John James Henry Miller

• Ph.D., Sc. D.
• Professor at Institute for Numerical Computation and Analysis (INCA) Dublin

Necessary and sufficient conditions for the uniform boundedness of families of matrices, including new proofs of the theorems of H. O. Kreiss.

We are concerned here with the qualitative theory of the zeros of polynomials and not with the quantitative problem of finding numerical approximations to zeros. We say that a polynomial is of type (PI' P2' P3) relative to the unit circle if it has PI zeros interior to, P2 on, and P3 exterior to the unit circle. The unit circle is appropriate for d...

We analyze the convergence, uniformly in e, of a standard and of an exponentially fitted finite difference scheme for a two point boundary singular perturbation problen without a first derivative tern. Sufficient conditions for the convergence, uniformly in e on the whole domain, are given for a general three point finite difference scheme. Let Q =...

In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation of reaction–diffusion type with a discontinuous source term is considered on the interval [0, 2]. A single discontinuity in the source term is assumed to occur at a point d∈(0,2). The leading term of the equation is multiplied by a small posi...

In this paper, a class of linear parabolic systems of singularly perturbed Robin problems is considered. The components of the solution v→ of this system exhibit parabolic boundary layers with sublayers. The numerical method suggested in this paper is composed of a classical finite difference scheme on a piecewise- uniform Shishkin mesh. This metho...

In this paper, a class of singularly perturbed coupled linear systems of second-order ordinary differential equations of convection–diffusion type is considered on the interval [0, 1]. Due to the presence of different perturbation parameters multiplying the diffusion terms of the coupled system, each of the solution components exhibits multiple lay...

In this paper, a weakly coupled partially singularly perturbed linear system of three second-order ordinary differential equations of convection–diffusion type with given boundary conditions is considered on the interval [0, 1]. In spite of coupling, only the components whose equations are perturbed exhibit boundary layers at the origin. A numerica...

A thin circular porous plate is submerged below the free surface of deep water. The problem is reduced to a hypersingular integral equation of the second kind over the surface of the unit disc. The solution is computed by a spectral method known to be efficient for the case of a solid plate. Numerical results are presented for the heave added mass...

A system of two coupled nonlinear initial value equations, arising in the mathematical modelling of enzyme kinetics, is examined. The system is singularly perturbed and one of the components will contain steep gradients. A priori parameter explicit bounds on the two components are established. A numerical method incorporating a specially constructe...

The parameter-uniform convergence of a fitted operator method for a singularly perturbed differential equation is normally available only for uniform meshes. Here we establish the parameter-uniform convergence of a fitted operator method on a non-uniform mesh for a singularly perturbed initial value problem. This is obtained by a new method of proo...

In this paper a survey is presented of the use of finite element methods for the simulation of the behaviour of semiconductor devices. Both standard and mixed finite element methods are considered. We indicate how the various mathematical models of semiconductor device behaviour can be obtained from the Boltzmann transport equation and the appropri...

In this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered. A parameter robust computational method is constructed and it is proved that it gives essentially first order parameter-uniform convergence in the maximum norm. Numerical results are presented in support of the theory.

In this paper, a class of linear parabolic singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The solution u of this equation is smooth, whereas the first derivative in the space variable exhibits parabolic boundary layers. A numerical method composed of a cl...

In this paper, a class of linear parabolic systems of singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The components of the solution $\vec u$ of this system exhibit parabolic boundary layers with sublayers. A numerical method composed of a classical finite...

In this paper, a boundary value problem for a singularly perturbed two-parameter delay differential equation is considered on the interval [0,2]. The solution of this problem exhibits boundary layers at x=0 and x=2 and interior layers at x=1. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin m...

In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The components of the solution u→ of this system are smooth, whereas the components of u→ x exhibit parabolic boundary layers. A numerical method composed...

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection-diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied...

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection- diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied...

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1]. The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is sugg...

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1].The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is sugge...

In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval OE0; 2�: The components of the solution of this system exhibit boundary layers at x D 0 and x D 2 and interior layers at x D 1. A numerical method composed of a classical finite difference scheme a...

A general parabolic system of singularly perturbed linear equations of reaction-diffusion type is considered. The components of the solution exhibit overlapping layers. A numerical method with the Crank-Nicolson operator on a uniform mesh for time and classical finite difference operator on a Shishkin piecewise uniform mesh for space is constructed...

In this paper, a boundary value problem for a semi-linear system of two singularly perturbed second order delay differential equations is considered on the interval (0, 2). The components of the solution of this system exhibit boundary layers at \(x=0\) and \(x=2\) and interior layers at \(x=1\). A numerical method composed of a classical finite di...

In this paper an initial value problem for a system of singularly perturbed first order delay differential equations with discontinuous source terms is considered on the interval (0, 2]. The source terms are assumed to have simple discontinuities at the point \(d \in (0,2)\). The components of the solution exhibit initial layers and interior layers...

In the first section we introduce a simple singularly perturbed initial value problem for a first order linear differential equation. We construct the backward Euler finite difference method for this problem. We then discuss continuous and discrete maximum principles for the associated continuous and discrete operators and we conclude the section b...

This book offers an ideal introduction to singular perturbation problems, and a valuable guide for researchers in the field of differential equations. It also includes chapters on new contributions to both fields: differential equations and singular perturbation problems. Written by experts who are active researchers in the related fields, the book...

In this paper, a numerical method is suggested for solving a mathematical model for the process of cell proliferation and maturation and a model for determining the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites. Both these models give rise to singularly perturbed delay differential e...

In this paper, a boundary value problem for a semi-linear system of two singularly perturbed second order delay differential equations is considered on the interval (0,2). The components of the solution of this system exhibit boundary layers at x = 0 and x = 2 and interior layers at x = 1. A numerical method composed of a classical finite differenc...

In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval [0, 2]. The components of the solution of this system exhibit boundary layers at x = 0 and x = 2 and interior layers at x = 1. A numerical method composed of a classical finite difference scheme ap...

In this paper an initial value problem for a coupled system of two singularly perturbed first-order delay differential equations is considered on the interval (0,2]. The components of the solution of this system exhibit initial layers at 0 and interior layers at 1. A numerical method composed of a classical finite difference scheme on a piecewise u...

In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation is considered. The solution of this problem exhibits boundary layers at x = 0 and x = 2 and interior layers at x = 1. A numerical method composed of a classical finite difference scheme applied on a piecewise-uniform Shishkin mesh is suggeste...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. A small positive parameter multiplies the leading term of each equation. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping...

In this paper an initial value problem for a semi-linear system of two singularly perturbed first order delay differential equations is considered on the interval (0,2]. The components of the solution of this system exhibit initial layers at 0 and interior layers at 1. A numerical method composed of a classical finite difference scheme on a piecewi...

In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can be presented in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes. In this work...

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first m equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularl...

System of equations with partially singularly perturbed on Shishkin meshes is presented by Valarmathi Sigamani

The aim of this work is to develop a simple nonlinear model of a wave energy converter (WEC) for capturing power from ocean waves and converting it into electrical power. A generic axisymmetric device is considered, which consists of a vertical circular cylinder surrounded by a circular annulus. The nonlinear system of equations of motion of this g...

Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an...

Incluye bibliografía e índice

This study presents a new automatic spike sorting method based on feature extraction by Laplacian eigenmaps combined with k-means clustering. The performance of the proposed method was compared against previously reported algorithms such as principal component analysis (PCA) and amplitude-based feature extraction. Two types of classifier (namely k-...

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the sol...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlap...

A coupled first order system of one singularly perturbed and one non-perturbed ordinary differential equation with prescribed initial conditions is considered. A Shishkin piecewise uniform mesh is constructed and used, in conjunction with a classical finite difference operator, to form a new numerical method for solving this problem. It is proved t...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise--uniform...

In this research, a mathematical model is used to simulate the pollution in the reservoir. The hydrodynamic model that provides the velocity field and elevation of the water. The dispersion model gives the pollutant concentration as a water quality measurement. The reservoir is connected to the outer water area is considered. In the simulating proc...

In this research, two mathematical models are used to simulate pollution due to sewage effluent in the uniform reservoir with varied current velocity. The first is the hydrodynamic model that provides the velocity field and elevation of the water flow. The second is the dispersion model that gives the pollutant concentration fields. In the simulati...

Parameter uniform finite difference for a system of equations extended for more than two equations is presented

In this paper we construct three new test problems, called Models A, B and C, whose solutions have two-dimensional boundary layers. Approximate analytic solutions are found for these problems, which converge rapidly as the number of terms in their expansion increases. The approximations are valid for ϵ=10−8ϵ=10−8 in practical computations. Surprisi...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. A Shishkin piecewise-uniform mesh is const...

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions. We transform the problem to an initial boundary value problem in dimensionless form. There are two parameters in the coefficients of the resulting linear parabolic partial differential equation. For a range of values of th...

In this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered. A parameter robust computational method is constructed and it is proved that it gives essentially first order parameter-uniform convergence in the maximum norm. Numerical results are presented in support of the theory.

In this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered. A parameter robust computational method is constructed and it is proved that it gives essentially first order parameter-uniform convergence in the maximum norm. Numerical results are presented in support of the theory.

System of first order singularly perturbed linear differential equations using upwind scheme on shishkin mesh is presented by J.J.H.Miller

The problem of periodic flow of an incompressible fluid through a pipe, which is driven by an oscillating pressure gradient (e.g. a reciprocating piston), is investigated in the case of a large Reynolds number. This process is described by a singularly perturbed parabolic equation with a periodic right-hand side, where the singular perturbation par...

A coupled system of two singularly perturbed ordinary differential equations of first order with the prescribed initial values are considered. The leading term of each equation is multiplied by a small positive parameter and the parameters may differ. The solution exhibits overlapping layers. A Shishkin mesh is constructed. A classical finite diffe...

Featuring international contributors from both industry and academia, Numerical Methods for Finance explores new and relevant numerical methods for the solution of practical problems in finance. It is one of the few books entirely devoted to numerical methods as applied to the financial field. Presenting state-of-the-art methods in this area, the b...

The numerical solution of water pollutant transport problems arises in many important applications in environmental sciences. In this research, the finite element method for solving the one-dimensional and two-dimensional steady state convection-diffusion equation with constant coefficients of nearly closed water area is presented. The presented ma...

We discuss a dimensionless formulation of the Black-Scholes equation for the value of a European call option. We observe that, for some values of the parameters, this may be a singularly perturbed problem. We demonstrate numerically that, in such a case, a standard numerical method on a uniform mesh does not produce robust numerical solutions. We t...

The Prandtl's layer equations was considered for laminar flow past a plate with suction/blowing of the flow rate density v 0(x) = ∓v i2 ∓1/2RE 1/2x ∓1/2. It was found that Reynolds number was large the solution of the problem had a parabolic boundary layer at the surface of the plate excluding its leading edge. A direct numerical method for computi...

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter ε, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates ε-uniformly convergent numerical approximations...

In the present paper we consider a boundary value problem on the semiaxis (0,∞) for a singularly perturbed parabolic equation with the two perturbation parameters ε1 and ε2 multiplying, respectively, the second and first derivatives with respect to the space variable. Depending on the relation between the parameters, the differential equation can b...

In this paper a singularly perturbed convection–diffusion equation with a discontinuous source term is examined. Boundary and weak interior layers appear in the solution. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singula...

The value of a European option satisfies the Black-Scholes equation with appro-priately specified final and boundary conditions. We transform the problem to an initial boundary value problem in dimensionless form. There are two free parameters in the coefficients of the re-sulting linear parabolic partial differential equation and none in the domai...

We consider Prandtl's boundary layer problem for incompressible laminar o w past a three dimensional yawed wedge. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. We construct a direct numerical method for computing approxi- mations to the solution of this problem using a compound piecewise-uniform mesh...

In this paper, we describe an experimental error analysis technique for computing realistic values of the parameter-uniform order of convergence and error constant in the maximum norm associated with a parameter-uniform numerical method for solving singularly perturbed problems. We then employ this technique to compute Reynolds-uniform error bounds...

This paper deals with grid approximations to Prandtl's boundary value problem for boundary layer equations on a flat plate in a region including the boundary layer, but outside a neighbourhood of its leading edge. The perturbation parameter ε= Re-1 takes any values from the half-interval (0,1] ; here Re is the Reynolds number. To demonstrate our nu...

Farrell et al. (Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC: Boca Raton, 2000) develop a Reynolds-uniform numerical method for the solution of the Prandtl equations in the case of flow past a flat plate. In this paper, we examine the applicability of this Prandtl method to the stagnation line flow problem in a domain tha...

In this paper, we deal with Prandtl's boundary layer problem for incompressible laminar flow past a wedge. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. We construct a direct numerical method for computing approximations to the solution of this problem using a piecewise uniform fitted mesh technique...

In this paper we deal with Prandtl's boundary layer problem for incompressible laminar ow past a wedge. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. We construct a direct numerical method for computing approximations to the solution of this problem using a piecewise uniform tted mesh technique appro...

This paper deals with grid approximations to Prandtl's boundary value problem for boundary layer equations on a at plate in a region including the boundary layer, but outside a neighbourhood of its leading edge. The perturbation parameter " = Re takes any values from the half-interval (0; 1]; here Re is the Reynolds number. To demonstrate our numer...

In this paper we describe an experimental technique for computing realistic values of the parameter-uniform order of convergence and error constant in the maximum norm associated with a parameter-uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds-uniform error bounds in the maximum norm f...

. We consider free convection near a semi-infinite vertical flat plate. This problem is singularly perturbed with perturbation parameter Gr, the Grashof number. Our aim is to find numerical approximations of the solution in a bounded domain, which does not include the leading edge of the plate, for arbitrary values of Gr 1. Thus, we need to determi...

The problem of constructing a parameter-uniform numerical method for a singularly perturbed self-adjoint ordinary differential equation is considered. It is shown that a suitably designed discrete Schwarz method, based on a standard finite difference operator with a uniform mesh on each subdomain, gives numerical approximations which converge in th...

A Dirichlet boundary value problem for a linear parabolic di#erential equation is studied on a rectangular domain in the x t plane. The coe#cient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical meth...

In this paper we solve numerically two singularly perturbed linear convection–diffusion problems for heat transfer in a fluid with an assumed flow field in the neighbourhood of a 180° bend in a channel. In the first problem the theoretical solution has a parabolic boundary layer and in the second problem there is both a parabolic and a regular boun...

We consider grid approximations of a boundary value problem for the boundary layer equations modeling flow along a flat plate in a region excluding a neighbourhood of the leading edge. The problem is singularly perturbed with the perturbation parameter ∈ = 1/Re multiplying the highest derivative. Here the parameter ∈ takes any values from the half-...

Singularly perturbed quasilinear boundary value problems exhibiting boundary layers are considered. Special piecewise-uniform meshes are constructed which are fitted to these boundary layers. Numerical methods composed of upwind difference operators and these fitted meshes are shown to be parameter robust, in the sense that the solutions satisfy an...

We consider grid approximations of a boundary value problem for the boundary layer equations modeling flow along a flat plate in a region excluding a neighbourhood of the leading edge. The problem is singularly perturbed with the perturbation parameter epsilon = 1/Re multiplying the highest derivative. Here the parameter epsilon takes any values fr...

In this paper we consider numerical methods for a singularly perturbed reaction–diffusion problem with a discontinuous source term. We show that such a problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-un...

Discrete approximations for certain classes of singularly perturbed boundary value problems are considered. We are interested in analyzing the special technique for the construction of finite difference approximations which converge uniformly with respect to the perturbation parameter ε (or, in short, converge ε-uniformly). The discrete approximati...

. The error generated by the classical upwind finite di#erence ethod on a unifor esh, when applied to a class of singularly perturbed odel ordinary di#erential equations with a singularly perturbed Neu ann boundary condition, tends to infinity as the singular perturbation para eter tends to zero. Note that the exact solution is unifor ly bounded wi...

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed modelo rdinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounde...

Plane stagnation point flow is one of a small class of problems for which a self-similar solution of the incompressible Navier-Stokes equations exists. The self-similar solution and its derivatives can be expressed in terms of the solution of a transformed problem comprising a partially coupled system of quasilinear ordinary differential equations...

We construct a new numerical method for computing reference numerical solutions to the self-similar solution to the problem of incompressible laminar flow past a thin flat plate with suction-blowing. The method generates global numerical approximations to the velocity components and their scaled derivatives for arbitrary values of the Reynolds numb...