Showing papers on "Method of matched asymptotic expansions published in 2012"
TL;DR: In this paper, a new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics, which fulfills local and global dissipation inequalities.
Abstract: A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation, we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.
295 citations
TL;DR: In this article, the authors present new results in numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years, mainly discussing layer-adapted meshes.
Abstract: We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.
208 citations
TL;DR: The diffusion of finite-sized hard-core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions, and a nonlinear diffusion equation for the one-particle distribution function is obtained.
Abstract: Excluded-volume effects can play an important role in determining transport properties in diffusion of particles. Here, the diffusion of finite-sized hard-core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions. The result is a nonlinear diffusion equation for the one-particle distribution function, with excluded-volume effects enhancing the overall collective diffusion rate. An expression for the effective (collective) diffusion coefficient is obtained. Stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples.
107 citations
TL;DR: Using a rigorous method of matched asymptotic expansions, the equation of motion of a small, compact body in an external vacuum spacetime is derived through second order in the body's mass (neglecting effects of internal structure).
Abstract: Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.
88 citations
TL;DR: In this article, the authors deal with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities and study the asymptotic of the solution with respect to the period and thickness of the inhomogenities using so-called matched ASM expansions.
73 citations
TL;DR: This model can explain two alternative notions of the diffusion coefficient that are often confounded, namely collective diffusion and self-diffusion.
Abstract: Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial differential equations. In this paper we consider multiple interacting subpopulations/species and study how the inter-species competition emerges at the population level. Each individual is described as a finite-size hard core interacting particle undergoing Brownian motion. The link between the discrete stochastic equations of motion and the continuum model is considered systematically using the method of matched asymptotic expansions. The system for two species leads to a nonlinear cross-diffusion system for each subpopulation, which captures the enhancement of the effective diffusion rate due to excluded-volume interactions between particles of the same species, and the diminishment due to particles of the other species. This model can explain two alternative notions of the diffusion coefficient that are often confounded, namely collective diffusion and self-diffusion. Simulations of the discrete system show good agreement with the analytic results.
57 citations
TL;DR: In this article, the authors consider multiple interacting sub-populations/species and study how the inter-species competition emerges at the population level, where each individual is described as a finite-size hard core interacting particle undergoing Brownian motion.
Abstract: Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial differential equations. In this paper we consider multiple interacting subpopulations/species and study how the inter-species competition emerges at the population level. Each individual is described as a finite-size hard core interacting particle undergoing Brownian motion. The link between the discrete stochastic equations of motion and the continuum model is considered systematically using the method of matched asymptotic expansions. The system for two species leads to a nonlinear cross-diffusion system for each subpopulation, which captures the enhancement of the effective diffusion rate due to excluded-volume interactions between particles of the same species, and the diminishment due to particles of the other species. This model can explain two alternative notions of the diffusion coefficient that are often confounded, namely collective diffusion and self-diffusion. Simulations of the discrete system show good agreement with the analytic results.
56 citations
15 Jun 2012
TL;DR: The theorems which state front motion description and stationary contrast structures formation are proved for parabolic, parabolic-periodic and integro-parabolic problems are proved.
Abstract: In the present paper we discuss father development of the general scheme of the asymptotic method of differential inequalities and illustrate it applying for some new important cases of initial boundary value problem for the nonlinear singularly perturbed parabolic equations,which are called in applications as reaction-diffusion-advection equations. The theorems which state front motion description and stationary contrast structures formation are proved for parabolic, parabolic-periodic and integro-parabolic problems.
47 citations
TL;DR: The problem of sliding mode control for singularly perturbed systems in the presence of matched bounded external disturbances is investigated and two proposed schemes that ensure the asymptotic stability of the system are presented.
Abstract: A variable structure system can be studied using the singular perturbation theory. The discontinuous control that leads to a finite-time reaching of the sliding surface creates fast-time transients analogous to the stable boundary layer dynamics of a singularly perturbed system. As the sliding mode is attained, the slow-time dynamics prevails, just as that of a singularly perturbed system after the boundary layer dynamics fades away. In this technical note, the problem of sliding mode control for singularly perturbed systems in the presence of matched bounded external disturbances is investigated. A composite sliding surface is constructed from solutions of algebraic Lyapunov equations which are derived from both the fast and the slow subsystems. The resultant sliding motion ensures Lyapunov stability with disturbance rejection. Two proposed schemes that ensure the asymptotic stability of the system are presented. The effectiveness of the proposed methods is demonstrated in a numerical example of a magnetic tape control system.
46 citations
TL;DR: In this paper, the authors studied Gevrey asymptotic properties of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain and constructed actual holomorphic solutions of these problems with the help of the Borel-Laplace transforms.
Abstract: We study Gevrey asymptotic properties of solutions of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain. We construct actual holomorphic solutions of these problems with the help of the Borel---Laplace transforms. Using the Malgrange---Sibuya theorem, we show that these holomorphic solutions have a common formal power series asymptotic expansion of Gevrey order 1 in the perturbation parameter.
36 citations
TL;DR: A numerical method named as Asymptotic Initial Value Technique (AIVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary delay differential equation with the discontinuous convection–diffusion coefficient term.
TL;DR: In this paper, a numerical solution of a class of non-linear fractional singularly perturbed two points boundary-value problem is discussed, which consists of solving reduced problem and boundary layer correction problem.
TL;DR: Several asymptotic expansions for the gamma function due to Laplace, Ramanujan–Karatsuba, Gosper, Mortici, Nemes and Batir are unify.
TL;DR: It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations, and the proposed control methodology can be applied to practical applications even if the value of the small parameter $\varepsilon$ is not precisely known.
Abstract: This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of $O(\sqrt{\varepsilon})$ approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an $O(\varepsilon)$ approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter $\varepsilon$ is not precisely known.
TL;DR: In this paper, the singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case, and the asymptotic behavior and existence of a solution with an internal transition layer are analyzed.
Abstract: A singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case. The asymptotic behavior and existence of a solution with an internal transition layer are analyzed. The asymptotics are substantiated using the asymptotic method of differential inequalities.
TL;DR: In this article, the differential transform method is used for solving linear singularly perturbed two-point boundary value problems with high accuracy, and four numerical examples are given to demonstrate the effectiveness of the present method.
Abstract: Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.
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16 Dec 2012
TL;DR: In this article, Gevrey Theory is applied to composite expansions and Singularly Perturbed Differential Equations (SPDE) in the context of singularly perturbed differential equations.
Abstract: Four Introductory Examples.- Composite Asymptotic Expansions: General Study.- Composite Asymptotic Expansions: Gevrey Theory.- A Theorem of Ramis-Sibuya Type.- Composite Expansions and Singularly Perturbed Differential Equations.- Applications.- Historical Remarks.- References.- Index.
TL;DR: In this paper, a numerical method to solve singularly perturbed delay differential equations is proposed, which works smoothly in both the cases, i.e., whether the delay is of
Abstract: In this paper, we propose numerical method to solve singularly perturbed delay differential equations which works smoothly in both the cases, i.e., whether the delay is of
TL;DR: In this paper, the authors considered zero-relaxation limits for periodic smooth solutions of Euler-Maxwell systems and proposed an approximate solution based on a new asymptotic expansion up to any order.
TL;DR: In this paper, the problem of spherical growth is generalized to include the case of a concentration dependent diffusion coefficient and solutions obtained for slow growth by the method of matched asymptotic expansions.
Abstract: Some moving boundary problems are considered for time fractional diffusion and explicit results obtained for the motion of planar boundaries, as well as cylindrical and spherical ones. The problem of spherical growth is generalized to include the case of a concentration dependent diffusion coefficient and solutions obtained for slow growth by the method of matched asymptotic expansions.
TL;DR: In this paper, the spline of degree four is used for the approximate solution of a third order self adjoint singularly perturbed boundary value problem, and convergence analysis is given and the method is proved to be second order convergent.
Abstract: Singularly perturbed boundary value problems are solved using various techniques. The spline of degree four is used for the approximate solution of a third order self adjoint singularly perturbed boundary value problem. Convergence analysis is given and the method is proved to be second order convergent. Two examples numerically illustrate the method.
References M. Cui and F. Geng, A computational method for solving third order singularly perturbed boundary value problems, Applied Mathematics and Computation , Vol. 198 , (2008) pp. 896--903, doi:10.1016/j.amc.2007.09.023 Zengji Du, Singularly perturbed boundary value problem for nonlinear systems, Applied Mathematics and Computation , Vol. 189 , (2007) pp. 869--877, doi:10.1016/j.amc.2006.11.167 F. A. Howers, Singular perturbation and differential inequalities, Memoris of the American Mathematical Socity, Providence, Rhode Island , Vol. 168 , (1976). Mohan K. Kadalbajoo and Kailash C. Patidar, Numerical soution of singularly perturbed two-point boundary value problem by spline in tension, Applied Mathematics and Computation , Vol. 131 , (2002) pp. 299--320, doi:10.1016/S0096-3003(01)00146-1 Petio Kelevedjiev, Existence of positive solutions to a singular second order boundary value problem, Nonlinear Analysis:Theory, Methods and Applications , Vol 50 (8), (2002) pp. 1107--1118, doi:10.1016/S0362-546X(01)00803-3 Arshad Khan, Islam Khan and Tariq Aziz, Sextic spline solution of singularly perturbed boundary value problem, Applied Mathematics and Computation , Vol. 181 , (2006) pp. 432--439, doi:10.1016/j.amc.2005.12.059 Manoj Kumar, A fourth order finite differeence method for a class of singular two point boundary value problems, Applied Mathematics and Computation , Vol. 133 , (2002) pp. 539--545, doi:10.1016/S0096-3003(01)00255-7 R. K. Mohanty and Navnit Jha, A class of variable mesh spline in compression methods for singularly perturberd two point singular boundary value problem, Applied Mathematics and Computation , Vol. 168 , (2005) pp. 704--716, doi:10.1016/j.amc.2004.09.049 J. Rashidinia, R. Mohammadi and M. Ghasemi, Cubic spline solution of singularly perturbed boundary value problem with significant first derivatives, Applied Mathematics and Computation , Vol. 190 , (2007) pp. 1762--1766, doi:10.1016/j.amc.2007.02.050, H. G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed difference equation, Springer verlag , (1996). Lin Su-rang, Tian Gen-bao and Lin Zong-chi, Singular perturbation of boundary value problem for Quasilinear third order ordinary differential equations involving two small parameters, Applied Mathematics and Mechanics , Vol. 22 (2), (2001) pp. 229--236, doi:10.1023/A:1015553219376 Muhammad I Syam and Basem S. Attili, Numerical solution of singularly perturbed fifth order two point boundary value problem, Applied Mathematics and Computation , Vol. 170 (2005), pp. 1085--1094, doi:10.1016/j.amc.2005.01.003 Ikram A. Tirmizi, Fazal-i-Haq and Siraj-ul-islam, Non-polynomial spline solution of singularly perturbed boundary-value problems, Applied Mathematics and Computation , Vol. 196 , (2008) pp. 6--16, DOI: 10.1016/j.amc.2007.05.029,. Wenyan Wang, Minggen Cui and Bo Han, A new method for solving a class of singular two-point boundary value problem, Applied Mathematics and Computation , Vol. 206 , (2008) pp. 721--727, doi:10.1016/j.amc.2008.09.019
TL;DR: It is shown how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others, including matched asymptotic, WKB, and multiple-scales expansions.
Abstract: This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain—and the connections between these equations for this particular class of solutions. Specifically, we consider “thin-layer” solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero. For the well-studied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail. Examples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes. It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation. We also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others. In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions.
TL;DR: The obtained solutions show that the OHAM is more effective, simpler and easier than other methods, and the results reveal that the method is explicit.
TL;DR: In this article, the authors consider the numerical solution of a initial boundary value problem with a time delay and construct a finite difference method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameter.
Abstract: We consider the numerical solution of a initial boundary value problem with a time delay. The problem under consideration is singularly perturbed from the mathematical perspective. Assuming that the coefficients of the differential equation are smooth, we construct and analyze the finite difference method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameter. The method permits its extension to the case of adaptive meshes, which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practice satisfies the theoretical predictions.
01 Sep 2012
TL;DR: In this paper, a numerical integration method to solve singularly perturbed delay differential equations is presented, in which linear interpolation is used to get three term recurrence relation which is solved easily by discrete invariant imbedding algorithm.
Abstract: In this paper, we present a numerical integration method to solve singularly perturbed delay differential equations. In this method, we first convert the second order singularly perturbed delay differential equation to first order neutral type delay differential equation and employ the numerical integration. Then, linear interpolation is used to get three term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and perturbation parameter.
TL;DR: In this article, the authors developed improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression, which can be extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients.
Abstract: In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Peclet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate.
TL;DR: In this paper, the structure of planar and adiabatic premixed flames is investigated for a simple two-step chain-branching model with finite activation energy within the diffusive-thermal pproximation.
TL;DR: It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters.
TL;DR: A mixed asymptotic-numerical method for the solution of singularly perturbed convection diffusion problems by careful factorization of original problem into two explicit problems which are independent of perturbation parameter contaminating the solution.