Table of Contents Author Guidelines Submit a Manuscript
Modelling and Simulation in Engineering
Volume 2019, Article ID 5157145, 9 pages
https://doi.org/10.1155/2019/5157145
Research Article

The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning Approach

Department of Informatics and Telematics, Harokopio University, Athens, Greece

Correspondence should be addressed to Anastasia-Dimitra Lipitakis; moc.liamg@ikatipilda

Received 30 August 2018; Accepted 4 December 2018; Published 2 January 2019

Academic Editor: Michele Calì

Copyright © 2019 Anastasia-Dimitra Lipitakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Critical comments on the complexity of computational systems and the basic singularly perturbed (SP) concepts are given. A class of several complex SP nonlinear elliptic equations arising in various branches of science, technology, and engineering is presented. A classification of complex SP nonlinear PDEs with characteristic boundary value problems is described. A modified explicit preconditioned conjugate gradient method based on explicit inverse preconditioners is presented. The numerical solution of a characteristic 3D SP nonlinear parabolic model is analytically given and numerical results for several model problems are presented demonstrating both applicability and efficiency of the new computational methods.

1. Introduction

1.1. Complexity of Computational Systems

A wide spectrum of complex computational problems can be found in computer science and information management, as well as in different disciplinary fields, such as applied mathematics, engineering, business, finance, medicine, computational biology, social networks, transportation, telecommunications, education, government, and healthcare.

In recent times, complex systems of all types, like web-based systems, cloud infrastructures and big data centres, social networks, peer-to-peer, mobile and wireless systems, cyber-physical systems, the Internet of things, and real-time and embedded systems, have increasingly distributed, and dynamic system architectures providing high flexibility also increase the complexity of managing end-to-end application performance.

1.2. Singular Perturbation Problems

Singular perturbation (SP) problem in computer mathematics is a computational problem containing small parameters that cannot be approximated by setting the parameter values to zero; that is, the problem solution cannot be uniformly approximated by asymptotic expansions. The term singular perturbation was introduced in the 1940s by Wasow [1], and SP problems are generally characterized by dynamics operating on multiple scales.

Basic SP methods include the method of matched asymptotic expansions, the Poincaré–Lindsted method, the method of multiple scales and periodic averaging, and the WKB approximation for spatial problems [25]. Numerical techniques for solving nonlinear elliptic SP problems have been developed in several branches of science, technology, and engineering.

1.3. Topics of Various SP Problems

Topics of various problems are raised in several mathematical models in mathematical physics, optimization, and economic, where nonlinear PDEs of elliptic type arise almost in every scientific field. Solutions of such equations occur in diverse fields of mathematics, such as functional analysis, algebraic topology, differential geometry, variational calculus, and potential theory, while SP problems can occur in various areas of applied mathematics and engineering, i.e., fluid dynamics, fluid mechanics, quantum mechanics, magnetohydrodynamics, elasticity, chemical reactor theory, and reaction-diffusion processes. The topics of boundary-layer theory and approximation of solutions of various computational problems, where SP parameters (large or small) exist in complex solutions, and other critical problems require their analyses of asymptotic methods.

The asymptotic analysis for differential operators refers to operative perturbations over (very) narrow regions across dependent variables with very rapid chances, and the small parameters multiply the highest derivatives. These are usually referred very difficult for numerical solving, such as boundary layers in fluid mechanics, skin layers in electrical applications, shock layers in fluid and solid mechanics, transition points in quantum mechanics, Stokes lines, and surfaces. The terminology boundary layer has been introduced by Prandl [6], and since then, several 2D nonlinear SP problems remain unsolved [7].

During the last decades, several approximate methods for analysing nonlinear SP problems have been developed including the boundary-layer method, the method of averaging, the method of matched asymptotic expansion, and multiple scales, while a class of SP nonlinear boundary value problems for ODEs, the reaction-diffusion equations [8], the shock layer solution of nonlinear equations for SP problems, and problems of atmospheric physics [9] have been also developed.

The numerical solution of SP elliptic PDEs plays an important role in computational fluid dynamics for the simulation of flow problems [10, 11], but the derivation of computational discretization schemes appropriate for all types of linear and nonlinear SP elliptic equations is still an open problem. Note that the SP problems can be classified in numerical and asymptotic problems. Numerical analysis provides quantitative information about the given problem, while asymptotic analysis provides elements of quantitative behavior of classes of problems by giving semiquantitative information about any particular member of this class of problems.

In this research work, a modified version of the explicit adaptable preconditioned conjugate gradient method based on a new explicit inverse preconditioner is presented. The application of the new adaptable modified explicit preconditioned method, based on the new explicit preconditioner, leads to faster numerical solution of singular perturbed parabolic differential equations, especially in the case of three space variable problems. Note that proposed methods based on EPCG and inverse preconditioners by choosing the appropriate preconditioner lead to more accurate numerical solution methods than these with no preconditioner for solving distinct complex inverse problems. The usage of such powerful explicit inverse preconditioner, in the case that the original coefficient matrix of the given discretized partial differential equation is a nonsingular (m × n) nonsymmetric matrix of irregular structure, for solving complex computational problems, yields efficient solutions of 3D singular perturbation time-dependent differential equations.

2. Classification of SP Nonlinear PDEs

Many physical phenomena in science and engineering can be modelled as boundary value problems associated with various types of PDEs or systems of PDEs. During the solution of these models, the important qualities can be retained by omitting negligible quantities involving (very) small parameters. A class of several complex SP nonlinear elliptic equations arising in various branches of science, technology, and engineering has been recently presented [12]. For the application point of view such models include the following physical phenomena: nonlinear waves arising in gas dynamics, water waves, flood waves in rivers, transport of pollutants, chemical reactions, traffic flow, chromatography, and various biological and ecological systems.

These classes of applications contain nonlinear elliptic/parabolic equations with singular perturbation, such as the following characteristic equations:(i)Reference [13] considered the problem arising in the study of reaction-diffusion systems with chemical or biological motivation:where is the bounded domain in Rn with smooth boundary , denotes the unit outer normal at , and .(ii)The standing waves of the nonlinear Schrodinger equation are considered by [14]where is subcritical and is the smooth bounded potential.(iii)A basic model is the system, according to [15], which models the densities of a chemical activator and an inhibitor and is used to describe experiments of regeneration of hydra:where , with the constraints .(iv)Nerve impulse application concerns the following nonlinear elliptic singular perturbation equation:on the smooth bounded domain . The perturbation parameter is positive and small.

Note that considerable research work in special topics of SP nonlinear differential equations has been recently presented [1626].

Conclusively, it is stated that a wide variety of important problems in science and engineering have been formulated in terms of nonlinear elliptic SP PDEs, which model nonlinear waves, arising in gas dynamics, water waves, chemical reactions, transport of pollutants, flood waves in rivers, chromatography, traffic flow, and a wide range of biological and ecological systems.

2.1. SP Nonlinear Parabolic/Elliptic Differential Equations

SP parabolic first BV problems have been discussed by several researchers [2729]. Various convection diffusion problems governed by second-order semilinear parabolic/elliptic equations with small parameters multiplying the second-order space derivatives subject to mixed types prescribed parabolic boundaries of the domains of the problems trying to determine how close the obtained approximate solution is to the actual solution of the problem [30, 31].

The differential equationswhen , which is sufficiently small, then the corresponding BV problem is a singular perturbation problem which arises in electrochemistry [32].

The area of SP is a field of increasing interest to applied mathematicians and computer mathematics scientists. During the last decades, considerable contributions on the topic and its applications have been made by several researchers [1, 3241]. Note that considerable research work in special topics of SP nonlinear differential equations has been recently presented [1621].

Conclusively, it is stated that a wide variety of important problems in science and engineering has been formulated in terms of nonlinear elliptic SP PDEs, which model nonlinear waves, arising in gas dynamics, water waves, chemical reactions, transport of pollutants, flood waves in rivers, chromatography, traffic flow, and a wide range of biological and ecological systems.

2.2. Solving Complex SP Nonlinear Problems: Recent Advances on Special Topics

Singular perturbation theory concerns the study of problem featuring parameters for which the solutions of problems at a limiting value of parameters are different in character from the limit of solutions of the general problem, i.e., the limit is singular. In contrast, for regular perturbation problems, the solutions of the general problem converge to solutions of the limit problem as the parameters approaches the limit values. Singular perturbation problems occur in a wide spectrum contexts, areas of applied mathematics, science and engineering, such as fluid mechanics (boundary-layer problems), elasticity (edge effort in shells), and quantum mechanics [19, 4254].

Perturbation theory is a collection of methods for obtaining approximate solutions to problems involving small parameters ε. These methods are very powerful; thus sometimes it is actually advisable to introduce a parameter ε temporarily into a difficult problem having no small parameter, and then finally to set ε = 1 to recover the original problem. The approach of perturbation theory is to decompose a difficult problem into a (infinite) number of relatively easy ones. The perturbation theory is most useful when the first few steps reveal the important features of the solution and the remaining ones give small corrections. Perturbation solutions can be classified into two types. A basic feature of regular perturbation problems is that the exact solution for small but nonzero ε smoothly approaches the unperturbed solution as ε ⟶ 0. A singular perturbation problem is defined as the one whose solution for ε = 0 is fundamentally different in character from the “neighbouring” solutions obtained in the limit ε ⟶ 0 [53, 5564].

Singular perturbation theory concerns the study of problems featuring parameters for which the solutions of problems at a limiting value of parameters are different in character from the limit of solutions of the general problem; that is, the limit is singular. In contrast, for regular perturbation problems, the solutions of the general problem converge to solutions of the limit problem as the parameters approach the limit values.

The computational singular perturbation (CSP) method [65] is a commonly used method for finding approximations of slow manifolds in systems of ordinary differential equations (ODEs) with multiple time scales. The validity of the CSP method was established for fast-slow systems with a small parameter [42, 44, 46, 5254, 61, 66, 67].

3. Numerical Solution of a 3D SP Nonlinear Parabolic Model

Let us consider the general SP quadratically nonlinear elliptic Dirichlet problem:where is an open and bounded set in Euclidean n-space ; is the boundary of are smooth functions, while the parameter is a very small, positive number [68].

Let us consider a class of singular perturbation (SP) nonlinear parabolic partial differential equation (PDE) in three space dimensions of the form:where is the operator (); is the real SP parameter; and are the real parameters; and is the time, subject to the boundary conditionsand the initial conditions

The nonlinear PDE can be solved, after the FD/FE discretization by a linearized inner-outer iterative scheme, where the outer iteration is carried out by a Newton iteration of the formwhere is the discretized differential operator. The backwards difference process can be used for the discretization of time.

The resulting nonlinear system is

The inner iterative scheme can be performed by using an explicit preconditioned conjugate gradient method with termination criterion:while the outer iteration termination criterion was chosen as

Note that m and p are the semibandwidths of the coefficient matrix; l1 and l2 are the width parameters in semibandwidths m and p, respectively; r1 and r2 are the fill-in parameters in semibandwidths m and p, respectively; and δli are the so-called “retention” parameters; that is, the number of diagonals retained in approximate inverse matrix [69]. For the numerical experimentation, the values of the fill-in parameters were chosen as r1 = r2 = 2 and the width parameters were chosen as l1 = l2 = 3.

4. Modified Explicit Preconditioned Conjugate Gradient Method

Let us assume that the coefficient matrix in Section 1.3 is in general a large nonsingular real unsymmetric matrix of semibandwiths m and p, respectively, retaining nonzero elements in widths l1 and l2 retaining r1 and r2 fill-in terms, respectively. The coefficient matrix A is considered to be a banded matrix of irregular structure. Let us also consider that there is a class of approximate inverses of A, with M the exact inverse of A [69].

Then, the following subclasses of approximate inverses, depending on the accuracy, storage, and computational work requirements, can be derived as it is shown in (14):where of subclass I is a banded form of the exact inverse retaining elements along each row and column, respectively, while its elements are equal to the corresponding elements of the exact inverse. The term of subclass II is a banded form of M, retaining only elements along each row and column during the computational procedure of the approximate inverse, and under certain hypotheses, it can be considered as a good approximation of the original inverse, while the entries of the approximate inverse in subclass III have been retained after computing and are less accurate than the corresponding entries of . Finally, in subclass IV, the elements of the approximate inverse can be computed [7072].

The SP nonlinear parabolic system can be solved by using an inner-outer iterative scheme. In this section, a modified explicit preconditioned conjugate gradient method is presented.

4.1. Adaptive Preconditioned Conjugate Gradient Method Using the Explicit Approximate Preconditioner

The PCG method can solve the problem , where R is the sparse, nonsingular QR factor, while the preconditioned CGLS method can solve the following equations: , , and . In order to compute efficiently the solution of the linear system , a modified explicit preconditioned conjugate gradient (mEPCG) method is applied in the format of Algorithm 1.

Algorithm 1: mEPCG ().

This algorithm requires the additional work that is needed to solve the linear systemonce per iteration. Therefore, the preconditioner () should be chosen such that the process can be done easily and efficiently.

The preconditioner () = G that results in a minimal memory use. The storage requirement was the vectors r, x, y, and p and the upper triangular matrix G, in the data implementation. The convergence rate of preconditioned CG is independent of the order of equations, and the matrix vector products are orthogonal and independent. The preconditioned CG method in not self-correcting and the numerical errors accumulate every iteration. Therefore, to minimize the numerical errors in the PCG, double precision variables were used at the cost of memory usage. An explicit PCG method of second order can be alternatively used in conjunction with the explicit approximate inverse for solving complex computational problems with the appropriate selection of the iterative parameters [71].

5. Numerical Results

Let us consider a characteristic singular perturbation (SP) nonlinear parabolic PDE in three space dimensions in a predetermined region. The FE discretization of this problem leads to the solution of a nonlinear system, where the coefficient matrix is a nonsingular, large sparse nonsymmetric (n × n) matrix of irregular structure [73]. In order to demonstrate both capabilities and efficiency of the proposed methods, a model problem has been selected and corresponding numerical results are indicatively presented.

Two hybrid inner-outer iterative methods have been considered, i.e., the Newton-mEPCG method, with inner iteration the modified explicit preconditioned conjugate gradient (mEPCG) method, and the Newton-EPBICG-STAB method, with inner iteration the explicit preconditioned biconjugate conjugate gradient (EPBICG-STAB) method.

The convergence behavior of Newton (outer iteration) and mEPCG/EPBICG-STAB (inner iteration) is shown in Tables 14 and Figures 1 and 2 for a model problem of n = 3375, m = 26, and , with several values of SP parameter εS, selected values of time step Δt, and several values of retention parameters δl.

Table 1: The convergence behavior of Newton-mEPCG method for solving the given nonlinear problem (time step Δt = 0.010).
Table 2: The convergence behavior of Newton-EPBICG-STAB method for solving the given nonlinear problem (time step Δt = 0.010).
Table 3: The convergence behavior of Newton−mEPCG method for solving the given nonlinear problem (time step Δt = 0.005).
Table 4: The convergence behavior of Newton-EPBICG-STAB method for solving the given nonlinear problem (time step Δt = 0.005).
Figure 1: Inner iterations (methods mEPCG (a) and EPBICG-STAB (b)) of the hybrid schemes Newton-mEPCG and Newton-EPBICG-STAB with Δt = 0.01.
Figure 2: Inner iterations (methods mEPCG (a) and EPBICG-STAB (b)) of the hybrid schemes Newton-mEPCG and Newton-EPBICG-STAB with Δt = 0.005.

The explicit preconditioned biconjugate conjugate gradient-STAB (EPBICG-STAB) method and variants have been presented in related research works [73, 74]. It should be noted that the explicit preconditioning methods and explicit approximate inverse combined with appropriate preconditioners can be applied in a wider application fields in applied and computer mathematics, while explicit approximate inverses can be combined with multigrid methods and other related hybrid computational techniques.

6. Conclusions

Basic elements of complexity and singularly perturbed concepts have been discussed. A classification of complex SP nonlinear elliptic equations arising in various branches of science and engineering in the form of a two-decade survey has been presented.

A modified explicit preconditioned conjugate gradient method based on explicit inverse preconditioners is introduced for solving complex nonlinear parabolic problems. The numerical solution of a characteristic SP nonlinear initial/boundary value is presented, and numerical results demonstrating both applicability and effectiveness of the derived new methods are given. Future research work is planned towards the implementation of the new computational methods in parallel computer environments.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

References

  1. W. Wasow, On boundary layer problems in the theory of ODE’s, Mathematics Research Center, University of Wisconsin-Madison, Madison, WI, USA, 1981.
  2. E. Hinch, Perturbation Methods, Cabridge University Press, Cambridge, UK, 1991.
  3. M. Holmes, Introduction to Perturbation Methods, Springer, Berlin, Germany, 1995.
  4. A. Tikhonov, “Systems of differential equations containing a small parameter multiplying the derivative,” Matematicheskii Sbornik, vol. 31, no. 73, pp. 575–586, 1952, in Russian. View at Google Scholar
  5. F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, Berlin, Germany, 2005.
  6. L. Prandtl, Uber Flussigkeits-Bewegung bei Kleiner Reibung in: Verhandlungen III, Int. Math. Kongresses, Tuebner, Leipzig, Germany, 1905.
  7. H. Roos, M. Stynes, and L. Tobiska, “Numerical methods for singularly perturbed differential equations,” International Journal of Nonlinear Science, vol. 17, no. 3, pp. 195–214, 1996. View at Google Scholar
  8. M. Jia-qi and Z. Jiang, “The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations,” Applied Mathematics and Mechanics, vol. 24, no. 5, pp. 527–531, 2003. View at Publisher · View at Google Scholar
  9. M. Jia-Qi, W. Hui, and L. Wan-Tao, “Singularly perturbed solution of a sea-air oscillator model for the ENSO,” Chinese Physics, vol. 15, no. 7, pp. 1450–1453, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. M. K. Kadalbajoo and K. C. Patidar, “A survey of numerical techniques for solving singularly perturbed ordinary differential equations,” Applied Mathematics and Computation, vol. 130, no. 2-3, pp. 457–510, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. M. K. Kadalbajoo and K. C. Patidar, “Singularly perturbed problems in partial differential equations: a survey,” Applied Mathematics and Computation, vol. 134, no. 2-3, pp. 371–429, 2003. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Kumar and A. Kumar Singh, “Singular perturbation problems in nonlinear elliptic partial differential equations: a survey,” International Journal of Nonlinear Science, vol. 17, no. 3, pp. 195–214, 2014. View at Google Scholar
  13. A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems, vol. 240, Progresses in Mathematics, Berlin Germany, 2006.
  14. A. Ambrosetti, A. Malchiodi, and S. Secchi, “Multiplicity results for some nonlinear schrödinger equations with potentials,” Archive for Rational Mechanics and Analysis, vol. 159, no. 3, pp. 253–271, 2001. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Gierer and H. Meinhardt, “A theory of biological pattern formation,” Kybernetik, vol. 12, no. 1, pp. 30–39, 1972. View at Publisher · View at Google Scholar · View at Scopus
  16. F. Geng and M. G. Cui, “A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4211–4215, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. A. Khan and P. Khandelwal, “Non-polynomial sextic spline solution of singularly perturbed boundary-value problems,” International Journal of Computer Mathematics, vol. 91, no. 5, pp. 1122–1135, 2013. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Pandit and M. Kumar, “Haar wavelet approach for numerical solution of two parameters singularly perturbed boundary value problems,” Applied Mathematics and Information Sciences, vol. 8, no. 6, pp. 2965–2974, 2014. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Rashidinia, M. Ghasemi, and Z. Mahmoodi, “Spline approach to the solution of a singularly-perturbed boundary-value problems,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 72–78, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. F. A. Shah, R. Abass, and J. Iqbal, “Numerical solution of singularly perturbed problems using Haar wavelet collocation method,” Applied and Interdisciplinary Mathematics, vol. 3, no. 1, 2016. View at Publisher · View at Google Scholar
  21. A.-M. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45–57, 2002. View at Publisher · View at Google Scholar · View at Scopus
  22. A. Blatov, “On the Galerkin finite-element method for elliptic quasilinear singularly perturbed boundary value problems I,” Differential Equations, vol. 28, no. 7, pp. 931–940, 1992. View at Google Scholar
  23. P. P. Chakravarthy, S. D. Kumar, and R. N. Rao, “An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays,” Ain Shams Engineering Journal, vol. 8, no. 4, pp. 663–671, 2017. View at Publisher · View at Google Scholar · View at Scopus
  24. M. Jia-Qi, “The singularly perturbed boundary value problems for semilinear elliptic equations of higher order,” Approximation Theory and its Application, vol. 16, no. 3, pp. 100–105, 2000. View at Google Scholar
  25. J. Miller, E. O’Riordan, and G. Shiskin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
  26. J. Rashidinia, R. Mohammadi, and R. Jalilian, “The numerical solution of non-linear singular boundary value problems arising in physiology,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 360–367, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. D. Aronson, “Linear parabolic differential equations containing a small parameter,” Indiana University Mathematics Journal, vol. 5, no. 6, pp. 1003–1014, 1956. View at Publisher · View at Google Scholar
  28. L. Bobisud, “Second-order linear parabolic equations with a small parameter,” Archive for Rational Mechanics and Analysis, vol. 27, pp. 385–397, 1968. View at Publisher · View at Google Scholar · View at Scopus
  29. N. Ramanujam and U. N. Srivastava, “Singular perturbation problems for systems of partial differential equations of elliptic type,” Journal of Mathematical Analysis and Applications, vol. 71, no. 1, pp. 18–35, 1979. View at Publisher · View at Google Scholar · View at Scopus
  30. C. Lam and R. B. Simpson, “Centered differencing and the box scheme for diffusion convection problems,” Journal of Computational Physics, vol. 22, no. 4, pp. 4486–4500, 1976. View at Publisher · View at Google Scholar · View at Scopus
  31. A. Van Harten, “Singularly perturbed nonlinear second order elliptic boundary value problems,” University of Utrecht, Utrecht, Netherlands, 1975, Ph.D thesis. View at Google Scholar
  32. W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin, Germany, 1970.
  33. G. D. Birkhoff, “Quantum mechanics and asymptotic series,” Bulletin of the American Mathematical Society, vol. 39, no. 10, pp. 681–701, 1933. View at Publisher · View at Google Scholar · View at Scopus
  34. L. Bobisud, “On the single first-order partial differential equation with a small parameter,” SIAM Review, vol. 8, no. 4, pp. 479–493, 1966. View at Publisher · View at Google Scholar
  35. I. Boglaev, “Domain decomposition in boundary layers for singularly perturbed problems,” Applied Numerical Mathematics, vol. 34, no. 2-3, pp. 145–166, 2000. View at Publisher · View at Google Scholar · View at Scopus
  36. F. O. Ilicasu and D. H. Schultz, “High-order finite-difference techniques for linear singular perturbation boundary value problems,” Computers and Mathematics with Applications, vol. 47, no. 2-3, pp. 391–417, 2004. View at Publisher · View at Google Scholar
  37. M. Inc and A. Akgül, “Numerical solution of seventh-order boundary value problems by a novel method,” Abstract and Applied Analysis, vol. 2014, Article ID 745287, 9 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  38. J. Kisynski, “Sur les equations hyperboliques avec petit parameter,” Colloquium Mathematicum, vol. 10, no. 2, pp. 331–343, 1963. View at Publisher · View at Google Scholar
  39. R. O’Malley, “Topics in singular perturbations,” Advances in Mathematics, vol. 2, no. 4, pp. 365–470, 1968. View at Publisher · View at Google Scholar · View at Scopus
  40. J. A. Smoller, “Singular perturbations of Cauchy's problem,” Communications on Pure and Applied Mathematics, vol. 18, no. 4, pp. 665–677, 1965. View at Publisher · View at Google Scholar · View at Scopus
  41. Y. L. Wang, H. Yu, F. G. Tan, and S. Qu, “Solving a class of singularly perturbed partial differential equation by using the perturbation method and reproducing kernel method,” Abstract and Applied Analysis, vol. 2014, Article ID 615840, 5 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  42. H. K. Mishra and S. Saini, “Various numerical methods for singularly perturbed boundary value problems,” American Journal of Applied Mathematics and Statistics, vol. 2, no. 3, pp. 129–142, 2014. View at Google Scholar
  43. R. N. Rao and P. P. Chakravarthy, “A fourth order finite difference method for singularly perturbed differential-difference equations,” American Journal of Computational and Applied Mathematics, vol. 1, no. 1, pp. 5–10, 2012. View at Publisher · View at Google Scholar
  44. M. Turkyilmazoglu, “Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems,” Applied Mathematical Modelling, vol. 35, no. 8, pp. 3879–3886, 2011. View at Publisher · View at Google Scholar · View at Scopus
  45. W. Zhang, “Numerical solutions of linear and nonlinear singular perturbation problems,” University of Wisconsin Milwaukee, Milwaukee, Wisconsin, 2006, Ph.D. thesis. View at Google Scholar
  46. W. Zhang, D. Schultz, and T. Lin, “Sharp bounds for semilinear singular perturbation problems,” International Journal of Applied Sciences and Computing, vol. 18, no. 1, 2012. View at Google Scholar
  47. J. Vigo-aguiar and S. Natesan, “A parallel boundary value technique for singularly perturbed two-point boundary value problems,” Journal of Supercomputing, vol. 27, no. 2, pp. 195–206, 2004. View at Publisher · View at Google Scholar · View at Scopus
  48. N. C. John, Singular Perturbation in the Physical Sciences, AMS University of California, Berkeley, CA, USA, 2015.
  49. R. S. Johnson, Singular Perturbation Theory, Springer, Berlin, Germany, 2004.
  50. J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, Berlin, Germany, 1981.
  51. H. Steinrück, Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances, Springer-Verlag, Berlin, Germany, 2010.
  52. D. A. Tursunov, “The asymptotic solution of the bisingular Robin problem,” Siberian Electronic Mathematical Reports, vol. 14, pp. 10–21, 2017, in Russian. View at Google Scholar
  53. B. J. Debusschere, Y. M. Marzouk, H. N. Najm, B. Rhoads, D. A. Goussis, and M. Valorani, “Computational singular perturbation with non-parametric tabulation of slow manifolds for time integration of stiff chemical kinetics,” Combustion Theory and Modelling, vol. 16, no. 1, pp. 173–198, 2012. View at Publisher · View at Google Scholar · View at Scopus
  54. A. Zagaris, H. G. Kaper, and T. J. Kaper, “Analysis of the computational singular perturbation reduction method for chemical kinetics,” Journal of Nonlinear Science, vol. 14, no. 1, pp. 59–91, 2004. View at Publisher · View at Google Scholar · View at Scopus
  55. Z. Bartoszewski and A. Baranowska, “Solving boundary value problems for second order singularly perturbed delay differential equations by Ε-approximate fixed-point method,” Mathematical Modelling and Analysis, vol. 20, no. 3, pp. 369–381, 2015. View at Publisher · View at Google Scholar · View at Scopus
  56. S. Cengizci, “An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations,” International Journal of Differential Equations, vol. 2017, Article ID 7269450, 8 pages, 2017. View at Publisher · View at Google Scholar · View at Scopus
  57. P. Chakravarthy, S. Kumar, and R. Rao, “Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension,” International Journal of Applied and Computational Mathematics, vol. 3, no. 3, pp. 1703–1717, 2017. View at Publisher · View at Google Scholar
  58. R. E. L. DeVille, A. Harkin, M. Holzer, K. Josić, and T. J. Kaper, “Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations,” Physica D: Nonlinear Phenomena, vol. 237, no. 8, pp. 1029–1052, 2008. View at Publisher · View at Google Scholar · View at Scopus
  59. N. Dogan, V. S. Erturk, and O. Akin, “Numerical treatment of singularly perturbed two-point boundary value problems by using differential transformation method,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 579431, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  60. A. S. Karth and P. M. Kumar, “A numerical technique for solving nonlinear singularly perturbed delay differential equations,” Mathematical Modelling and Analysis, vol. 23, no. 1, pp. 64–78, 2018. View at Publisher · View at Google Scholar · View at Scopus
  61. H. G. Kaper, T. J. Kaper, and A. Zagaris, “Geometry of the computational singular perturbation method,” Mathematical Modelling of Natural Phenomena, vol. 10, no. 3, pp. 16–30, 2015. View at Publisher · View at Google Scholar · View at Scopus
  62. F. Mohamed and M. F. Abd Karim, “Solving singular perturbation problem of second order ordinary differential equation using the method of matched asymptotic expansion (MMAE),” in Proceedings of AIP Conference Proceedings, Selangor, Malaysia, October 2015.
  63. R. Narasimhan, “Singularly perturbed delay differential equations and numerical methods,” in Differential Equations and Numerical Analysis: Tiruchirappalli, V. Sigamani, J. J. H. Miller, R. Narasimhan, P. Mathiazhagan, and F. Victor, Eds., pp. 41–62, Springer India, Berlin, Germany, 2016. View at Google Scholar
  64. R. Rao and P. Chakravarthy, “A numerical patching technique for singularly perturbed nonlinear differential-difference equations with a negative shift,” Journal Applied Mathematics, vol. 2, no. 2, pp. 11–20, 2012. View at Publisher · View at Google Scholar
  65. D. Lam and S. Goussis, Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, USA, 1988.
  66. K. Alymkulov and D. A. Tursunov, “A method for constructing asymptotic expansions of bisingularly perturbed problems,” Russian Mathematics, vol. 60, no. 12, pp. 1–8, 2016. View at Publisher · View at Google Scholar · View at Scopus
  67. K. Keldibay Alymkulov and D. Dilmurat Adbillajanovich Tursunov, Perturbed Differential Equations with Singular Points, Perturbation Theory Dimo I, Uzunov, Intech Open, London, UK, 2017.
  68. A. J. DeSanti, “Singularly perturbed quadratically nonlinear Dirichlet problems,” Transactions of the American Mathematical Society, vol. 298, no. 2, p. 733, 1986. View at Publisher · View at Google Scholar
  69. E. A. Lipitakis, “Numerical solution of three-dimensional boundary-value problems by generalized approximate inverse matrix techniques,” International Journal of Computer Mathematics, vol. 31, no. 1-2, pp. 69–86, 2007. View at Publisher · View at Google Scholar · View at Scopus
  70. K. M. Giannoutakis, “A study of advanced computational methods of high performance: Preconditioned methods and inverse matrix techniques,” Democritus University of Thrace, Komotini, Greece, 2008, Doctoral thesis. View at Google Scholar
  71. A. D. Lipitakis, “A generalized exact inverse solver based on adaptive algorithmic methodologies for solving complex computational problems in three space dimensions,” Applied Mathematics, vol. 7, no. 11, pp. 1225–1240, 2016. View at Publisher · View at Google Scholar
  72. E. A. Lipitakis and D. J. Evans, “Explicit semi-direct methods based on approximate inverse matrix techniques for solving boundary-value problems on parallel processors,” Mathematics and Computers in Simulation, vol. 29, no. 1, pp. 1–17, 1987. View at Publisher · View at Google Scholar · View at Scopus
  73. A. G. Gravvanis, “Solving initial value problems by explicit domain decomposition approximate inverses,” in Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2000), Barcelona, Spain, September 2000.
  74. H. A. van der Vorst, “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, vol. 13, no. 2, pp. 631–644, 1992. View at Publisher · View at Google Scholar