• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.83-92
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• 2021

This paper studies approximate solutions for a class of nonlinear diffusion population models. Our methods are to use the fundamental solution of heat equations to construct integral forms of the models and the well-known Banach compression map theorem to prove the existence of positive solutions of integral equations. Non-steady-state local approximate solutions for suitable harvest functions are obtained by utilizing the approximation theorem of multivariate continuous functions.

• Journal for History of Mathematics
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• v.25 no.3
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• pp.1-14
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• 2012

Since JiuZhang SuanShu(九章算術), the basic field of the traditional mathemtics in Eastern Asia is the field of rational numbers and hence irrational solutions of equations should be replaced by rational approximations. Thus approximate solutions of equations became a very important subject in theory of equations. We first investigate the history of approximate solutions in Chinese sources and then compare them with those in Chosun mathematics. The theory of approximate solutions in Chosun has been established in SanHakWonBon(算學原本) written by Park Yul(1621 - 1668) and JuSeoGwanGyun(籌書管見, 1718) by Cho Tae Gu(趙泰耉, 1660-1723). We show that unlike the Chinese counterpart, Park and Cho were concerned with errors of approximate solutions and tried to find better approximate solutions.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.22 no.7
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• pp.468-473
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• 2010

This paper deals with approximate integral solutions to the one-dimensional model describing the charging process of stratified thermal storage tanks. Temperature is assumed to be the form of Fermi-Dirac distribution function, which can be separated to two sets of cubic polynomials for each hot and cold side of thermal boundary layers. Proposed approximate integral solutions are compared to the previous works of the approximate analytic solutions and show reasonable agreement. The approach, however, has benefits in mathematical difficulties, complicated solution form and unstable convergence of series solution founded in the previous analytic solutions. Solutions for a semi-infinite region, which have simple closed form solutions, give close agreement to those for a finite region. Thermocline thickness is obtained in closed form and shows proportional behavior to the square root of time and inverse proportional behavior to the square root of flow rate.

• Journal of Mechanical Science and Technology
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• v.15 no.9
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• pp.1339-1345
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• 2001

Laminar film condensation of a saturated pure vapor in forced flow over a flat plate is analyzed as boundary layer solutions. Similarity solutions for some real fluids are presented as a function of modified Jakob number (C$\_$pι/ ΔΤ/Prh$\_$fg/) with property ratio (No Abstract.see full/text) and Pγ as parameters and compared with approximate solutions which were obtained from energy and momentum equations without convection and inertia terms in liquid flow. Approximate solutions agree well with the similarity solutions when the values of modified Jakob number are less then 0.1 near 1 atmospheric pressure.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.149-159
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• 2009

In this paper, we introduce some approximate optimal solutions and an augmented Lagrangian function in nonlinear programming, establish dual function and dual problem based on the augmented Lagrangian function, discuss the relationship between the approximate optimal solutions of augmented Lagrangian problem and that of primal problem, obtain approximate KKT necessary optimality condition of the augmented Lagrangian problem, prove that the approximate stationary points of augmented Lagrangian problem converge to that of the original problem. Our results improve and generalize some known results.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.631-642
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• 2016

This article deals with one-dimension backward heat conduction problem (BHCP). A new approach based on least squares support vector machines (LS-SVM) is proposed for obtaining their approximate solutions. The approximate solution is presented in closed form by means of LS-SVM, whose parameters are adjusted to minimize an appropriate error function. The approximate solution consists of two parts. The first part is a known function that satisfies initial and boundary conditions. The other is a product of two terms. One term is known function which has zero boundary and initial conditions, another term is unknown which is related to kernel functions. This method has been successfully tested on practical examples and has yielded higher accuracy and stable solutions.

• Communications of the Korean Mathematical Society
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• v.22 no.1
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• pp.145-155
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• 2007

In this paper, we show the convergence of approximate solutions to the convective porous media equation using methodology developed in [8]. First, we obtain the approximate transport equation for the given convective porous media equation. Then using the averaging lemma, we obtain the convergence.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.375-381
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• 2008

Boundedness for the set of all the $\epsilon$-approximate solutions for convex optimization problems are considered. We give necessary and sufficient conditions for the sets of all the $\epsilon$-approximate solutions of a convex optimization problem involving finitely many convex functions and a convex semidefinite problem involving a linear matrix inequality to be bounded. Furthermore, we give examples illustrating our results for the boundedness.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.749-762
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• 2018

In this paper, the new optimal perturbation iteration method has been applied to solve the generalized regularized long wave equation. Comparing the new analytic approximate solutions with the known exact solutions reveals that the proposed technique is extremely accurate and effective in solving nonlinear wave equations. We also show that,unlike many other methods in literature, this method converges rapidly to exact solutions at lower order of approximations.

• Structural Engineering and Mechanics
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• v.59 no.1
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• pp.1-14
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• 2016

In this paper, an alternative analytical method is presented to evaluate the nonlinear vibration behavior of single and double tapered cantilever beams. The admissible lateral displacement function satisfying the geometric boundary conditions of a single or double tapered cantilever beam is derived by using Rayleigh-Ritz method. Based on the Lagrange method and the Newton Harmonic Balance (NHB) method, analytical approximate solutions in closed and explicit form are obtained. These approximate solutions show excellent agreement with those of numeric method for small as well as large amplitude. Moreover, due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large amplitude vibration response of tapered beams.