Method For Solution Of Problems Research Articles

The effect of problem-based learning model to students’ cognitive achievement on high and low students’ problem-solving abilities

This study aimed to determine the effect of PBL model on students’ cognitive achievement on high and low students’ problem solving abilities. This study was quasi experiment research by using posttest only non-equivalent control group design with 22 students of X(3) grader science class as control class (conventional model) and 21 students of X(4) as experiment class (using PBL model) at one of senior high school in Surakarta. Students’ problem solving abilities illustrated as abilities in four aspects: identification the problem, identification the cause of problem, create the method of problem solution, examine the result of problem solving by using 40 items of multiple choice question refers to Problem Solving Skill Test [12]. Students’ problem solving abilities measured before the research is done and divided into high and low categories. Students’ cognitive achievement illustrated as students’ concept understanding and measured by using essay test was given after the PBL model used. Instrument had been validated by expert judgment and students. Data analyze by using t-test. The result showed: 1) Sig. (0,00<0,05) Ho rejected, there was difference of students’ achievement between experiment class and control class; 2) Sig (0,683 >0,05) H0 accepted, there wasn’t difference students’ achievement with high problem solving abilities between low problem solving abilities.

CSG: A new stochastic gradient method for the efficient solution of structural optimization problems with infinitely many states

This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.

Generalized Second Derivative Linear Multistep Methods Based on the Methods of Enright

The Adams-type second derivative multistep methods of W. H. Enright is generalized to a class of boundary value methods for the numerical solution of initial value problems in ordinary differential equations. The root distribution of the stability polynomials of the new class of methods are given for the purpose of their correct implementation as boundary value methods. Computational results on some stiff problems are compared with those from some existing methods, while ODE15s of MATLAB is used as a reference numerical solution.

A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically.

New Possibilities of Application of Artificial Intelligence Methods for High-Precision Solution of Boundary Value Problems

One of the main problems in modern mathematical modeling is to obtain high-precision solutions of boundary value problems. This study proposes a new approach that combines the methods of artificial intelligence and a classical analytical method. The use of the analytical method of fictitious canonic regions is proposed as the basis for obtaining reliable solutions of boundary value problems. The novelty of the approach is in the application of artificial intelligence methods, namely, genetic algorithms, to select the optimal location of fictitious canonic regions, ensuring maximum accuracy. A general genetic algorithm has been developed to solve the problem of determining the global minimum for the choice and location of fictitious canonic regions. For this genetic algorithm, several variants of the function of crossing individuals and mutations are proposed. The approach is applied to solve two test boundary value problems: the stationary heat conduction problem and the elasticity theory problem. The results of solving problems showed the effectiveness of the proposed approach. It took no more than a hundred generations to achieve high precision solutions in the work of the genetic algorithm. Moreover, the error in solving the stationary heat conduction problem was so insignificant that this solution can be considered as precise. Thus, the study showed that the proposed approach, combining the analytical method of fictitious canonic regions and the use of genetic optimization algorithms, allows solving complex boundary-value problems with high accuracy. This approach can be used in mathematical modeling of structures for responsible purposes, where the accuracy and reliability of the results is the main criterion for evaluating the solution. Further development of this approach will make it possible to solve with high accuracy of more complicated 3D problems, as well as problems of other types, for example, thermal elasticity, which are of great importance in the design of engineering structures.

Implicit Methods for Numerical Solution of Singular Initial Value Problems

Abstract Various order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result.

Integration of carrier selection and supplier selection problem in humanitarian logistics

Carrier selection is a crucial issue in humanitarian relief operations, and the shortage of transportation vehicles can considerably affect relief distribution. In this paper, the significance of commodity and vehicle suppliers and guaranteeing prevention of shortage via on-time supplying of each relief commodity by the suppliers is emphasized by suggesting the integration of carrier selection and supplier selection for developing contractual agreements in humanitarian logistics. For this close relationship to be described, a scenario-based two-stage stochastic programming model for the establishment of joint decision-making in preparedness and response phases is developed so that a contract can be created on the quantity of reservation of both commodities and vehicles by the suppliers, as well as the transmission of the reserved commodities in the form of commodity purchase and vehicle rent in the response phase. An efficient two-step solution technique is suggested as a solution method for real scale problems, the effectiveness of which can be seen through the results. To illustrate the applicability of the model, a case study is suggested. The application of the model is discussed via numerical studies as well as a sensitivity analysis. The results have provided an appropriate managerial insight for responding to disaster for relief agencies.

A Variational Perturbative Approach to Planning in Graph-Based Markov Decision Processes

Coordinating multiple interacting agents to achieve a common goal is a difficult task with huge applicability. This problem remains hard to solve, even when limiting interactions to be mediated via a static interaction-graph. We present a novel approximate solution method for multi-agent Markov decision problems on graphs, based on variational perturbation theory. We adopt the strategy of planning via inference, which has been explored in various prior works. We employ a non-trivial extension of a novel high-order variational method that allows for approximate inference in large networks and has been shown to surpass the accuracy of existing variational methods. To compare our method to two state-of-the-art methods for multi-agent planning on graphs, we apply the method different standard GMDP problems. We show that in cases, where the goal is encoded as a non-local cost function, our method performs well, while state-of-the-art methods approach the performance of random guess. In a final experiment, we demonstrate that our method brings significant improvement for synchronization tasks.

Fast implicit solvers for phase-field fracture problems on heterogeneous microstructures

We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memory-efficient FFT-based solution techniques, and identify the heavy ball scheme as particularly powerful. We discuss the memory-efficient implementation and present demonstrative numerical examples.

Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction

We consider the problem of determination of a magnetic field from three dimensional polarimetric neutron tomography data. We see that this is an example of a non-Abelian ray transform and that the problem has a globally unique solution for smooth magnetic fields with compact support, and a locally unique solution for less smooth fields. We derive the linearization of the problem and note that the derivative is injective. We go on to show that the linearised problem about a zero magnetic field reduces to plane Radon transforms and suggest a modified Newton–Kantarovich method (MNKM) for the numerical solution of the non-linear problem, in which the forward problem is re-solved but the same derivative is used each time. Numerical experiments demonstrate that MNKM works for small enough fields (or large enough velocities) and we show an example where it fails to reconstruct a slice of the simulated data set. Lastly we show that, viewed as an optimization problem, the inverse problem is non-convex so we expect gradient based methods may fail.

Numerical solution of singular boundary value problems using advanced Adomian decomposition method

In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method which is the modification of the Adomian decomposition method. In this method, we use all the boundary conditions to obtain the coefficient of the approximate series solution. Moreover, convergence analysis and an error bound for the approximate solution are discussed. This method overcomes the singular behaviour of problems and illustrates the approximations of high precision with a large effective region of convergence. To prove the robustness and effectiveness of the proposed method, various examples are considered and the obtained results are compared with the other existing methods.

Acceleration of uncertainty propagation through Lagrange multipliers in partitioned stochastic method

A partitioned stochastic method (PSM) is proposed for the solution of static structural mechanics problems with uncertainties, whose solution vectors are the displacements for each partition and Lagrange multipliers along with the partition interfaces. The proposed partitioned stochastic method employs three stochastic basis selection steps: an arbitrary initial choice of displacement random bases, a set of conjugate bases for the Lagrange multipliers, and finally modification of the displacement bases affected by those of the Lagrange multipliers. The present PSM thus propagates the uncertainties instantly across partitioned substructures, resulting in an improved rate of convergence. Numerical experiments illustrate the proposed PSM outperforms a conventional partitioned solution method for structural mechanics problems.

A novel iterative integration regularization method for ill-posed inverse problems

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

A method for solving a boundary value problem in a multilayered area

A mathematical model of thermal process in an electrical machine was built as an example, presented as a three-layer cylinder where internal heat sources operate in one of the layers and heat is submitted to the other two by means of heat conduction. A method of solving the boundary-value problems for heat conduction equation in a complex area – a multi-layered cylinder with internal heat sources operating in one part of the layers and external ones in another part, is proposed. A method of problem solution in conditions of uncertainty of one of the boundary condition at the layers interface with conductive heat exchange between the layers is reviewed. The principle of method lies in the averaging of temperature distributions radially in the internal layers. As a result of transformations at the layers interface a boundary condition of the impedance-type conjugation appears. The analytical and numeric-analytical solutions of simplified problems were obtained.

REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION

Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

РЕАЛІЗАЦІЯ МОЖЛИВОСТЕЙ STEM-ОСВІТИ ЗАСОБАМИ ІНТЕГРАЦІЇ КРЕАТИВНИХ МЕТОДІВ НАВЧАННЯ

The article considers the problem of activation of the creative potential of students, their involvement in the searching activity using the integrated methods as the instrument to solve the contradictories between the requirements of innovative training at higher educational establishments and the practice of conventional methods application. The research argues the expediency of use of the opportunities of STEM education at technical higher educational establishments to train students to be able to make non-standard and adequate decision depending on the situation, to see the prospects and plan the strategies, etc. Creativity, cooperation and critical thinking are the key competences to get success in the 21st century, which secure development of the principal skills and provide knowledge not separately, but as an integration of courses into an unified educational system. The work outlines the main opportunities of STEM-education at technical higher educational establishments, particularly mastering tools of analysis and synthesis for specification of the problem and its solution; learning the mathematic apparatus for solution of the set professional tasks; readiness to integrate different aspects of knowledge and activities; ability to make situational integration of means, knowledge and actions in the process of problem solution; ability to creative integration in its dynamic development, focus on optimization of the professional activity, development of critical thinking as a principal feature of specialists; ability to set the problem correctly, etc. The article supplies examples of use of the cognitive teaching methods of propaedeutic block for the creative methods application. The research describes the opportunities to use such cognitive methods as empathy; semantic imagination; imaginary thinking; symbolic thinking; heuristic issues; comparison; heuristic observation; facts; investigation; constructing of notions, norms and theories; assumptions and trips into the future; forecasting, etc. The research considers peculiarities of the creative methods of training at technical higher educational establishments on the example of teacher’s proficiency in physics, which are based on the integrated approach by shaping of a logical chain of methods for solution of specific problems (methods of agglutination, synectics, analogy, inversion, excogitation, etc.). The conclusion is that the main task of introduction of the STEM-education at technical higher educational establishments expects appropriate pedagogical conditions to secure development of the creative potential of a personality, independent critical thinking, system of values, formation of the range of professional competences of the specialist.

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

Hermite wavelet method for the numerical solution of nonlinear singular initial value problems

A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Abstract The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.

Direct methods for finding optimal solution of assignment problems are not always dependable

Direct methods for finding optimal solution of assignment problems are not always dependable

A Numerical Method for the Solution of Two Inverse Problems in the Mathematical Model of Redox Sorption

For a mathematical model with internal-diffusion kinetics and an oxidation-reduction reaction, we consider two inverse problems of determining the sorption isotherm or the redox reaction rate constant from the output dynamic curve. A numerical method for the solution of these inverse problems is proposed, computational results are reported, and its potential is investigated.