G. I. Shishkin

• Professor
• Leading Scientific Researcher at Russian Academy of Sciences

The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x -derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval {(0,1]} . For small ε, the problem involves a bounda...

On the set \(\overline{G} =G \cup S\), \(G=(0,d]\times (0,T]\) with the boundary \(S=S_0 \cup S^{\,\ell }\), we consider an initial-boundary value problem for the singularly perturbed transport equation with a perturbation parameter \(\varepsilon \) multiplying the spatial derivative, \(\varepsilon \in (0,1]\). For small values of the perturbation...

In a space-time domain \(\overline{G}=\overline{D} \times [0,T]\), where \(\overline{D}\) is a doubly connected domain in space—a rectangle \(\overline{D}_1\) with a removed circle \(D_2\), we consider the Dirichlet initial–boundary value problem for a singularly perturbed parabolic reaction–diffusion equation. As \(\varepsilon \rightarrow 0\), bou...

For a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter \(\varepsilon \), \(\varepsilon \in (0,1]\), multiplying the highest-order derivative in the equation, we construct an improved computer difference scheme (with approximation of the first-order spatial derivative in the convective term by the central di...

Grid approximation of the Cauchy problem on the interval D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem...

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set Ḡ = G ∪ S, where Ḡ = D̅ × [0 ≤ t ≤ T], D̅ = {0 ≤ x ≤ d}, S = Sl ∪ S, and Sl and S0 are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open...

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is anal...

Finding numerical solutions to the Troesch’s problem is known to be challenging, especially when the sensitivity parameter \(\lambda \) is large. In this manuscript, we propose a numerical method for solving the Troesch’s problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite d...

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction–diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter ε, ε ∈ (0, 1] (i.e., ε-uniformly) with order of accur...

A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not ε-uniformly stable with respect to perturbations. The conditions impos...

In this paper, for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε ², ε ∈ (0,1], multiplying the highest-order derivative in the equation, an initial-boundary value Dirichlet problem is considered. For this problem, a standard difference scheme constructed by using monotone grid approximations...

An initial–boundary value problem is considered for a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) multiplying the highest order derivative. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform mesh is analyzed, and the behavior of dis...

In this paper we consider a 1D parabolic singularly perturbed reaction-convection-diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter ε2) and the convection term (multiplied by the parameter μ) in the differential equation (ε∈(0,1], μ∈[0,1], μ≤ε). Moreover, the convective term degenerates inside th...

Experimental results of the study on causes of the difference in thermal conductivity coefficient of water under water heating by a biological object (operator hand) compared to heating by an electric radiator of the same temperature are given. Two possible causes of the observed effect, which are associated with the difference in the spectral comp...

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if t...

The reasons for differences in the temperature coefficients of the electrical conductivity of water during its heating by a biological object (e.g., an operator’s hand) in comparison with an electric heater of the same temperature were investigated. Two possible explanations for the observed effects, viz., the difference in the spectral composition...

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0,...

A Dirichlet problem is considered for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter \(\varepsilon\) (\(\varepsilon \in (0,1]\)) multiplying the highest-order derivative in the equation. This problem is approximated by the standard monotone finite difference scheme on a uniform grid. Such a...

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter E > (that takes arbitrary values from the half-open interval (0, 1]) is considered. For this problem, an approach to the construction of a numerical method based on a standard difference scheme on uniform meshes is deve...

A new technique to study special difference schemes numerically for a Dirichlet problem on a rectangular domain (in x, t) is considered for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε; ε ∈ (0, 1]. A well known difference scheme on a piecewise-uniform grid is used to solve the problem. Such a scheme c...

We consider a technique to construct ε-uniformly convergent in the maximum norm grid approximations of higher accuracy order on uniform grids for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ε (ε ∈.(0; 1]) multiplying the highest-order derivative, the solution of which has a parabolic boundary layer in...

We consider a Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter \(\varepsilon\) (\(\varepsilon \in (0,1]\)) multiplying the highest-order derivative in the equation. This problem is approximated by the standard monotone finite difference scheme on a uniform grid. Such a sc...

In this paper we consider the efficient numerical approximation of a singularly perturbed parabolic convection-diffusion problem having a convective term which degenerates inside the domain, in the case that the right-hand side of the differential equation is discontinuous on the degeneration line. For small values of the diffusion parameter ε 2 (ε...

An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter ε, ε ∈ (0;1]. We consider a standard difference scheme, which is a monotone scheme on a uni...

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid...

For a singularly perturbed ordinary differential convection-diffusion equation with a perturbation parameter ϵ (ϵ∈(0,1]), we analyze the stability of a standard finite difference scheme based on monotone approximations of the problem on a uniform mesh. This scheme does not converge ϵ-uniformly in the maximum norm and (in the case of its convergence...

For a Dirichlet problem for an one-dimensional singularly perturbed parabolic convection-diffusion equation, a difference scheme of the solution decomposition method is constructed. This method involves a special decomposition based on the asymptotic construction technique in which the regular and singular components of the grid solution are soluti...

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the de...

We consider a numerical approximation of the classical problem of the turbulent free jet. The same boundary layer equation is used for the laminar jet, but with the introduction of the turbulent viscosity. This viscosity depends on the kinematic momentum and the slenderness parameter and varies in space. Here, the problem under consideration is tak...

A phenomenon of superspiralization was revealed in homochiral solutions of biomimetics with strings lengths from 10(2)nm to 10(2)μ and longer: strings of greater size spontaneously formed in solution are twined of the smaller strings, which also have a helical structure. The chiral pitch depends on the conditions of formation of a particular string...

A research of non-equilibrium thermal and biothermal radiation generated by heated solid materials and hematothermal living organisms is performed by water conductometric sensors. Procedure and measuring technique are given, experimental results are described and analysed.

We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the stream is parallel to its axis. When the Reynolds number is large, this problem is known to exhibit boundary layers which grow downstream and eventually shows a ‘‘3-D character’’ because of the large curvature of the body in the trans...

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a small parameter ɛ (ɛ ∈ (0, 1]) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relat...

A finite difference scheme on special piecewise-uniform grids condensing in the interior layer is constructed for a singularly perturbed parabolic convection-diffusion equation with a discontinuous right-hand side and a multiple degenerating convective term (the convective flux is directed into the domain). When constructing the scheme, monotone gr...

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

For a model of two-dimensional boundary value problem for a second-order parabolic equation, finite difference schemes on the base of a domain decomposition method, oriented on modern parallel computers, is constructed. In the used finite difference schemes iterations at time levels are not applied; some subdomains overlap. We study two classes of...

In the case of the Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with a small parameter ε multiplying the higher order derivative, a finite difference scheme of improved order of accuracy that converges almost ε-uniformly (that is, the convergence rate of this scheme weakly depends on ε) is constructed. When ε...

Grid approximations of the Dirichlet problem are considered in a vertical strip for the semilinear elliptic convectiondiffusion equation; for this problem, the nonlinear difference scheme based on the classic approximation of the problem on a piecewise-uniform grid refined in a layer converges μ-uniformly in the uniform norm with the convergence ra...

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter...

A Stefan-type problem is considered. This is an initial-boundary value problem on a composite domain for a parabolic reaction-diffusion equation with a moving interface boundary. At the moving boundary between the two subdomains, an interface condition is prescribed for the solution of the problem and its derivatives. A finite difference scheme is...

For an initial-boundary value problem for a singularly perturbed parabolic reaction-diffusion equation on a composed domain, a conservative flux difference scheme on flux piecewise-uniform grids is constructed whose solution and also normalized diffusion flux converge (in the maximum norm) independent of the perturbation parameter e\epsilon at th...

For a Dirichlet problem for a singularly perturbed parabolic convection-diffusion equation with small parameter e\epsilon multiplying the highest-order derivative, a finite difference scheme with improved accuracy is constructed that converges almost e\epsilon -uniformly with order of the convergence rate close to 2 for fixed values of e\epsi...

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this d...

A grid approximation is considered for the Dirichlet problem in a rectangular domain for the one-dimensional singularly perturbed parabolic reaction-diffusion equation. A parabolic boundary layer appears in the neighbourhood of the lateral part of the domain boundary for small values of the perturbation parameter ε, ε ∈ (0, 1]. For the differential...

A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appea...

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost...

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes that converge uniformly with respect to the perturbation parameter ɛ, ɛ ∈ (0, 1]. The approach is based on the decomposition of a discrete solution int...

An initial boundary value problem for a singular perturbed parabolic reaction–diffusion equation is considered in a domain unbounded in x on the real axis; the leading derivative of the equation contains the parameter ε 2; ε ∈ (0, 1]. The right-hand side of the equation and the initial function indefinitely grow as 𝒪(x 2) for x→ ∞, which leads to a...

The initial-boundary value problem in a domain on a straight line that is unbounded in x is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ2, where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as x → ∞...

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ɛ2, where ɛ ∈ (0, 1]. When ɛ is small, a boundary and an interior layer (with the characteristic width ɛ)...

A class of singularly perturbed convection–diffusion problems is considered which contain a mixed derivative term. We consider the case when exponential boundary layers are present in the solutions of problems from this class. Under appropriate assumptions on the data of the problem, we construct a decomposition of the solution into regular and lay...

On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction‐diffusion equations connected only by terms that do not involve derivatives. The highest‐order derivatives in the equations, having divergent form, are multiplied by the perturbation parameter e ; ϵ ∈ (0,1]. When ϵ → 0, the...

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is...

An initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm...

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform nu...

This paper deals with the measurement of characteristics of electromagnetic emission of electric propulsions as applied to the EMC problems. Electric propulsion systems on the basis of pulsed plasma thrusters, which are used for the orbit correction of small satellites, are considered as the sources of unintended noise of artificial origin. Test re...

Preface Part I: Grid Approximations of Singular Perturbation Partial Differential Equations Introduction Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Smooth Boundaries Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Piecewise-Smooth Boundaries Generalizations for Elliptic Reacti...

The grid approximation of an initial‐boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation. The second‐order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters å 2 1 and å 2 2, respectively, that take arbitrary values in the open‐closed interval (0...

Parameter-uniform numerical methods for a singularly perturbed elliptic problem with parabolic boundary layers in the solution are analyzed. A priori parameter explicit bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. Based on this decomposition a numerical met...

On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction-diffusion equations. The highest-order derivatives in the equations are multiplied by the perturbation parameter epsilon(2); epsilon is an element of (0, 1]. When epsilon -> 0, the parabolic boundary layer appears. Using the...

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary lay...

The electrodynamic models of scattered fields of moving inhomogeneous weakly ionized plasma media are based on the study of its gas dynamic and plasma physical properties. The solution of different radio physical (radar) problems requires the determination of the electron density distribution over jet volume. Such distributions have been obtained b...

Because of emerging multimedia applications, such as video-on-demand, video conferencing, interactive gaming, IPTV and e-learning, bandwidth demands from end users are constantly increasing. The copper wire technologies (e.g. cable and DSL) bridging ...

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered in a rectangular domain in x and t; the perturbation parameter ɛ multiplying the highest derivative takes arbitrary values in the half-open interval (0,1]. For the boundary value problem, we construct a scheme based on the method of lines in x...

A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction‐diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ϵ tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obta...

A Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with a piecewise-continuous initial condition. For this problem, using the method of additive splitting of singularities (generated by discontinuities of the initial function and its lowest derivatives) and the Richardson method, a finite difference s...

We consider a Dirichlet problem on a ball for a singularly perturbed parabolic reaction-diffusion equation. The Laplacian in the equation is multiplied by a perturbation parameter ε 2, where ε ∈ (0,1]. The solution of such a problem exhibits the parabolic boundary layer in a neighbourhood of the ball boundary as ε→0. Using the integro-interpolation...

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solutio...

Singularly perturbed heat equations are studied in the case when the heat/diffusion flux is given on the domain boundary. This paper shows that (depending on the value of perturbation parameter) the errors in the approximate solution, computed by a classical finite difference scheme, can exceed the exact solution many times. The computed normalised...

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. When ɛ vanishes, the system of parabolic equations degenerates into a syst...

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic convection‐diffusion equation. For this problem, upwind difference schemes on the well‐known piecewise‐uniform meshes converge ϵ‐uniformly in the maximum discrete norm at the rate O(N− 1lnN + N0 −1 ), where N + 1 and N 0 + 1 are the number of mesh po...

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condi...

Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the e-uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods. The first part of the book explores boundary value p...

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...

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ɛ 2, ɛ ε (0, 1]. For small values of the parameter ɛ, in a neighborhood of the lateral part of the bou...

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving bound...

A priori parameter explicit bounds on the solution of singularly perturbed elliptic problems of convection–diffusion type are established. Regular exponential boundary layers can appear in the solution. These bounds on the solutions and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. By...

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.

A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ 2; ɛ ϵ (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a nei...

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ɛ i 2 (i = 1, 2). The parameters ɛi take arbitrary values in the half-open interval (0, 1]. When the vector parameter ɛ = (ɛ...

A problem for the black-Scholes equation that arises in financial mathematics is reduced, by a transformation of variables, to the Cauchy problem for a singularly perturbed parabolic equation with the variables x, t and a perturbation parameter ɛ, ɛ ∈ (0, 1]. This problem has several singularities such as the unbounded domain, the piecewise smooth...

IntroductionNumerical Solutions of the Diffusion Equation with Prescribed Values on the BoundaryNumerical Solutions of the Diffusion Equation with Prescribed Diffusion Fluxes on the BoundaryDiffusion Equations with Concentrated SourcesApplication to Heat Transfer in Some Technologies

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 grid approximation of the Dirichlet problem is considered on a segment for a parabolic convection-diffusion equation; the high derivative of the equation contains a parameter ε taking arbitrary values from the half-interval (0,1]. A difference scheme on a posteriori adaptive grids is constructed for the boundary value problem. The classic approxi...

The objective of this paper is to construct some high order uniform numerical methods to solve linear singularly perturbed delay parabolic differential equations, which present only parabolic layers in their solution. First, we use the Crank-Nicolson method to discretize the time variable and the central finite difference scheme, defined on a piece...

Dirichlet problem is considered for a singularly perturbed semilinear parabolic convection-diffusion equation on a rectangular domain. The solution of the classical finite difference scheme on a uniform mesh converges at the rate O((ε + N−1)−1 N−1 + N0−) where N + 1 and N0 + 1 denote the numbers of mesh points with respect to χ and t respectively,...

We investigate the model problem of flow of a viscous incompressible fluid past a symmetric curved surface when the flow is parallel to its axis. This problem is known to exhibit boundary layers. Also the problem does not have solutions in closed form, it is modelled by boundary‐layer equations. Using a self‐similar approach based on a Blasius seri...

The Dirichlet problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation on a rectangular domain. For this problem classical finite difference (nonlinear) schemes on piecewise uniform meshes condensing in the boundary layer converge e-uniformly at a rate that is at best first-order. Using a Richardson extra...

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε2 multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval...

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small pa-rameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decom-posed into a sum of regular and singular componen...

A priori parameter explicit bounds on the derivatives of the solution of a two parameter singularly perturbed elliptic problem in two space dimensions are presented. These bounds are used to establish parameter uniform error bounds for a numerical method consisting of upwinding on a tensor product of two piecewise uniform meshes. First Published O...

Neumann problems for singularly perturbed parabolic equations are considered on a segment and on a rectangle. The second-order derivatives are multiplyed by a small parameter 2. When =0, the parabolic equation degenerates, and only the time derivative remains. The normalized diffusion flux, i.e., the product of and the derivative in the direction...

We construct a finite difference method for boundary value problems modelling heat and mass transfer for fast-running processes. The dimensionless form of the equation in these problems is singularly perturbed, i.e., the highest derivatives are multiplied by a parameter 2 which can take any values from the interval (0,1]. The equation involves con...

A numerical method is developed for a time dependent reaction diffusion two dimensional problem. This method is deduced by combining an alternating direction technique and the central finite difference scheme on some special piecewise uniform meshes. We prove that this method is uniformly convergent with respect to the diffusion parameter , achievi...