• Korea-Australia Rheology Journal
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• 제20권2호
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• pp.59-63
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• 2008

A theoretical model was developed to describe the flow behavior of a filled polymer in the packing stage of reaction injection molding and predict the residual stress distribution of thin injection-molded parts. The model predictions were compared with experiments performed for phenol-formaldehyde resin filled with wood dust and cured by urotropine. The packing stage of reaction injection molding process presents a typical example of complex non-isothermal flow combined with chemical reaction. It is shown that the time evolution of pressure distribution along the mold cavity that determines the residual stress in the final product can be described by a single 1D partial differential equation (PDE) if the rheological behavior of reacting liquid is simplistically described by the power-law approach with some approximations made for describing cure reaction and non-isothermality. In the formulation, the dimensionless time variable is defined in such a way that it includes all necessary information on the cure reaction history. Employing the routine separation of variables made possible to obtain the analytical solution for the nonlinear PDE under specific initial condition. It is shown that direct numerical solution of the PDE exactly coincides with the analytical solution. With the use of the power-law approximation that describes highly shear thinning behavior, the theoretical calculations significantly deviate from the experimental data. Bearing in mind that in the packing stage the flow is extremely slow, we employed in our theory the Newtonian law for flow of reacting liquid and described well enough the experimental data on evolution of pressure.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권2호
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• pp.108-120
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• 2022

We considered qualitative behaviour of the generalization of Einstein's model of Brownian motion when the key parameter of the time interval of free jump degenerates. Fluids will be characterised by number of particles per unit volume (density of fluid) at point of observation. Degeneration of the phenomenon manifests in two scenarios: a) flow of the fluid, which is highly dispersing like a non-dense gas and b) flow of fluid far away from the source of flow, when the velocity of the flow is incomparably smaller than the gradient of the density. First, we will show that both types of flows can be modeled using the Einstein paradigm. We will investigate the question: What features will particle flow exhibit if the time interval of the free jump is inverse proportional to the density and its gradient ? We will show that in this scenario, the flow exhibits localization property, namely: if at some moment of time t₀ in the region, the gradient of the density or density itself is equal to zero, then for some T during time interval [t₀, t₀ + T] there is no flow in the region. This directly links to Barenblatt's finite speed of propagation property for the degenerate equation. The method of the proof is very different from Barenblatt's method and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. From PDE point of view it assumed that solution exists in appropriate Sobolev type of space.

• Structural Engineering and Mechanics
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• 제41권2호
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• pp.263-284
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• 2012

This paper aims to propose a computational technique for estimating the region of attraction (RoA) for autonomous nonlinear systems. To achieve this, the collocation method is applied to approximate the Lyapunov function by satisfying the modified Zubov's partial differential equation around asymptotically stable equilibrium points. This method is formulated for n-scalar differential equations with two classes of basis functions. In order to show the efficiency of the suggested approach, some numerical examples are solved. Moreover, the estimated regions of attraction are compared with two similar methods. In most cases, the proposed scheme can estimate the region of attraction more efficient than the other techniques.

• Journal of applied mathematics & informatics
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• 제14권1_2호
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• pp.97-113
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• 2004

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.315-327
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• 2007

Mathematical aspects of the electromagnetic surface-wave propagation are examined for the dielectric core consisting of multiple sub-layers, which are embedded in the gap between the two bounding cladding metals. For this purpose, the linear problem with a partial differential wave equation is formulated into a nonlinear eigenvalue problem. The resulting eigenvalue is found to exist only for a certain combination of the material densities and the number of the multiple sub-layers. The implications of several limiting cases are discussed in terms of electromagnetic characteristics.

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.315-328
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• 2005

In this paper we present a new method for designing a nozzle. In fact the problem is to find the optimal domain for the solution of a linear or nonlinear boundary value PDE, where the boundary condition is defined over an unspecified domain. By an embedding process, the problem is first transformed to a new shape-measure problem, and then this new problem is replaced by another in which we seek to minimize a linear form over a subset of linear equalities. This minimization is global, and the theory allows us to develop a computational method to find the solution by a finite-dimensional linear programming problem.

• Korea-Australia Rheology Journal
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• 제14권4호
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• pp.161-174
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• 2002

A new technique for numerical calculation of viscoelastic flow based on the combination of Neural Net-works (NN) and Brownian Dynamics simulation or Stochastic Simulation Technique (SST) is presented in this paper. This method uses a "universal approximator" based on neural network methodology in combination with the kinetic theory of polymeric liquid in which the stress is computed from the molecular configuration rather than from closed form constitutive equations. Thus the new method obviates not only the need for a rheological constitutive equation to describe the fluid (as in the original Calculation Of Non-Newtonian Flows: Finite Elements St Stochastic Simulation Techniques (CONNFFESSIT) idea) but also any kind of finite element-type discretisation of the domain and its boundary for numerical solution of the governing PDE's. As an illustration of the method, the time development of the planar Couette flow is studied for two molecular kinetic models with finite extensibility, namely the Finitely Extensible Nonlinear Elastic (FENE) and FENE-Peterlin (FENE-P) models.P) models.

• Biotechnology and Bioprocess Engineering:BBE
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• 제11권2호
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• pp.110-115
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• 2006

We conducted a sensitivity analysis of the simulated moving bed (SMB) chromatography with the case model of the separation of two amino acids phenylalanine and tryptophan. We consider a four-zone SMB chromatography where the triangle theory is used to determine the operating conditions. Competitive Langmuir isotherm model was used to determine the adsorption isotherm. The finite difference method is used to solve nonlinear partial differential equation (PDE) systems numerically. We examined the effects of alterations in the operating conditions(feed-extract, feed-raffinate, eluent-extract, eluent-raffinate, recycle, and switching time) and the adsorption isotherm parameters (Langmuir isotherm parameters a and b) on SMB efficiency. The variation range of operating conditions and Langmuir isotherm a was between -50 and 50% of original value and the variation range of the Langmuir isotherm b was between $2.25^{-5}$ and $2.25^5$ times of original value.

• Journal of applied mathematics & informatics
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• 제30권5_6호
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• pp.719-728
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• 2012

In this paper, we develop a numerical method to solving an optimal control problem, which is governed by a parabolic partial differential equations(PDEs). Our approach is to approximate the PDE problem to initial value problem(IVP) in $\mathbb{R}$. Then, the homogeneous part of IVP is solved using semigroup theory. In the next step, the convergence of this approach is verified by properties of one-parameter semigroup theory. In the rest of paper, the original optimal control problem is solved by utilizing the solution of homogeneous part. Finally one numerical example is given.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제27권3호
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• pp.180-193
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• 2023

Approximating the implied volatilities and estimating the model parameters are important topics in quantitative finance. This study proposes an approximation formula for short-maturity near-the-money implied volatilities in stochastic volatility models. A general second-order nonlinear PDE for implied volatility is derived in terms of time-to-maturity and log-moneyness from the Feyman-Kac formula. Using regularity conditions and the Taylor expansion, an approximation formula for implied volatility is obtained for short-maturity nearthe-money call options in two stochastic volatility models: Heston model and SABR model. In addition, we proposed a novel numerical method to estimate model parameters. This method reduces the number of model parameters that should be estimated. Generating sample data on log-moneyness, time-to-maturity, and implied volatility, we estimate the model parameters fitting the sample data in the above two models. Our method provides parameter estimates that are close to true values.