Abstract
In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are 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, , and mesh size, The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and -uniformly convergent for where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.
1. Introduction
Singular perturbation problems (SPPs) are differential equations with a small positive parameter multiplying the highest derivative term. These problems arise in several branches of applied mathematics, including fluid dynamics, quantum mechanics, elasticity, chemical reactor theory, and gas porous electrodes theory. Examples of SPPs include the Navier-Stokes equation of fluid flow at high Reynolds number, the equation governing flow in porous media, the drift-diffusion equation of semiconductor devices, physics and mathematical models of liquid crystal material, and the convection-diffusion and reaction-diffusion equations to mention but a few [1, 2].
Such equations typically exhibit solutions with layers, which cause severe computational difficulties for standard numerical methods. Consequently, a variety of different numerical strategies have been devoted [1–3] and the references are therein to the construction and analysis of accurate numerical methods for SPPs. Recently, authors [4–8] have considered SPPs for second-/third-order ordinary differential equations (ODEs) with discontinuous source term and/or discontinuous convection coefficient. The novel aspect of the problem under consideration is that we take a source term in the differential equation which has a jump discontinuity at one or more points in the interior of the domain. This gives rise to an interior layer in the exact solution of the problem, in addition to the boundary layer at the outflow boundary point.
A single discontinuity is assumed to occur at a point . The solution of this problem has a boundary layer at and interior layers with different widths at It is convenient to introduce the notation and and to denote the jump at in any function with .
Recently, Chandru and Shanthi [9] presented a fitted mesh method to solve singularly perturbed robin-type boundary value problems with discontinuous source term. But still, there is a room to increase the accuracy and develop its convergence for the developed scheme because the treatment of singular perturbation problem is not trivial distributions and the solution pended on perturbation parameter and mesh size [10]. Due to this, numerical treatment of singularly perturbed boundary value problems needs improvement.
Different authors in [11–23] have applied different families of spline method for second- and higher order singular and regular problems. As far as the researchers’ knowledge is concerned, numerical solution of fitted nonpolynomial cubic spline method for the problem under consideration is first being considered. Hence, in the present paper, motivated by the works of [9], we developed a fitted nonpolynomial cubic spline method for singularly perturbed robin-type boundary value problems with discontinuous source term. Therefore, it is important to develop more accurate and convergent numerical method for solving singularly perturbed robin boundary value problems. Thus, the purpose of this study is to develop stable, convergent, and more accurate numerical method for solving singularly perturbed robin boundary value problems with discontinuous source term.
In this article, we consider the following class of problems: subject to the boundary conditions, where and are a singular perturbation parameter; and are sufficiently smooth functions on and , respectively; and and . It is assumed that is a sufficiently smooth function in ; the left and right limits of and their derivatives are assumed to exist at . The function is assumed to have simple discontinuity at . Hence, the solution of equations (1) and (2) does not necessarily have a continuous second derivative at the point ; that is, does not belong to the class of functions . Hence, the class of functions, where belongs to it, is taken as . Further, the SPPs in equations (1) and (2) have a unique solution [24].
Boundary value problems of the type in equations (1) and (2) are model confinement of a plasma column by reaction pressure and geophysical fluid dynamics [25].
2. Properties of Continuous Solution
The following lemmas are necessary for the existence and uniqueness of the solution and for the problem to be well-posed.
Lemma 1 (continues minimum principle). Suppose that the function satisfies and and , and .
Proof. For the proof, refer to [26].
Lemma 2 (stability result). Consider the boundary value problem of equations (1) and (2) subject to the condition If then
Proof. For the proof, refer to [26].
Lemma 3. For each integer , satisfying , the solution of equations (1) and (2) satisfy the bounds
Proof. To establish the parameter robust properties of the numerical methods involved in this paper, the following decomposition of into smooth and singular components is required.
The smooth component is defined as the solution of
Note that , where and
As in [27], we can obtain the following bounds on the derivatives of for :
Note also that . Define the singular components of the decomposition as follows. Find such that
We can further decompose as where is the boundary layer function satisfying and is the interior layer function satisfying
Lemma 4. Let be the solution of . Then, for
Proof. For the proof, refer to [26].
3. Formulation of the Numerical Method
Consider the following singularly perturbed robin-type boundary value problems with discontinuous source terms of the form
Rearranging equation (13), we obtain where and .
Under boundary conditions,
Consider a uniform mesh with interval in which , where.
For each segment , the nonpolynomial cubic spline has the following form: where are unknown coefficients and arbitrary parameter which will be used to increase the accuracy of the method of Bawa [28].
To determine the unknown coefficients in equation (17) in terms of , first, we define
By some algebra, the coefficients in equation ((17)) becomes where
Using the continuity condition of the first derivative at , , we have
Reducing indices of equation (19) by one and substituting into equation (20), we have
Multiplying both side of equation (21) by , we obtain where
As
Substituting in equation (14), we get
Substituting equation (24) into equation (22), we obtain and the following approximations for the first derivatives of by Bawa [28]: where is the parameter used to raise the accuracy of the method.
Substituting equations (26) into equation (25) and rearranging, we get
From the theory of singular perturbation, it is known that the solution of equation (14) and equation (15) is of the form (O’Malley [29]) where is the solution of the reduced problem (obtained by setting of equation (14) which is given by
Expanding and about point “” up to the first term using Taylor’s series, the discretized form of equation (28) gives
Therefore, where .
To handle the effect of the perturbation parameter in equation (27), artificial viscosity () is multiplied on the term containing the perturbation parameter as
Multiplying equation (32) by and taking limit as we get the fitting factor which is a required fitting factor in the left boundary layer. From equation (32), we get where
Then, we get the three term recurrence relation of the form where
To treat the boundary condition, we used forward finite difference formula for and backward difference formula for , respectively, for the first derivative term.
That is, for , from equation (2), we have
Similarly, for , from equation (2), we have
Therefore, the problem in equation (1) with given boundary condition in equation (2) can be solved using the scheme equations (36), (38), and (39) which form an system of algebraic equations.
4. Truncation Error
From equation (26), we have
From equation (24) and equation (22),
Putting the exact solution in equation (41), we have where
Subtract equation (41) from equation (42) and putting , we get
Using equation (40), we have
Let
Substituting equation (46) into equation (45), we have where
is a local truncation error associated with the scheme developed in equation (36). Thus, for different values of in the scheme of equation (36), the following different orders are indicated: (1)For , and for any value of , the scheme of equation (36) gives the second-order method(2)For , and , from equation (36), the fourth-order method is derived
5. Stability and Convergence Analysis
Writing the tridiagonal system in equation (36) in matrix vector form, we have where is a tridiagonal matrix of order . Multiplying both sides of equation (34) by (-1), we get
is the column vector, where
is an approximation solution; for , , , and .
Now, considering the above system with exact solution , we have where
is a local truncation error.
From equation (49) and equation (52), we obtain
This gives the error equation where
Let where are positive constants.
Let be the element of the matrix ; then, for ,
Thus, for sufficiently small , we have
Hence, the matrix is irreducible (Varga [30]).
From equation (36), for sufficiently small , and we get .
Hence, is diagonally dominant. Under this condition, Thomas algorithm method is stable (Kadalbajoo and Reddy [31]).
Let be the sum of the elements of the row of the matrix ; then, we have
Let , then and
For sufficiently small , is monotone (Young [32]).
Hence, exist and . From equation (63), we have
For sufficiently small , we have where
Let be the element of , and we define
Since , then from the theory of matrices, we have
Hence,
From equations (54)–(68) and (48), we get where which is independent of mesh size . This establishes that the present method is fourth-order convergent for .
6. Numerical Examples and Results
To validate the applicability of the method, two model problems are considered for numerical experimentation for different values of the perturbation parameter and mesh points. The numerical results are tabulated in terms of maximum absolute errors, numerical rate of convergence, and uniform errors (see Tables 1–4) and compared with the results of the previously developed numerical methods existing in the literature (Tables 2 and 4). Further, behavior of the numerical solution without and with fitted operator (Figures 1 and 2), point-wise absolute errors (Figure 3), and the -uniform convergence of the method are shown by the log-log plot of the -uniform error (Figure 4).
Having (the approx. solution obtained via the present method) for different values of and since the exact solution is not available, the maximum errors (denoted by ) are evaluated using the formula given by the double-mesh principle:
Further, we will tabulate the errors
The numerical rates of convergence are computed using the formula and the numerical rate of “-uniform convergence” is computed using
Example 1. where
Example 2. where
7. Discussion and Conclusion
This study introduces uniformly convergent nonpolynomial cubic spline method based on exponential fitted operator for solving singularly perturbed second-order ODEs of Robin-type BVPs with discontinuous source term. Due to discontinuity in the source term, there is an interior layer occurring. To fit the interior and boundary layer, a suitable fitted operator finite difference method on uniform mesh is constructed. The behavior of the continuous solution of the problem is studied and shown that it satisfies the continuous stability estimate and the derivatives of the solution are also bounded. The numerical scheme is developed on uniform mesh. The Robin-type BVPs are treated using numerical finite difference techniques, and the results are compared accordingly. The stability of the developed scheme is established, and its uniform convergence is proved. Unlike other fitted operator finite difference methods constructed in standard ways, the method that we presented in this paper is fairly simple to construct.
Data Availability
All data generated or analyzed during this study are included.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
HGD proposed the main idea of this paper. HGD and GFD prepared the manuscript and performed all the steps of the proofs in this research. Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
Acknowledgments
The authors wish to express their thanks to Jimma University, College of Natural Sciences, for the technical support and the authors of literatures for the provided scientific aspects and idea for this work.