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A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition
- Mustafa Kudu^{1}Email author
https://doi.org/10.1186/s13662-018-1620-0
© The Author(s) 2018
Received: 20 December 2017
Accepted: 26 April 2018
Published: 8 May 2018
Abstract
We consider a uniform finite difference method on a Bakhvalov mesh to solve a quasilinear first order parameterized singularly perturbed problem with integral boundary conditions. Uniform first order error estimates in the discrete maximum norm have been established. Numerical results that demonstrate the sharpness of our theoretical analysis are presented.
Keywords
- Singular perturbation
- Finite difference scheme
- Uniform convergence
- Parameterized problem
- Bakhvalov mesh
- Integral boundary condition
MSC
- 65L11
- 65L12
- 65L20
1 Introduction
Singularly perturbed differential equations are typically characterized by a small parameter ε multiplying some or all of the highest order terms in the differential equations as normally boundary layers occur in their solutions. These equations play an important role in today’s advanced scientific computations. Many mathematical models starting from fluid dynamics to the problems in mathematical biology are modeled by singularly perturbed problems. Typical examples include high Reynold’s number flow in the fluid dynamics, heat transport problem, etc. For more details on singular perturbation, one can refer to the books [1–4] and the references therein. The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such a problem undergoes rapid changes within very thin layers near the boundary or inside the problem domain [2, 3]. It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter is small. Therefore, it is important to develop suitable numerical methods to the problems, whose accuracies do not depend on the parameter value, i.e., methods that are convergent ε-uniformly. For the various approaches on the numerical solutions of differential equations with steep, continuous solutions, we may refer to the monographs [1, 4, 5].
Parameterized boundary value problems have been considered by many researchers for many years. Such problems arise in physical chemistry and physics, describing the exothermic and isothermal chemical reactions, the steady-state temperature distributions, the oscillation of a mass attached by two springs leading to a differential equation with a parameter [5, 11, 12]. An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in [11–14](see also references therein). In [11, 12, 14, 15], the authors have also considered some approximating aspects of this kind of problems. But in the above-mentioned papers, algorithms are only concerned with the regular cases (i.e., when the boundary layers are absent). In recent years, many researchers presented the numerical methods for the singular perturbation cases of parameterized problems. Uniform convergent finite-difference schemes for solving parameterized singularly perturbed two-point boundary value problems have been considered in [8, 10, 15–22] (see also references therein). In [8, 10, 16, 17, 19, 20] authors used the boundary layer technique for solving an analogous problem. A methodology based on the homotopy analysis technique to approximate the analytic solution was investigated in [15, 21, 22].
Also it is well known that nonlinear differential equations with integral boundary conditions have been used in description of many phenomena in the applied sciences, e.g., heat conduction, chemical engineering, underground water flow, and so on [23–25]. Therefore, boundary value problems involving integral boundary conditions have been studied by many authors [6, 7, 26–31] (see also references therein). Some approximating aspects of this kind of problems in the regular cases, i.e., in the absence of layers, were investigated in [7, 26, 32]. In recent years, many researchers have considered the singularly perturbed cases for these problems. In [9] authors developed a finite difference scheme on Shishkin mesh for a problem with integral boundary conditions and proved that the method is nearly first order convergent except for a logarithmic factor. A hybrid scheme, which is second order convergent on Shishkin mesh, was discussed in [30] (see also [20, 33]). For the numerical methods concerning second order singularly perturbed differential equations with integral boundary conditions, one can see, e.g., [28, 29, 31].
In this paper, as far as we know, the numerical solution of the singularly perturbed boundary value problem containing both control parameter and integral condition is first being considered. For the numerical solution of such problems, a specific approach is required to construct the appropriate difference scheme and examining the error analysis. The scheme is constructed by the method of integral identities with the use of appropriate quadrature rules with the remainder terms in integral form. The aim here is to construct an ε-uniformly numerical method which gives ε-uniformly convergent numerical approximations to solve problem (1)–(3). For this, we use a finite difference scheme on a Bakhvalov mesh which is dense in the initial layer. The Bakhvalov mesh is dependent on ε and mesh points have to be condensed in a neighborhood of \(t=0 \) in order to resolve the initial layer. In the Bakhvalov mesh, basically half of the mesh points are concentrated in \(O(\varepsilon \vert \ln \varepsilon \vert )\) neighborhood of the point \(t=0\) and the remaining half forms a uniform mesh on the rest of \([ 0, T ] \) (see [2, 4, 10, 30, 32, 34, 35]). We show that the proposed scheme is uniformly convergent in the discrete maximum norm accuracy of \(O(N^{-1})\) on Bakhvalov meshes. Note that, in [10], the first order convergent difference scheme in Bakhvalov type mesh under the first type boundary conditions for equation (1.1) was presented. Also, in the above-mentioned work [9] that includes integral boundary condition, while conditions (2.1) and (4.8) are generally provided for sufficiently small values of ε, as the integral boundary condition of our work is more general, and the convergence is uniform for both small and moderate values of perturbation parameter ε.
The paper is organized as follows. In Sect. 2, the difference scheme constructed on the non-uniform mesh for the numerical solution (1)–(3) is presented and graded mesh is introduced. The uniform convergence of the difference scheme is investigated and error of the difference scheme is evaluated in Sect. 3. Finally, in Sect. 4 some numerical results are presented to confirm the theoretical analysis. The paper ends with conclusions.
Henceforth, C and c denote the generic positive constants independent of both the perturbation parameter ε and the mesh parameter N. Such a subscripted constant is fixed. We also will use \(\Vert g\Vert _{\infty }=\max_{0\leq t\leq T} \vert g(t)\vert \) for any \(g\in C[0,T]\).
2 The finite difference scheme
To construct the numerical method and for convergence analysis, we need the asymptotic estimates for the differential solution \(\lbrace u(t), \lambda \rbrace \).
Lemma 2.1
Proof
One can prove this result following the method given in [9], Lemma 2.1, and in [10], Lemma 2.1. □
3 Uniform convergence and error estimates
Lemma 3.1
Proof
Finally, applying the maximum principle for the difference operator \(L^{N}z_{i}^{N}:=\varepsilon z_{\bar{t},i}^{N}+a_{i}z_{i}^{N}\), \(1\leq i\leq N\), to Eq. (25) immediately leads to (24). □
Lemma 3.2
Proof
Theorem 3.1
Proof
This follows immediately by combining the previous lemmas. □
4 Algorithm and numerical results
The results of the numerical experiment are presented in this section, which confirms the theoretical bounds established in the previous section.
Errors \(e_{u}^{ \varepsilon,N}\) computed ε-uniform errors \(e_{u}^{N}\) and convergence rates \(p_{u}^{ \varepsilon,N}\) on \(\omega_{N}\)
ε | N = 64 | N = 128 | N = 256 | N = 512 | N = 1024 |
---|---|---|---|---|---|
2^{0} | 0.00464342 | 0.00249526 | 0.00128984 | 0.00065893 | 0.00033245 |
0.896 | 0.952 | 0.969 | 0.987 | ||
2^{−4} | 0.00692191 | 0.00372224 | 0.00192142 | 0.00065782 | 0.00033143 |
0.895 | 0.954 | 0.974 | 0.989 | ||
2^{−8} | 0.00460314 | 0.00247876 | 0.00128398 | 0.00065412 | 0.00032979 |
0.893 | 0.949 | 0.973 | 0.988 | ||
2^{−12} | 0.00460181 | 0.00247290 | 0.00128183 | 0.00065348 | 0.00032947 |
0.896 | 0.948 | 0.972 | 0.988 | ||
2^{−16} | 0.00460557 | 0.00247492 | 0.00128110 | 0.00065175 | 0.00032837 |
0.896 | 0.950 | 0.975 | 0.989 | ||
\(e_{u}^{\varepsilon,N}\) | 0.00464342 | 0.00372224 | 0.00128984 | 0.00065893 | 0.00033245 |
\(p_{u}^{\varepsilon,N}\) | 0.895 | 0.954 | 0.969 | 0.987 |
Errors \(e_{ \lambda }^{ \varepsilon,N}\) computed ε-uniform errors \(e_{ \lambda }^{N}\) and convergence rates \(p_{ \lambda }^{ \varepsilon,N}\) on \(\omega_{N}\)
ε | N = 64 | N = 128 | N = 256 | N = 512 | N = 1024 |
---|---|---|---|---|---|
2^{0} | 0.00749478 | 0.00402193 | 0.00204612 | 0. 00102448 | 0.00051242 |
0.898 | 0.975 | 0.998 | 1.000 | ||
2^{−4} | 0.00742715 | 0.00399393 | 0.00203893 | 0.00102159 | 0.00051114 |
0.895 | 0.970 | 0.997 | 0.999 | ||
2^{−8} | 0.00723501 | 0.00391767 | 0.00202090 | 0.00101748 | 0.00051051 |
0.885 | 0.955 | 0.990 | 0.995 | ||
2^{−12} | 0.0070768 | 0.00385598 | 0.00200291 | 0.00102333 | 0.00051019 |
0.876 | 0.945 | 0.983 | 0.990 | ||
2^{−16} | 0.00692575 | 0.00379203 | 0.00198065 | 0.00100764 | 0.00050909 |
0.869 | 0.937 | 0.975 | 0.985 | ||
\(e_{\lambda }^{\varepsilon,N}\) | 0.00749478 | 0.00402193 | 0.00204612 | 0. 00102448 | 0.00313856 |
\(p_{\lambda }^{\varepsilon,N}\) | 0.898 | 0.975 | 0.998 | 1.000 |
5 Conclusion
We have considered the numerical approximations of a class of quasilinear singularly perturbed first order parameterized differential problems with integral boundary conditions, which serves as the model for many scientific applications. For the numerical solution of this problem, we proposed a uniform convergent finite difference scheme on the graded Bakhvalov mesh. The ideas presented here can be easily applied for solving more complicated initial value problems for parameterized singularly perturbed equations with integral boundary conditions, and the technique presented in the paper can also be applied to high-dimensional systems.
Declarations
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
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Authors’ Affiliations
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