No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Uniform numerical methods for problems with initial and boundary layers. (Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem). Transl. from the English
... Herbert Keller [R15] made an overview of numerical methods used in boundary-layer theory. Many scholars continue these efforts offering new computing techniques [5][6][7][8] and programming codes [R16, R17] with adaptations to new complex practical applications. ...
... The numerical analysis of such problems by traditional box-schemes [4] is limited by non-uniform convergence or even divergence of numerical solutions. In this study, the numerical solutions of the model singular ordinary differential equation [5] have been evaluated for the linear boundary value problem. The developed numerical method is used for the analysis of gas flow parameters in boundary and viscous shock layers under the conditions of gas injection from the body surface. ...
... The linear boundary value problem is studied for the following model singular ordinary differential equation [5] and boundary conditions: ...
Students will use computer programs (or create their own programming code) based on exponential box-scheme approximations for solving systems of nonlinear differential equations that contain small parameters for the highest derivative terms or singularities in boundary conditions. The uniform second-order accuracy is obtained for functions and derivatives in this approach. The methodology is widely applied to practical studies of boundary layers with gas injection and combustion.
... The assumption (2) causes the solution to have a boundary layer at x = 1. The first order partial derivative (with respect to x) of the exact solution satisfies the bound lu'(x,e)l<C*(l+e-lexp(-fl(1-x)/e)) for0<x< 1, ...
... Classical numerical methods are in general unsuitable for such problems--see for instance [2,6,10]. When e << 1, they are stable only if the mesh width h satisfies unrealistically strong conditions. ...
... Due to the difficulty of discretizing the operator part of the norm II1' Ill, we compute only the error in the e-weighted piecewise Ht-norm II" I1~-As usual (see, e.g., [2]) we use the double mesh principle to determine the numerical order of convergence: ...
... This case study introduces you to the development and applications of numerical methods for solving singular (ordinary or partial) differential equations with small coefficients for the highest derivative terms. This singularity leads to the formation of regions with small linear dimensions where gradients of functions are large, thus making numerical steps unsteady [1,[4][5][6]. (The analytical analyses of these zones do not provide reliable quantitative data estimations [1][2][3][4][5]). The numerical analysis of such problems by traditional box-schemes is restricted by non-uniform convergence or even divergence of numerical solutions [4][5][6]. ...
... This singularity leads to the formation of regions with small linear dimensions where gradients of functions are large, thus making numerical steps unsteady [1,[4][5][6]. (The analytical analyses of these zones do not provide reliable quantitative data estimations [1][2][3][4][5]). The numerical analysis of such problems by traditional box-schemes is restricted by non-uniform convergence or even divergence of numerical solutions [4][5][6]. ...
... (The analytical analyses of these zones do not provide reliable quantitative data estimations [1][2][3][4][5]). The numerical analysis of such problems by traditional box-schemes is restricted by non-uniform convergence or even divergence of numerical solutions [4][5][6]. In this case study, you will find the numerical solutions of the model's singular ordinary differential equation evaluated for the linear boundary value problem [5]. ...
Students will use computer programs (or create their own programming code) based on exponential box-scheme approximations for solving systems of nonlinear differential equations that contain small parameters for the highest derivative terms or singularities in boundary conditions. The uniform second-order accuracy is obtained for functions and derivatives in this approach. The methodology is widely applied to practical studies of boundary layers with gas injection and combustion.
... Singular perturbation problems model convection-diffusion processes in applied mathematics that arise in diverse areas, including linearized Navier-Stokes equation at high Reynolds number and the drift-diffusion equation 352 Abagero, Duressa and Debela of semiconductor device modeling, heat and mass transfer at high Pe'clet number, and so on; see [6,13,18,19]. The novel aspect of the problem under consideration is that we take a source term in the differential equation that has a jump discontinuity at one or more points in the interior of the domain. ...
... Recently, Shandru and shanthi [3] presented a fitted mesh method to solve singularly perturbed robin type boundary value problems with discontinuous source terms. Indeed, still, there is a room to increase the accuracy and show the parameter uniform convergence because the treatment of singular perturbation problem is not trivial distributions and the solution is pended on perturbation parameter, ε and mesh size, h; see [6]. Due to this, the numerical treatment of singularly perturbed boundary value problems is need improvement. ...
... and it satisfies the following minimum principle for boundary value problems. The following lemmas [6] are necessary for the existence and uniqueness of the solution and for the problem to be well-posed. ...
Singularly perturbed robin type boundary value problems with discontinuous source terms applicable in the geophysical fluid are considered. Due to the discontinuity, interior layers appear in the solution. To fit the interior and boundary layers, a fitted nonstandard numerical method is constructed. To treat the robin boundary condition, we use a finite difference formula. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε, and mesh size, h. The numerical result is tabulated, and it is observed that the present method is more accurate and uniformly convergent with an order of convergence of O(h).
... The classical methods fail to prevent the rapid change in the boundary layer region. Therefore, a special deliberation is required to construct an appropriate numerical methods for these problems whose accuracy does not depend on singular perturbation parameter (ε), i.e., the methods that are ε-uniformly convergent [4,6,22]. A hybrid method with Shishkin mesh for solving singularly perturbed delay differential equations in [10]. ...
... It is assumed that the solution u(x) of the problem (5.1)-(5.2) is continuous on [0, 1] and differentiable on (0, 1). Also assume that F (x, u, u ′ (x − δ)) as F (x, u, z) is a smooth function satisfying For δ = 0, under these conditions the problem (5.1)-(5.2) has a unique solution (see, [4]). ...
... In this section, we have solved linear and nonlinear problems to validate the efficiency and applicability of the proposed method. The exact solution for the following problems are not known so we use the double mesh principle [4] for the maximum point-wise error and rate of convergence are defined as ...
This article presents a hybrid numerical scheme for a class of linear and nonlinear singularly perturbed convection delay problems on piecewise uniform. The proposed hybrid numerical scheme comprises with the tension spline scheme in the boundary layer region and the midpoint approximation in the outer region on piecewise uniform mesh. Error analysis of the proposed scheme is discussed and is shown -uniformly convergent. Numerical experiments for linear and nonlinear are performed to confirm the theoretical analysis.
... The fundamental idea used in this method is the Boundary Value Technique (BVT) discussed by many authors for second order, third order and fourth order ODEs [21,32,27] in which the authors divided the interval [0, 1] into two subintervals namely [0, kε] and [kε, 1] where kε is taken as the approximate width of the boundary layer. In the inner region [0, kε] they applied an EFFD scheme of [1] and a classical finite difference scheme for the outer region [kε, 1]. They also presented error estimates for the numerical solution. ...
... From the stability result as given in Doolen [1] we have, ...
... where b(x) and f (x) are sufficiently smooth and b(x) ≥ β, β > 0, 0 ≥ c(x) ≥ −γ, γ > 0. Proof. Please refer [1]. ...
In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter {\varepsilon} multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.
... then after putting Equation (12) into Equation (11) and making rearrangements, the following equation is generated: ...
... , so as to ensure that the sum of all terms in the right of equal sign in Equation (14) is positive. The reason for this condition is that: when the sum of all terms in the right of equal sign in Equation (14) is greater than or equal to zero, this implies that ) ( x u is smaller than or equal to the exact solution "zero", yet Expression (12) indicates that only a negative ) ( x u can make the obtained approximate solution ) ( x u be smaller than or equal to the exact solution ) (x u . And similarly, this is also applicable in obtaining the upper solution. ...
... Secondly, although the maximum possible errors of approximate solutions increase as 2 decreases, the method mentioned in this paper still can be effective in defining the maximum possible errors of solutions under conditions of unknown exact solutions. Besides, if based on the double mesh principle (Dollan et al. [12]), the approximate solutions are considered as the exact solutions, the exact maximum error obtained in this paper is far smaller than the estimated maximum possible error, as displayed in Table 2. This indicates that the obtained upper and lower approximate solutions can not only define the error range of approximate solutions, but also the mean approximate solution derived from the upper and lower approximate solutions is characterized by more accuracy. ...
This paper seeks to use the proposed residual correction method in coordination with the monotone iterative technique to obtain upper and lower approximate solutions of singularly perturbed non-linear boundary value problems. First, the monotonicity of a non-linear differential equation is reinforced using the monotone iterative technique, then the cubic-spline method is applied to discretize and convert the differential equation into the mathematical programming problems of an inequation, and finally based on the residual correction concept, complex constraint solution problems are transformed into simpler questions of equational iteration. As verified by the four examples given in this paper, the method proposed hereof can be utilized to fast obtain the upper and lower solutions of questions of this kind, and to easily identify the error range between mean approximate solutions and exact solutions.
... It is well-known fact that the solution of SPPs exhibits a multi-scale character that is; there are thin layer(s) where the solution varies rapidly, while away from the layer(s) the solution behaves regularly and varies slowly. For detailed discussion on the analytical and numerical treatment of SPPs we may refer the reader to the books of O'Malley [2], Doolan et al. [3], Roos et al. [4] and Miller et al. [5]. ...
... Then L ε satisfies the following minimum principle on Ω. Using the techniques adapted in [3,5] the following theorem can be proved. ...
... where . is the maximum norm, Y m (x) is the m th iterative value of u and λ is a prescribed error tolerance. In the interval (0, kε), (1 − kε, 1), (kε, d − kε) and (d + kε, 1 − kε) the problem (3.1) is smooth and the following theorem can be easily proved by [2,3,4,5]. ...
In this paper a second-order Singularly Perturbed Ordinary Differential Equation (ODE) of Reaction-Diffusion type Boundary Value Problems (BVPs) with discontinuous source term is considered. A numerical method is suggested in this paper to solve such problems. The domain of definition of the differential equation (a closed interval) is divided into five non-overlapping subintervals, which we call "Inner Region" (Boundary Layers) and "Outer Region". Then, the Differential Equation is solved in these intervals separately. The solutions obtained in this region are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval), we mostly use zero-order asymptotic expansion of the solution of the BVPs. Error estimates of the solution and numerical examples are provided.
... It is wellknown fact that the solution of SPPs exhibits a multi-scale character that is; there are thin layer(s) where the solution varies rapidly, while away from the layer(s) the solution behaves regularly and varies slowly. For detailed discussion on the analytical and numerical treatment of SPPs we may refer the reader to the books of O'Malley [2], Doolan et al. [3], Roos et al. [4] and Miller et al. [5]. ...
... Then L ε satisfies the following minimum principle on Ω . Using the techniques adopted in [3,5] the following theorem can be proved. ...
... Analogous to the discontinuous problem of (1) and (2), we can give definitions and results for the discrete schemes of above (i) and (ii). The following theorems can be easily obtained from the procedure given in [3,5,[14][15]. 1) and (2) is discussed in the whole domain by various authors [4,[16][17], due to the parallel computing interest we present the following BVT method to the Eqs. (1) and (2). ...
In this paper a second-order Singularly Perturbed Ordinary Differential Equation(ODE) of Reaction-Diffusion type Boundary Value Problems (BVPs) with discontinuous source term is considered. A numerical method is suggested in this paper to solve such problems. The domain of definition of the differential equation (a closed interval) is divided into five non-overlapping subintervals, which we call "Inner Region" (Boundary Layers) and "Outer Region". Then, the Differential Equation is solved in these intervals separately. The solutions obtained in this region are combined to give a solution in the entire interval. To obtain terminal boundary conditions (boundary values inside this interval), we mostly use zero-order asymptotic expansion of the solution of the BVPs. Errorestimates of the solution and numerical examples are provided.
... This interval (x j;0:5 x j+0:5 ) is refered as the control volume associated with the grid point x j (the j-th cell). To derive a di erence equation associated with the grid point x j we i n tegrate the self-adjoint f o r m o f v ector di erential equation (1) @( @u=@x)@ x= G(x) (2.1) in the intervals (x j;0:5 x j+0:5 ) : W j+0:5 ; W j;0:5 = Z xj xj;0:5 G j dz + Z xj+0:5 xj G j+1 dz (2.2) where W j 0:5 = W j x=xj 0:5 x j 0:5 = ( x j + x j 1 )=2 h j = x j ; x j;1 ...
... nite{ di erence approximation of systems (1) on nonuniform grid with grid points x j the system can be written in the following local selfadjoint form (4) and the monotone vector-di erence equations of 3-point s c heme can be represented in the form(6). By calculation the matrix-function g(s) on the spectrum of the matrix s = ;the matrix-function g(s) ensures the exact discrete approximation of boundary value problem for equation(1). ...
We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.
First Published Online: 14 Oct 2010
... ISSN 1992-5980 eISSN 1992-6006 The presence of ε small parameter in the Poisson equation (34) means that the boundary problem is singularly perturbed. This significantly complicates its numerical solution, since such problems are stiff [26]. φ potential of the electric field and 1 2 , С С ion concentrations change very quickly in a narrow boundary layer whose thickness is equal to D l Debye length [5]. ...
... To solve this problem, it is advisable to compact the computational grid in the boundary layer and use special methods for solving stiff problems [26]. ...
Introduction. The theoretical description of the ion transport in membrane systems in the galvanostatic mode is presented. A desalting channel of the electrodialysis apparatus is considered as a membrane system. The work objectives are the development and verification of a two-dimensional mathematical model of the stationary transport of salt ions in the desalting channel of the electrodialysis apparatus for the galvanostatic mode.
Materials and Methods. A new model of ion transfer is proposed. It is based on the Nernst –Planck – Poisson equations for the electric potential and on the equation for the electric current stream function. A numerical solution to the boundary value model problem by the finite element method is obtained using the Comsol Multiphysics software package.
Research Results. The developed mathematical model enables to describe the stationary transfer of binary salt ions in the desalting channel of the electrodialysis apparatus. Herewith, the violation of the solution electroneutrality and the formation of the dilated domain of space charge at overlimiting currents in the galvanostatic mode are considered. A good agreement between the physicochemical characteristics of the transfer calculated by the models for the galvanostatic and potentiostatic modes implies adequacy of the constructed model.
Discussion and Conclusions. The developed model can interpret the experimental study results of ion transfer in membrane systems if this process takes place in the galvanostatic mode. Some electrokinetic processes are associated with the appearance of a dilated domain of space charge at overlimiting currents. When describing the formation of this domain, it is possible to find out how the processes dependent on it affect the ion transfer in the galvanostatic mode.
... The boundary value problem (1)- (2), under the condition (3), has a unique solution (see [15]). Numerical treatment of the problem (1), has been considered by many authors, under different condition on the function f, and made a significant contribution. ...
... the rate of convergence Ord we calculate in the usual way Remark 6.1. In a case when the exact solution is unknown we use the double mesh method, see [2,17,18] for details. ...
In this paper we consider two difference schemes for numerical solving of a one--dimensional singularly perturbed boundary value problem. We proved an --uniform convergence for both difference schemes on a Shiskin mesh. Finally, we present four numerical experiments to confirm the theoretical results.
... Singular Perturbation Problems (SPPs) (differential equations with small positive parameter ε multiplying the highest derivatives) arise in many branches of applied mathematics like fluid flow problems involving high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, control theory, electrical networks etc. [4,8]. The presence of small parameter in these problems prevents us from obtaining 212 N. Geetha, A. Tamilselvan and J. Christy Roja parameter uniform numerical solutions. ...
... Therefore we seek a numerical method, which is uniformly convergent with respect to the parameter. We refer to [8,10,18,24,26] for detailed review of SPPs. ...
... Numerical treatment of singular perturbation problems has received a great deal of attention in the past. This type of problems arise in various fields of applied mathematics, mechanics and physics, see [5,6,11,12,15] and also references therein. ...
... For each value of n the algorithm (4.1)-(4.3) has been solved by the periodical factorization procedure (see [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16]), with the initial guess y ...
... Numerical treatment of singular perturbation problems has received a great deal of attention in the past. This type of problems arise in various fields of applied mathematics, mechanics and physics, see [5,6,11,12,15] and also references therein. ...
... For each value of n the algorithm (4.1)-(4.3) has been solved by the periodical factorization procedure (see [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16]), with the initial guess y ...
This work deals with the singularly perturbed periodical boundary value problem for a quasilinear
second-order differential equation. The numerical method is constructed on piecewise uniform Shishkin
type mesh, which gives first-order uniform convergence in the discrete maximum norm. Numerical results
supporting the theory are presented.
... Here, the error is estimated using using double mesh principle [18,19] as ...
This paper deals the design and analysis of a parameter uniform finite difference algorithm for a class of parabolic singularly perturbed convection-diffusion problems. The discrete nearby problem is designed using the backward Euler scheme in time and upwind scheme on mesh in space. We present the convergence analysis of the algorithm, which proves that it is parameter uniform in norm. The numerical experiments support the theoretical estimates and demonstrates the efficiency of mesh with the existing meshes, like Shishkin and Bakhvalov type meshes.
... Consider the following problem[25] ...
... Consider the following problem[25] ...
This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.
... Singularly Perturbed Differential Equations(SPDEs) appear in several branches of applied mathematics. Analytical and numerical treatment of these equations have drawn much attention of many researchers [1,3,2,4,5]. In general, classical numerical methods fail to produce good approximations for these equations. ...
We consider a system of two singularly perturbed Boundary Value Problems (BVPs) of convection-diffusion type with discontinuous source terms and a small positive parameter multiplying the highest derivatives. Then their solutions exhibit boundary layers as well as weak interior layers. A numerical method based on finite element method (Shishkin and Bakhvalov-Shishkin meshes) is presented. We derive an error estimate of order in the energy norm with respect to the perturbation parameter. Numerical experiments are also presented to support our theoretical results.
... Standard numerical methods like finite difference and finite element methods on uniform mesh for solving this type of euations fail to produce good approximations to exact solutions. Many authors [2,3,12,13,14] have developed efficient numerical methods to resolve boundary and interior layers. A good number of articles have been appearing in the past three decades on non-classical methods which cover mostly single second order equation. ...
In this paper, we consider boundary value problem for a weakly coupled system of two singularly perturbed differential equations of con-vection diffusion type with discontinuous source term. In general, solution of this type of problems exhibits interior and boundary layers. A numerical method based on streamline diffusiom finite element and Shishkin meshes is presented. We derive an error estimate of order O(N −2 ln 2 N) in the maximum norm with respect to the perturbation parameters. Numerical experiments are also presented to support our theoritical results.
... Using classical numerical methods such are finite difference schemes and finite element methods, which do not take into account the appearance of the boundary or inner layer, we found the results unacceptable from the standpoint of stability, the value of the error or the cost of calculation. Then, classical difference schemes do not converge to exact solution [23,24]. To give accurate results when the perturbation parameter is small. ...
A uniform finite difference method on a B‐mesh is applied to solve the initial‐boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point‐wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.
... For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see [1], [2], [8] and [9]. In [3], a Dirichlet boundary value problem for a linear parabolic singularly perturbed differential equation is studied and a numerical method comprising of a standard finite difference operator on a fitted piecewise uniform mesh is considered and it is proved to be uniform with respect to the small parameter in the maximum norm. ...
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 of this system exhibit parabolic boundary layers with sublayers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters
... For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see [1], [2], [8] and [9]. In [3], a Dirichlet boundary value problem for a linear parabolic singularly perturbed differential equation is studied and a numerical method comprising of a standard finite difference operator on a fitted piecewise uniform mesh is considered and it is proved to be uniform with respect to the small parameter in the maximum norm. ...
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 classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.
... For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see [1], [2], [8] and [9]. In [3], a Dirichlet boundary value problem for a linear parabolic singularly perturbed differential equation is studied and a numerical method comprising of a standard finite difference operator on a fitted piecewise uniform mesh is considered and it is proved to be uniform with respect to the small parameter in the maximum norm. ...
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 of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.
... In this section, we shall consider two test examples and the computational results obtained by the numerical method based on special mesh. Since the exact solution for the given examples are not known, we use the double mesh principle [11] for calculating the maximum absolute errors. The maximum absolute error is defined by ...
In this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.
... N,ε is calculated by using double mesh principle [8], Table 1 presents the maximum absolute error for different values of λ 1 , λ 2 , ε and N . It can be observed from the results, the error decreases as the mesh size increases. ...
This paper presents a numerical technique for solving nonlinear singu- larly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a se- quence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.
... and theorem 2.2. can be found [6]. In order to obtain zeroth order asymptotic expansion appro- ...
In this paper, a new method is given for solving singularly perturbed convection-diffusion problems. The present method is based on combining the asymptotic expansion method and the variational iteration method (VIM) with an auxiliary parameter. Numerical results show that the present method can provide very accurate numerical solutions not only in the boundary layer, but also away from the layer. Кеуwords-Convection-diffusion problems, boundary layer, varia-tional iteration method, asymptotic expansion, auxiliary parameter..
... The parameter c can take any value in (0,1]. When c tends to zero in a neighbourhood of the boundary r a boundary layer appears. ...
... Since the accuracy of classical finite-difference methods for singularly perturbed problems generally depends not only on the mesh size but also on the value of the small parameter, special numerical methods that are uniformly convergent with respect to the small parameter are required for solving such problems[4]. One approach is to use classical difference schemes on grids specially designed to condense in boundary layers (see[5,6]). ...
***The file is available at http://www.staff.ul.ie/natalia/pubs.html***
... Let u 01 (x) be the reduced problem solution of the BVP (12)- (13) . From the Theorem 5 we get |y 1 ...
A singularly perturbed reaction-diffusion fourth-order ordinary differential equation(ODE) with discontinuous source term is considered. Due to the discontinuity, interior layers also exist. The considered problem is converted into a system of weakly coupled system of two second-order ODEs, one without parameter and another with parameter ? multiplying highest derivatives and suitable boundary conditions. In this paper a computational method for solving this system is presented. A zero-order asymptotic approximation expansion is applied in the second equation. Then, the resulting equation is solved by the numerical method which is constructed. This involves non-overlapping Schwarz method using Shishkin mesh. The computation shows quick convergence and results presented numerically support the theoretical results.
... To illustrate the theory given in the present study and examine the performance of the proposed numerical scheme a set of numerical experiments is carried out. Since exact solution is not known for the considered problem a double mesh principle [6] is used to tabulate the maximum pointwise error and the order of convergence of the proposed method. Maximum pointwise error E N,∆t ε at all the mesh points are evaluated using the formula . ...
In this article we propose an efficient numerical scheme based on a Shishkin mesh for a class of singularly perturbed parabolic convection-diffusion problems with boundary turning point and retarded arguments. The solution of the considered problem exhibit a boundary layer on the left side of the domain. The continuous problem is semidiscretized by means of backward Euler finite difference method in time to get a system of ordinary differential equations at each time level. This system of differential equations is discretized by using the standard upwind finite difference scheme on a nonuniform mesh of Shishkin type. It has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. Numerical experiments supporting the theoretical results are given.
...  çîíàõ ñaeàòèÿ ( ∇·v < 0) è ñâåðõçâóêîâîãî òå÷åíèÿ ( M > 1) êîýôôèöèåíò óðàâíîâåøèâàþùåé âÿçêîñòè ñïåöèàëüíî óìåíüøàåòñÿ âäâîå, ÷òî óëó÷øàåò ðàñ÷åò ñêà÷êîâ. Ôèçè÷åñêàÿ ñîñòàâëÿþùàÿ âÿçêîñòè ïðè ýòîì êîððåêòèðóåòñÿ (óìåíüøàåòñÿ ñ ðîñòîì èñêóññòâåííîé âÿçêîñòè) ïî ìåòîäó ýêñïîíåíöèàëüíîé ïîäãîíêè [11]. ...
Hybrid numerical method for unsteady continua is based on through calculation of continuous medium motion using modification of SUPG FEM, ALE and overlapping grids. Example calculations for supersonic gas flow and elastic plastic body forming are presented.
... We use the double mesh method [65] to compute the experimental rates of convergence. To compute the double mesh error we need each node of the coarse mesh X N s to coincide with a unique node in the fine mesh X 2N s . ...
This article (author's post-print) is available at my website:
http://nadukandi.es/publications.html
In this paper we present an accurate stabilized FIC-FEM formulation for the 1D advection-diffusion-reaction equation in the exponential and propagation regimes using two stabilization parameters. Both the steady-state and transient solutions are considered. The stabilized formulation is based on the standard Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method. The steady-state problem is considered first. The optimal value of the two stabilization parameters ensuring an exact (nodal) FEM solution using uniform meshes of linear 2-noded elements is obtained. In the absence of the absorption term the formulation simplifies to the standard one-parameter Petrov-Galerkin method for the advection-diffusion problem. For the diffusion-reaction case one stabilization parameter is just needed and the diffusion-type stabilization term is identical to that obtained by Felippa and Oñate (2007) using a variational FIC approach. A procedure for computing the stabilization parameters for the transient problem is proposed. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of steady-state and transient 1D advection-diffusion-radiation problems for a the range of physical parameters and boundary conditions.
... As most of these differential equations exclude analytical solution, developing parameter uniform numerical methods to derive numerical approximations to the solution is an important area of research. Fitted operator methods [3] and fitted mesh methods [7] are robust and most popular numerical methods reported in the literature to solve these problems. Of these, fitted mesh methods are preferred because these methods resolve layers exhibited by the solutions of singularly perturbed differential and delay differential equations. ...
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 equations. The former is an initial value problem for a first order singularly perturbed delay partial differential equation, while the latter is a boundary value problem for a second order singularly perturbed delay differential equation. Numerical illustrations are provided.
... Since the accuracy of classical finite-difference methods for singularly perturbed problems generally depends not only on the mesh size but also on the value of the small parameter, special numerical methods that are uniformly convergent with respect to the small parameter are required for solving such problems [4]. One approach is to use classical difference schemes on grids specially designed to condense in boundary layers (see [5,6]). ...
***The file is available at http://www.staff.ul.ie/natalia/pubs.html***
A weighted two-layer difference scheme for the one-dimensional time-dependent convec-tion-diffusion equation is examined. In this scheme, the first spatial derivative is approximated by the central divided difference, It is shown that, on the Shishkin piecewise uniform grid condensing in the -b, oundary layer, the scheme is uniformly convergent for 0' ?: 0,5 with respect to a small parameter in the .sense of the grid L: -norm with an O(N-2In2N + (0' -0.5)'C+ 'C2)convergence rate, where 0' is the scheme parameter, N is the number of grid points, and 'c is the time step.
In this work we consider the singularly perturbed one-dimensional semi-linear reaction-diffusion problem " y (x) = f (x; y); x 2 (0; 1) ; y(0) = 0; y(1) = 0; where f is a nonlinear function. Here the second-order derivative is multiplied by a small positive parameter and consequently, the solution of the problem has boundary layers. A new difference scheme is constructed on a modified Shishkin mesh with O(N) points for this problem. We prove existence and uniqueness of a discrete solution on such a mesh and show that it is accurate to the order of N^{-2} ln^{2} N in the discrete maximum norm. We present numerical results that verify this rate of convergence.
A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε−uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.
The main aim of this article is to present an a priori estimate of an exponential layer and thereby a new mesh (G-mesh) for a class of singularly perturbed problems with discontinuous data. We have also proved that the standard upwind scheme on the proposed mesh is parameter uniformly convergent and is of almost first order. To demonstrate the efficiency of the proposed mesh we carried out a few numerical experiments comparing the performance of the proposed G-mesh with other standard meshes available in the literature.
In this article, we consider singularly perturbed differential-difference equation
with delay and advance arguments (i.e) negative and positive shifts, in which the solution of the problem exhibit layer
or oscillatory behavior depending on certain conditions. But we restrict our studies
only to layer phenomena (boundary or interior layers). We construct the variational
formulation of the above problem with suitable solution space and appropriate energy norm.
We also prove the existence and uniqueness of the solution of the weak
problem by means of the Lax-Milgram lemma. The discretization of the weak problem
along with the layer adapted meshes like Shishkin and Bakhvalov-Shishkin meshes gives
almost second-order convergence of the solution of the problem in the maximum norm. Numerical results are also supported by our theoretical findings.
In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction.
An asymptotic numerical method for third order singularly perturbed delay differential equation of reaction diffusion type is presented in this article. The third order equation is transformed into equivalent weakly coupled system, one of them is first order differential equation with suitable initial condition and another one is second order delay differential equation with suitable boundary condition. Further, the zeroth order asymptotic expansion approximation to the solution is constructed for the weakly coupled system. Using the zeroth order asymptotic expansion an auxiliary problem is constructed. A fitted finite difference scheme and the Euler scheme is suggested to solve the auxiliary and reduced problem respectively, over the Shishkin mesh in the domain [0, 2]. An error estimate is derived by using supremum norm and it is of almost first order convergence.
In this article we present the analysis of generalized S-mesh (denoted by ) based hybrid algorithm for a class of second-order singularly perturbed differential equations with discontinuous convection coefficient. The solution of the problem exhibits interior layer because of the discontinuity in convection coefficient. We have demonstrated the generation of mesh for a domain with interior layer and also estimated that the algorithm is perturbation parameter () uniformly convergent with error asymptotic to where . The numerical experiments of the algorithm on Shishkin mesh, B-type mesh and mesh are carried out for couple of examples, to demonstrate the efficiency of the proposed hybrid algorithm on mesh. It was observed that upwind algorithm on B-type mesh is more efficient than upwind algorithm on Shishkin or mesh and the proposed hybrid algorithm on mesh is of second order and observed to be efficient.
This paper presents a higher order uniformly convergent discretization for second order singularly perturbed parabolic differential equation with delay and advance terms. The retarded terms are approximated by the Taylor series which give rise to a nearby singularly perturbed parabolic differential equation. A hybrid numerical algorithm based on implicit Euler scheme is proposed to discretize the time variable and a combined finite difference scheme made out of modified upwind and central difference schemes to discretize the spatial variable on a piecewise uniform mesh of Shishkin type in space and a uniform mesh in time. The existence and uniqueness of a solution for the proposed hybrid algorithm is analyzed. It is proved that the algorithm is -uniformly convergent of almost second order in space, and first order in time, . A few numerical experiments supporting the results are presented. The efficiency of the proposed hybrid algorithm is demonstrated by comparing upwind and modified upwind algorithm.
Диссертация на соискание ученой степени доктора физико-математических наук по специальности "Механика деформируемого твердого тела"
The book describes the statements and solution methods for a wide range of nonlinear continuum mechanics problems, including contact problems in solid and fluid mechanics, quasistatics and dynamics of elastic-plastic media, the problems of the fracture and damage of solids and the consolidation (cold pressing and hot sintering) of powder composites, mold filling problems, internal and external flows of compressible/incompressible viscous fluids with movable contact, free and phase transition boundaries such as water-ice and melt-crystal interfaces.
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 error estimate in the maximum norm which is independent of the value of the singular perturbation parameter. Numerical results supporting the theory are presented.
We consider the singularly perturbed Riccati equation and present a new finite difference scheme, which is exact when the coeffcientsa, b andc are constant. We show that, under certain restrictions, the solution,u i h , of this difference scheme converges, uniformly in ε, tou(x i), the solution of the differential equation on the uniform meshx i=ih. Numerical results are presented.
ResearchGate has not been able to resolve any references for this publication.