Showing papers on "Method of matched asymptotic expansions published in 2015"

Conditional Mean First Passage Times to Small Traps in a 3-D Domain with a Sticky Boundary

Abstract: Calculating the time required for a diffusing object to reach a small target within a larger domain is a feature of a large class of modeling and simulation efforts in biology. Here, we are motivated by the motion of a T cell of the immune system seeking a particular antigen-presenting cell within a large lymph node. The precise nature of the cell motion at the outer boundary of the lymph node is not completely understood in terms of how cells choose to remain within a given lymph node, or exit. In previous work, we and others have studied diffusive motion to a small trap. We extend this previous work to analyze models where the diffusing object may exit the outer boundary of the domain (in this case, the lymph node). This is modeled by a Robin boundary condition on the surface of the lymph node. For the general problem of small traps inside a three-dimensional domain that has a partially sticky or absorbent domain boundary, the method of matched asymptotic expansions is used to calculate the mean and variance of the conditional first passage time for the T cell to reach a specific target trap. Our results are illustrated explicitly for the idealized situation of a spherical lymph node containing small spherically shaped traps, and are verified for a radially symmetric geometry with one trap at the origin where exact solutions are available. Mathematically, our analysis extends previous work on the calculation of the mean first passage time by allowing for a sticky boundary and by calculating conditional statistics of the diffusion process. Finally, our results are interpreted and applied to the context of T cell biology.

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

Abstract: In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.

Singular Perturbation in the Physical Sciences

Abstract: * What is a singular perturbation?* Asymptotic expansions* Matched asymptotic expansions* Matched asymptotic expansions in PDE* Prandtl boundary layer theory* Modulated oscillations* Modulation theory by transforming variables* Nonlinear resonance* Bibliography* Index

Self-electrophoresis of spheroidal electrocatalytic swimmers

Abstract: Using the method of matched asymptotic expansions, we derive a general expression for the speed of a prolate spheroidal electrocatalytic nanomotor in terms of interfacial potential and physical properties of the motor environment in the limit of small Debye length and Peclet number. This greatly increases the range of geometries that can be handled without resorting to numerical simulations, since a wide range of shapes from spherical to needle-like, and in particular the common cylindrical shape, can be well-approximated by prolate spheroids. For piecewise-uniform distribution of surface cation flux with fixed average absolute value, the mobility of a prolate spheroidal motor with a symmetric cation source/sink configuration is a monotonically decreasing function of eccentricity. A prolate spheroidal motor with an asymmetric sink/source configuration moves faster than its symmetric counterpart and can exhibit a non-monotonic dependence of motor speed on eccentricity for a highly asymmetric design.

Asymptotic expansions and numerical simulations of I-V relations via a steady state Poisson-Nernst-Planck system

Abstract: A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

Some Open Questions in the Numerical Analysis of Singularly Perturbed Differential Equations

Abstract: Abstract Several open questions in the numerical analysis of singularly perturbed differential equations are discussed. These include whether certain convergence results in various norms are optimal, when supercloseness is obtained in finite element solutions, the validity of defect correction in finite difference approximations, and desirable adaptive mesh refinement results that remain to be proved or disproved.

A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

Abstract: In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of lin- ear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefficients of the sought-for spectral solutions. Linear differential equations are treated by applying the Petrov-Galerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specific illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efficiency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

Abstract: A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius $\epsilon\ll 1$ are coupled through a passive bulk diffusion field. The method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of these solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell-bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that the ODE-PDE system can be approximated by a finite-dimensional dynamical system, which is studied both analytically and numerically. For one illustrative example of the theory it is shown that when the number of cells exceeds some critical number, the bulk diffusion field can trigger oscillations that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a clustered spatial configuration of cells inside the domain leads to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to that of localized spot patterns for activator-inhibitor reaction-diffusion systems in the limit of long-range inhibition and short-range activation.

Finite Difference Method for a Singularly Perturbed Differential Equations with Integral Boundary Condition

Abstract: Consider the singularly perturbed second order ordinary differential equation with integral boundary condition Before presenting the numerical method for its solution necessary asymptotic bounds have been obtained Uniform convergence of the approximate solution on an uniform mesh is proved Numerical example supporting the theorical analysis is presented

Asymptotic methods in mechanics of solids

Abstract: Asymptotic Estimates.- Asymptotic Estimates for Integrals.- Regular Perturbation of ODE's.- Singularly Perturbed Linear ODE's.- Linear ODE's with Turning Points.- Asymptotic Integration of Nonlinear ODE's.- Bibliography.- Index.

The perturbation of electromagnetic fields at distances that are large compared with the object's size

Abstract: We rigorously derive the leading order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared to its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading order terms in expansions for the fields for a low frequency problem when the scatterer is a non�lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case we express our results in terms of polarisation tensors.

On the solutions of a three-dimensional system of difference equations

Abstract: In this paper, we obtain the explicit solutions of the three-dimensional system of difference equations with multiplicative terms, which extended some results in literature. Also, by using explicit forms of the solutions, we study the asymptotic behaviour of well-defined solutions of the system.

Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method

Abstract: In this paper, we are concerned with a singularly perturbed higher-order KdV equation, which is considered as a paradigm in nonlinear science and has many applications in weakly nonlinear and weakly dispersive physical systems. Based on the relation between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, the persistence of the solitary wave solution for the singularly perturbed KdV equation is investigated by using the geometric singular perturbation theory and dynamical systems approach when the perturbation parameter is suitably small.

Second-order gravitational self-force

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.

Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem

Abstract: A singularly perturbed periodic problem for a parabolic reaction-advection-diffusion equation at low advection is studied. The case when there is an internal transition layer under unbalanced nonlinearity is considered. An asymptotic expansion of a solution is constructed. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is studied; the proof uses the Krein-Rutman theorem.

Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains

Abstract: Full asymptotic expansions are constructed and justified for two series of eigenvalues and the corresponding eigenfunctions of the spectral Steklov problem in a domain with a singular boundary perturbation having the form of a small cavity. The terms of those series are of type λk+o(1) and ε −1(μm+o(1)), where λk and μm are the eigenvalues of the Steklov problem in a bounded domain without cavity and the exterior Steklov problem for a cavity of unit size. A similar problem of the surface wave is also treated. The smoothness requirements on the boundary are discussed and unsolved problems are stated.

New representations of the Maslov canonical operator and localized asymptotic solutions for strictly hyperbolic systems

Abstract: A procedure for constructing asymptotic solutions of strictly hyperbolic systems of partial differential equations is described in general terms. On the basis of the obtained results, an asymptotic solution of a three-dimensional wave equation with variable velocity is constructed. The case of a cylindrically symmetric velocity is considered.

Asymptotic expansions of solutions in a singularly perturbed model of virus evolution

Abstract: An initial boundary value problem for a singularly perturbed system of partial integro-differential equations involving two small parameters multiplying the derivatives is studied. The problem arises in a virus evolution model. An asymptotic solution of the problem is constructed by the Tikhonov-Vasil’eva method of boundary functions. The analytical results are compared with numerical ones.

A new quartic B-spline method for third–order self-adjoint singularly perturbed boundary value problems

Abstract: In this paper, a reliable algorithm is presented to develop approximate solution of third-order self-adjoint singularly perturbed two-point boundary value problem in which the highest order derivative is multiplied by a small parameter. In this method, first we introduce the Quartic B-spline basis function in the form of ( ) i B x . After that we use the linear sequence of Quartic B-spline to get the numerical solution of system of equations. These systems of equations are solved by using MATLAB. Two Examples are illustrated to understand the present method. The results of these examples are also compared with the available results obtained by Ghazala Akram [1]. Mathematics Subject Classification: 65N20

Model reduction for a class of singularly perturbed stochastic differential equations

Abstract: A class of singularly perturbed stochastic differential equations (SDE) with linear drift and nonlinear diffusion terms is considered We prove that, on a finite time interval, the trajectories of the slow variables can be well approximated by those of a system with reduced dimension as the singular perturbation parameter becomes small In particular, we show that when this parameter becomes small the first and second moments of the reduced system's variables closely approximate the first and second moments, respectively, of the slow variables of the singularly perturbed system Chemical Langevin equations describing the stochastic dynamics of molecular systems with linear propensity functions including both fast and slow reactions fall within the class of SDEs considered here We therefore illustrate the goodness of our approximation on a simulation example modeling a well known biomolecular system with fast and slow processes

Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics

Abstract: A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in the separation coordinate between two possible reactants. This boundary condition can be interpreted as an idealization of a reactive interaction potential model, in which a potential barrier must be surmounted before reactions can occur. In this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation, where molecules react instantly upon reaching the peak of the barrier. The limiting boundary condition is derived by the method of matched asymptotic expansions, and shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased. Limiting boundary conditions for the same interaction potential in both the overdamped Fokker-Planck equation (Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are investigated. It is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the Kramers equation solution converges to the solution of the Fokker-Planck equation.

On the Stability and the Attraction Domain of the Stationary Solution of a Singularly Perturbed Parabolic Equation with a Multiple Root of the Degenerate Equation

Abstract: We construct and justify the asymptotics of a boundary layer solution of a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the degenerate (finite) equation has an identically double root. A specific feature of the asymptotics is the presence of a three-zone boundary layer. The solution of the boundary value problem is a stationary solution of the corresponding parabolic equation. We prove the asymptotic stability of this solution and find its attraction domain.

A fitted numerical scheme for second order singularly perturbed delay differential equations via cubic spline in compression

Abstract: This paper deals with the singularly perturbed boundary value problem for the second order delay differential equation. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. An exponentially fitted difference scheme on a uniform mesh is accomplished by the method based on cubic spline in compression. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter, which is illustrated with numerical results.

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

Abstract: In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented. Keywords : Reaction-diffusion systems, singular perturbations, fitted operator finite difference methods, convergence analysis.

The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results

Abstract: After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , ie as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1 Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations” This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses