs. Multigraphs as a modelling tool are able to capture different available trade-offs between objectives for a given section of a route. For this reason, they are increasingly popular in modelling transportation problems with multiple conflicting objectives (e.g., travel time and fuel consumption), such as time-dependent vehicle routing, multi-modal transportation planning, energy-efficient driving, and airport operations. The multigraph MSPP is more complex than the NP-hard simple graph MSPP. Therefore, approximate solution methods are often needed to find a good approximation of the true Pareto front in a given time budget. Evolutionary algorithms have been successfully applied for the simple graph MSPP. However, there has been limited investigation of their applications to the multigraph MSPP. Here, we extend the most popular genetic representations to the multigraph case and compare the achieved solution qualities. Two heuristic initialisation methods are also considered to improve the convergence properties of the algorithms. The comparison is based on a diverse set of problem instances, including both bi-objective and triple objective problems. We found that the metaheuristic approach with heuristic initialisation provides good solutions in shorter running times compared to an exact algorithm. The representations were all found to be competitive. The results are encouraging for future application to the time-constrained multigraph MSPP.

Quasi-static quintication method for periodic solution of strong nonlinear oscillators

In this paper, a quasi-static quintication method for approximate periodic solution of strong nonlinear oscillators was presented. The basic idea of the method is to replace the original nonlinear oscillator with an equivalent cubic-quintic oscillator, whose stiffness constants are derived based on quasi-static equilibrium principles. The exact solution of the equivalent cubic-quintic oscillator was then applied to obtain the approximate solution to the original oscillator. The advantage of the present method is that its quintication constants (i.e. stiffness constants of the equivalent oscillator) are always in the form of elementary functions and are simpler than the corresponding constants derived by the Chebyshev quintication method. The present quintication method was verified using examples of strong nonlinear oscillators and was found to give very accurate results.

An Efficient Approach for Time-Fractional Damped Burger and Time-Sharma-Tasso-Olver Equations Using the FRDTM

The objective of this paper is to use the fractional reduced differential transform method (FRDTM) to find approximate analytical solutions to the time-fractional Sharma-Tasso-Olver equation and the time-fractional damped Burger equation. The fractional derivatives are described in the Caputo sense. We compare our results with those from existing methods such as the homotopy analysis method (HAM), variation iteration method (VIM) and the Adomian decomposition method (ADM). Also, the results we obtained in this paper are in a good agreement with the exact solutions; hence, this technique is powerful and efficient as an alternat ive method for finding approximate and exact solutions for nonlinear fract ional PDEs.

Iterative Method for a System of Nonlinear Fredholm Integral Equations

In this paper, the iteration method is proposed to solve a class of system of Fredholm-type nonlinear integral equations. First, the existence and uniqueness of solution are theoretically proven by the fixed-point theorem. Second, the approximation solution method is given by using the appropriate integration rule. The error analysis for the approximated solution with the exact solution is discussed for infinity-norm, and the rates of convergence are obtained. Furthermore, an iteration algorithm is constructed, and the convergence of the proposed numerical method is rigorously derived. Finally, some numerical examples are given to illustrate the theoretical results.

Analytic insights into nonlocal energy transport. III. steady state Fokker Planck theory in spherical and planar geometry

This paper develops an approximate steady state Fokker Planck theory for energetic electron transport in a spherical laser fusion target. First, we apply the theory to only a small population of electrons, which is specified at the outset. Then, one determines the nonlocal electron energy flux of these electrons at a particular position. These energetic electrons may either come from the tail of the thermal distribution function or be generated by an instability. This paper develops two approximate methods of solution, which we call the “characteristic method” and “sparse eigenfunction.” The former works only in planar geometry and the latter in both planar and spherical geometry. Comparison of the two methods in planar geometry shows that even though the approximations are very different, they give about the same result, increasing their credibility. For the example we have chosen, spherical effects are not important for electrons from the tail of the distribution function but may well be for instability generated electrons, which have much higher energy. Comparing planar to spherical, one finds an additional spherical barrier protecting the fuel. It turns out that the associated fuel preheat a Fokker Planck model predicts is considerably less than that predicted by the Krook models as developed at both NRL and other places.

Modification of the Marquardt method for training a neural network predictor in eddy viscosity models

The subject of this article is the numerical optimization techniques used in training neural networks that serve as predicate components in certain modern eddy viscosity models. Qualitative solution to the problem of training (minimization of the functional of neural network offsets) often requires significant computational costs, which necessitates to increase the speed of such training based on combination of numerical methods and parallelization of calculations. The Marquardt method draws particular interest, as it contains  the parameter that allows speeding up the solution by switching the method from the descent away from the solution to the Newton’s method of approximate solution. The article offers modification of the Marquardt method, which uses the limited series of random samples for improving the current point and calculate the parameter of the method. The author demonstrate descent characteristics of the method in numerical experiments, both on the test functions of Himmelblau and Rosenbrock, as well as the actual task of training the neural network predictor applies in modeling of the turbulent flows. The use of this method may significantly speed up the training of neural network predictor in corrective models of eddy viscosity. The method is less time-consuming in comparison with random search, namely in terms of a small amount of compute kernels; however, it provides solution that is close to the result of random search and is better than the original Marquardt method.

Method for finding approximate solutions of elasticity equations using spline wavelets

Method for finding approximate solutions of elasticity equations using spline wavelets

Iterative methods for approximate solution of the Ornstein-Uhlenbeck process with normalised brownian motion

This work considers the concept of the Normalised Brownian motion for the solutions of the Ornstein-Uhlenbeck process using the Daftardar-Jafari Method (DJM) and Picard Iterative Method (PIM) as the approximate-analytical methods of solutions. The results obtained from DJM are compared with those of the PIM. The obtained results, therefore, show the effectiveness of the proposed methods.

Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения

Studying heat transfer processes in periodic media containing vacuum interlayers or cavities, heat through which is transferred by radiation, is of significant interest for applications. Direct numerical solution of such problems involves considerable computational efforts and becomes almost impossible for systems containing a large number of heat conducting elements, especially for 2D and 3D structures. Therefore, it is of issue to develop effective approximate solution methods for such problems. This publication continues a series of studies on developing and substantiating special discrete and asymptotic approximations of radiant-and-conduction heat transfer problems in periodic systems of heat conducting elements separated by vacuum. In this study, the stationary radiant-and-conduction heat transfer problem in a system of absolutely black square rods is considered. The sought quantity is the absolute temperature, which is found from the solution of the boundary-value problem for the stationary heat conduction equation with nonlinear nonlocal boundary conditions describing radiant heat transfer between the rods through vacuum interlayers. A special discrete approximation of this problem leading to the system of linear algebraic equations with respect to the fourth power of the temperature is presented. The solution of this system as approximation of the mean temperature over the rod cross-section is described. The discrete approximation error estimate as a function of the square rod side length (the small parameter of the problem) and the thermal conductivity coefficient has been obtained. The obtained error estimate proves applicability of the discrete approximation for materials with a high thermal conductivity coefficient.

Algorithm of the systemic solutions choosing for the creation (development) of hierarchical information and telecommunication systems

A structural diagram of a hierarchical information and telecommunications system, the most general formulation, as well as a general approach to solving the choosing problem of the systems engineering solutions for the creation (development) of a hierarchical information and telecommunications system, taking into account resource constraints at individual planning stages, are adduced. The complexity estimation of the exact algorithm for solving the assigned task is carried out, the application of an approximate solution method, on the basis of which an algorithm is proposed for solving the problem is justified. Keywords hierarchical information and telecommunication system; complex of automation devices; universal modular communication center

Approximate iterative solutions of Hamilton–Jacobi equations for nonlinear systems

In this paper, we present computational methods for approximate solution to the time-dependent first-order Hamilton–Jacobi equations (HJEs) arising in deterministic optimal control of nonlinear systems in state-space, as well as other areas of science and engineering. We establish convergence of the methods for optimal control problems and for the case of general nonlinear systems under fairly mild assumptions. Examples are then solved to demonstrate the effectiveness of the methods and the special case of affine nonlinear systems is also considered. Results are then extended to more general initial value problems of the HJE arising in other areas of science and engineering.

Computational and Experimental Study of Time-Averaged Characteristics of Positive and Negative DC Corona Discharges in Point-Plane Gaps in Atmospheric Air

The use of stationary solvers instead of approximate solution methods or time-dependent solvers, which are standard tools in gas discharge modeling, allows one to develop a very fast and robust numerical model for studying the time-averaged characteristics of dc corona discharges. Such an approach is applied to dc corona discharges in point-plane gaps in ambient air. A wide range of currents of both voltage polarities and various gap lengths are investigated, and the simulation results are validated by comparing the computed current–voltage characteristics and spatial distributions of the radiation intensity with experimental results. Specific features of the numerical and experimental results at both polarities are discussed.

Solving dynamic discrete choice models using smoothing and sieve methods

We propose to combine smoothing, simulations and sieve approximations to solve for either the integrated or expected value function in a general class of dynamic discrete choice (DDC) models. We use importance sampling to approximate the Bellman operators defining the two functions. The random Bellman operators, and therefore also the corresponding solutions, are generally non-smooth which is undesirable. To circumvent this issue, we introduce smoothed versions of the random Bellman operators and solve for the corresponding smoothed value functions using sieve methods. We also show that one can avoid using sieves by generalizing and adapting the “self-approximating” method of Rust (1997b) to our setting. We provide an asymptotic theory for both approximate solution methods and show that they converge with N-rate, where N is number of Monte Carlo draws, towards Gaussian processes. We examine their performance in practice through a set of numerical experiments and find that both methods perform well with the sieve method being particularly attractive in terms of computational speed and accuracy.

Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations

Abstract We construct and study a class of numerically implementable iteratively regularized Gauss–Newton type methods for approximate solution of irregular nonlinear operator equations in Hilbert space. The methods include a general finite-dimensional approximation for equations under consideration and cover the projection, collocation and quadrature discretization schemes. Using an a posteriori stopping rule for the iterative processes and the standard source condition on the solution, we establish accuracy estimates for the approximations generated by the methods. We also investigate projected versions of the processes which take into account a priori information about a convex compact containing the solution. An iteratively regularized quadrature process is applied to an inverse 2D problem of gravimetry.

Solutions of fractional-stochastic Bao’s system

Systems of high-dimensional nonlinear ordinary differential equations play a significant role in Physics and applied sciences including big-data optimization, financial models, epidemic disease models. In this paper, we are concerned with numerical solutions of Bao’s system that is a 4-dimensional hyperchaotic system introduced by Bo-Cheng and Zhong (2008). We solve the Bao’s system with both the Crank-Nicolson and power series methods. Crank-Nicolson method is eventually evolved into a new system whose solution is presented in a quite neat algorithmic manner. By adding standard Brownian motion to each term in the model, we express the Bao’s system as a system of stochastic differential equations. We solve the stochastic system with an Euler-type approximate solution method. By adding noise and expressing time derivatives with Caputo-type fractional derivative, we study on synchronization and parameter estimation of the models. To the best of our knowledge, Bao’s system has not been numerically solved with the methods employed in this paper previously, and this paper considers fractional and stochastic Bao’s system for the first time in the history of research. Techniques employed by us in this paper may serve as a framework for solutions of many other systems of ordinary differential equations including Lorenz types and epidemic models.

Flow modeling and experimental verification of flow resistors used in microfluidic chips driven by capillary force

A microfluidic chip driven by capillary force has the advantage of low cost and ease of manufacturing in batches, and its flow behavior is dominated by the geometry and surface characteristics of the microchannel. The design of mirochannel structures is very important for the microfluidic chips. This study presents a semi analytical method for the design of capillary microchannel. In this method, the quasi-steady state approximate solution method of the Young–Laplace equation is used to derive the capillary surface tension, and the parallel flow assumption based on the Reynolds equation is used to derive the resistance of the channel wall. A computational fluid dynamics simulation is used to provide the inlet effect coefficient and channel shape effect coefficient of this model. The availability of the semi analytical model is verified by the experiment. This model realizes the flow analysis of two-dimensional capillary flow channel with a continuous shape of the flow channel wall, providing a fast and accurate method for the structural design of the microfluidic chip driven by capillary force.

Extending meshless method of approximate particular solutions (MAPS) to two-dimensional convection heat transfer problems.

Natural, forced and mixed convection heat transfer problems are solved by the meshless Method of Approximate Particular Solutions (MAPS). Particular solutions of Poisson and Stokes equations are employed to approximate temperature and velocity, respectively. The latter is used to obtained a closed expression for pressure particular solution. In both cases, the source terms are multiquadric radial basis functions which allow to obtain analytical expressions for these auxiliary problems. In order to couple momentum and energy equations, a relaxation strategy is implemented to avoid convergence problems due to the difference between successive temperature and velocity changes when solving the steady problem from an initial guess. The developed and validated numerical scheme is used to study flow and heat transfer in two two-dimensional problems: natural convection in concentric annulus between a square and a circular cylinder and non-isothermal flow past a staggered tube bundle. Numerical solutions obtained by MAPS are comparable in accuracy to solutions reported by authors who uses denser nodal distributions, showing the capability of the present method to accurately solve heat convection problems with temperature and heat flux boundary conditions as well as curve geometries and internal flow situations.