A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

Gelu, Fasika Wondimu; Duressa, Gemechis File

doi:https://doi.org/10.1155/2021/8835595

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2021 | Article ID 8835595 | https://doi.org/10.1155/2021/8835595

A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

Fasika Wondimu Gelu¹and Gemechis File Duressa²

Academic Editor: Victor Kovtunenko

Received06 Sept 2020

Accepted22 Feb 2021

Published13 Mar 2021

Abstract

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be -uniformly convergent of , where and denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter . Some numerical results are carried out to support the theory.

1. Introduction

Let , , and , where and are the left and right sides of the rectangular domain corresponding to and , respectively, and is the base of the domain and given by =. Note that and =. In this work, we consider the following singularly perturbed delay parabolic reaction-diffusion problem subject to the initial condition and boundary conditions

where is a diffusion parameter whose presence makes the problem singularly perturbed and is a delay parameter. For the uniqueness of the solution, we assume that the functions involved in problem (1)–(3) are sufficiently smooth and bounded functions satisfying the following conditions

This type of problem arises in various areas of science and engineering. The problem with delay occurs in the field of many biological models like epidemiology and population ecology. A typical example of delay parabolic partial differential equation is the temporal Wazewska-Czyzewska and Lasota equation which describes the survival of red blood cells in animals. This equation may be extended by incorporating a spatial component. The spatiotemporal delay reaction-diffusion equation of the following form is discussed in [1, 2] where is a bounded domain and . The state variable denotes the number of red blood cells located at at time . The constant time delay denotes the time needed to produce blood cells. The parameter is the death rate of red blood cells, where the parameters and are related to the generation of red blood cells.

The numerical solution of delay partial differential equation depends not only on the solution at a present stage but also at some past stages. Many researchers have proposed different numerical methods to solve singularly perturbed time delay parabolic reaction-diffusion equations; for instance, see [3–8]. Singularly perturbed two-parameter time delay parabolic problems are studied in [9] using hybrid method based on a uniform mesh in time and a layer-adapted Shishkin mesh in space. In [10], the authors studied singularly perturbed differential-difference convection-diffusion equations using a higher order numerical method. But all the mentioned authors considered singularly perturbed time delay parabolic reaction-diffusion equations subject to Dirichlet boundary conditions. The authors in [11] studied singularly perturbed time delay parabolic reaction-diffusion equation subject to mixed-type boundary conditions, yet to the best of our knowledge, no further study has been done. In this work, we apply the extended form of cubic B-spline collocation method for singularly perturbed time delay parabolic reaction-diffusion problem subject to mixed boundary conditions.

B-spline functions have emerged as powerful techniques in the numerical solution of linear and nonlinear partial differential equations. B-spline functions are piecewise polynomial or nonpolynomial functions. B-spline functions were first formulated in 1946. Extended cubic B-spline is an improved version of cubic B-spline, where its basis is constructed in such a way that one free parameter, , is included and the degree of the piecewise polynomial is increased but the continuity of the extended cubic B-splines remains in order three. In [12], the authors proposed an extension of cubic B-spline of degree four with one free parameter, , which is called a shape parameter. This parameter is introduced within the basis function so as to change the shape of the spline. The authors in [13] generalized the extension to degrees five and six. Different researchers used extended cubic B-spline basis function to solve linear and nonlinear ordinary and partial differential equations; for example, see [14–16]. The authors in [17, 18] studied singularly perturbed semilinear differential equation of reaction-diffusion and convection-diffusion type, respectively, using cubic B-spline collocation method on a piecewise Shishkin mesh. Recently, cubic B-spline collocation method has been developed for time-dependent singularly perturbed differential-difference equations; see [19, 20]. An extended cubic B-spline collocation method has been developed for time-dependent singularly perturbed partial differential equations with time lag in [21] and singularly perturbed parabolic differential-difference equation arising in computational neuroscience in [22]. Recently, scholars in [23–26] studied time-dependent singularly perturbed parabolic partial differential equations.

This article is structured as follows. The properties of continuous problem are given in Section 2. In Section 3, the description of numerical scheme and bounds of error for time semidiscretization followed by spatial discretization using extended B-spline collocation method are discussed. Error analysis in spatial direction and the overall error bounds are given in Section 4. Some numerical computations are given in Section 5. The conclusion is given in Section 6.

2. Properties of Continuous Problem

In the study of the numerical aspects of singularly perturbed problems, their analytical aspects play an important role. Setting the value , the reduced problem corresponding to (1)–(3) is

The reduced problem (6) is an initial value problem which will not make use of the two boundary conditions. As a result, the solution of problem in (1)–(3) will have left and right boundary layers. The characteristic curve of the reduced problem in equation (6) is the vertical lines , which implies that boundary layers arising in the solution are of parabolic type. The problem in (1)–(3) satisfies the following continuous maximum principle.

Theorem 1. Assume thatand letbe a sufficiently smooth function defined onsuch that, , , andwhere. Then,, for all.

Proof. The details of the proof are found in [11].

Stability and -uniform bound for problem (1) is established in the following theorem in the sense of the maximum norm which follows from Theorem 1.

Theorem 2. Letbe any function in the domain of problem. Then, we have the bound

Proof. See the details of the proof in [11].

The existence and uniqueness for a solution of (1)–(3) can be established under the assumption that the data are Hlder continuous and also satisfy an appropriate compatibility conditions at the corner points and . The boundary functions are said to satisfy the order compatibility condition at the initial function if

Therefore, the problem in (1)–(3) will have a unique solution which exhibits parabolic boundary layers at and ; see the details in [3, 11]. Now, we establish the classical bounds on the solution and its derivatives.

Theorem 3. Let the coefficients and source function. Assume that the compatibility conditions forare fulfilled. Then, the problem has a unique solution and the derivatives of the solutionsatisfy the boundwhere the constant is independent of .

Proof. The proof of the first part is given in [27]. The bounds on the solution and its derivatives are explained in [11].

The classical bounds in Theorem 3 are not adequate for the proof of -uniform error estimate. Thus, the nonclassical bounds in singular and regular components and its derivatives are established in the following theorem.

Theorem 4. Let the coefficients and source function. Under the smoothness and compatibility conditions, we have the bounds for

Proof. The details of the proof are found in [11, 27].

3. Description of the Numerical Method

In this section, we utilize the implicit Euler method to discretize time derivative and then we introduce a piecewise uniform Shishkin mesh to discretize the space derivative using the extended form of cubic B-spline collocation method for the linear differential equations resulted from the time semidiscretization.

3.1. Time Semidiscretization

We discretize time derivative in (1)–(3) by means of the implicit Euler scheme on a uniform mesh with step length of defined by where denotes the number of mesh elements in temporal direction. Uniform meshes with step size and with and mesh elements are used on the interval and , respectively. The mesh size is chosen in such a way that the delay parameter , where is a positive integer, , . We obtain the following system of ordinary differential equations subject to the boundary conditions where and is the numerical solution at the th time level. For each time step, equations (12)–(13) can be rewritten as subject to the initial and boundary conditions, respectively,

Here, and .

By using the initial condition, we can evaluate the right-hand side as

The local truncation error of an implicit Euler scheme for the temporal semidiscretization is given by . This error measures the contribution of each time step to the global error of the time semidiscretization.

Lemma 5. Ifthen the local error bound in the temporal direction is given by

Proof. For the proof of the lemma, the readers can refer to [28].
The global error is the measure of the contribution of the local error estimate at each time step and is given by .

Lemma 6. Under the hypothesis of Lemma5, the global error estimate atis given by

We conclude that time semidiscretization is first-order uniformly convergent.

3.2. Spatial Discretization

We first construct nonequidistant (layer adapted) Shishkin mesh as follows. We divide the three nonoverlapping subintervals , , and into , , and equidistant subintervals. We define the transition parameter as . Let be the set of mesh points. Now, we define piecewise uniform mesh points as with piecewise uniform mesh spacing , if , and , if . Now, we apply the extended cubic B-spline collocation method to find the approximate solution for problem (14)–(16). Let be the spatial domain with a piecewise uniform mesh spacing . The extended form of cubic B-spline of degree 4, , is defined by [21, 29] where and is a free parameter which is used to change the shape of the B-spline curve and is the degree of extended cubic B-spline. The variation in gives different forms of extended cubic B-spline functions [29]. The extended cubic B-spline function has one free parameter , when the free parameter tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions. For , cubic B-spline and extended cubic B-spline share the same properties such as local support, nonnegativity, partition of unity, and continuity; the parameter controls the tension of the solution curve. The shape of extended cubic B-spline functions forces us to add two fictitious points and to satisfy the boundary conditions. Since B-splines of degree are times continuously differentiable piecewise polynomials that form a basis of the space of splines, let be the space of twice continuously differentiable piecewise extended cubic B-spline on . Since each is also a piecewise cubic with knots at , each . Suppose that span . Since the functions are linearly independent on , is an -dimensional. Let be the B-spline interpolating function for at the nodal points and . Therefore, we seek an approximate solution of the problem (14)–(16) which is given by where are unknown real coefficients to be determined by requiring that satisfies (14)–(16) at collocation points and boundary conditions. The values of extended B-splines and its derivatives at the nodal points can be calculated from (23) and depicted in Table 1.

An approximate solution over typical subinterval can be defined as

Now, substituting the values of for and for as stated in Table 1 in equation (14), we get linear equations in unknowns as where the coefficients are given by

Boundary conditions in (16) at and must be imposed to the system of equations in (25) to obtain the unique solution. Thus, we obtain the approximate solution at two boundary points as follows: where the coefficients are given by . Equations (25) and (27) lead to linear systems with unknowns . Excluding the unknowns and from (27) for and , then (25) becomes solvable system of linear equations in unknowns in matrix form as where the entries of the tridiagonal matrix are given by

The entries of the column vector are given as

Since , it is easily seen that for the matrix is strictly diagonally dominant and hence nonsingular. Since is nonsingular, we can solve the system (28) for . Hence, the extended cubic B-spline collocation method applied to problem (14) has a unique solution .

4. Error Analysis

This section proves the -uniformly convergence of the proposed method in the spatial direction. For this, we use the following lemma.

Lemma 7. The extended cubic B-splinesdefined in (22) satisfy the following inequality

Proof. We know that

Extended cubic B-spline is nonzero at only three nodal points. Thus, at any nodal value , from Table 1 we obtain

From Table 1, for , we have

Similarly, for , we have that

Now, for any point for , we get

Since , thus and this completes the proof.

Let be the unique cubic spline interpolate from an approximate solution of the problem (14)–(16) to the solution which is given by

Lemma 8. Letbe the cubic spline interpolant associated with a solution. If, it follows from the estimate of Hall [30] that the standard cubic spline interpolation error estimate holds, forwhere are constants independent of and .

Lemma 9. [31]. If matrixis strictly diagonally dominant by rows and constant. Then, we have the bound

Theorem 10. Letbe an extended cubic B-spline collocation approximation from the space of extended B-splinesto the solution of (14)–(16) andis the analytical solution to the problem. If, then the parameter-uniform error estimate satisfies the boundwhere is a constant independent of and .

Proof. To estimate the error , it follows immediately from the estimates in Lemma 8 and equation (37) that from (37).

Therefore, rearrangement of the above expression gives

Since the argument depends on whether or , there arises two cases.

Case 1. When , the mesh is uniform with spacing , that is, and gives . From this, we get . In this case, we use a classical analysis to prove convergence. Using the classical bound in Theorem 3, that is, together with equation (42) yields

Since , we obtain the following estimate:

Case 2. If lies in the boundary layer regions, then mesh spacing . Using the bound in the layer regions together with the estimates in equation (42), we have

Since , we obtain the following estimate:

On the other hand, for the subinterval , that is, for the outer region, the mesh spacing is . Using this in equation (42) together with the bounds gives us

Since is very small number, we obtain the following estimate:

Combining the above estimates for both the cases, we have

Therefore, we have that

We know that with the boundary conditions and leads to the linear system . Assume that with the boundary conditions and leads to the linear system . It follows that where and

Since implies that , from (50) we have

It can be seen that for reasonable large , the matrix is strictly diagonally dominant and thus nonsingular. From the estimate in Lemma 9, we get

Combining the bounds in equations (51)–(54), we obtain

Let , where . Now, from (51), we have

Using (53) and (54) in (56), we have the following estimate:

We have where , and are defined in (27). From this, it is a simple task to obtain and . Therefore, we have the following estimation from boundary conditions as

Therefore, the above inequality enables us to estimate as

Using (56) and Lemma 7, we obtain

Using triangle inequality and the results in (49) and (61) gives

Hence, this completes the proof.

Theorem 11. Letbe the extended B-spline collocation approximation to the solutionof the problem (14)–(16). Then, the parameter-uniform error estimate of fully discrete scheme is given bywhere is a constant independent of mesh parameters and .

Proof. The result of Lemma 6 and Theorem 10 proves this theorem.

5. Numerical Results and Discussions

To see the applicability and efficiency of the proposed method, an example is considered from the literature. Computations are done for reasonable value of the free parameter , which gives minimum error. Since the exact solution for the test example is unknown, we use the double mesh principle to calculate absolute errors. For each , we can determine the maximum point-wise errors using the formula as where denote the numerical solution obtained at mesh points whereas denote the numerical solution at mesh points. The uniform rate of convergence is calculated by the formula

Example 1. Consider the following singularly perturbed delay problem [11].

The computed maximum point-wise errors are given in Table 2. From this result, it is clear that the proposed method gives an -uniform convergence.

Computational results in Table 3 confirm that the present method has improved the finite difference method in the literature.

Table 4 displays the comparison of computational results using classical cubic B-spline method for and extended B-spline method for . Both the particular selections of the extension parameter give numerical results, but the later value of gives better result than the former one as well as improved results of finite difference method in the literature. It is clear from Table 4 that extended B-spline performs good result than classical cubic B-spline.

Figure 1 depicts the numerical simulation of solution profile at , and , which indicates parabolic boundary layers at and .

The maximum point-wise error values are plotted using log-log scale as can be seen in Figure 2.

To compare the numerical simulation for both classical B-spline and extended B-spline, solution profile is plotted at , and in Figure 3, which shows the solution profile of classical B-spline.

Figure 4 is the numerical simulation using line graph at , and , which indicates parabolic-type boundary layers at and .

6. Conclusion

This article proposed an implicit Euler method for time derivative with uniform mesh and extended cubic B-spline collocation method for space derivative on Shishkin mesh to solve singularly perturbed delay parabolic reaction-diffusion problem subject to mixed boundary conditions. The proposed method is shown to be accurate of by preserving an -uniform convergence. The suitable choice of the extension parameter minimizes the error. To see the effect of delay and extension parameter in the boundary regions, graphs are plotted on the solution profile for .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

Acknowledgments

This work was partially supported by Jimma University, College of Natural Sciences, Ethiopia.

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Copyright © 2021 Fasika Wondimu Gelu and Gemechis File Duressa. 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.

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