1. Introduction
Solving singular perturbation problems (SPPs) is very tedious due to the presence of layer phenomena in the solution domain. They have a wide range of applications in the field of applied mathematics and engineering due to their complex boundary layer phenomena. Their application is prevalent in areas such as aerodynamics to examine boundary layer behavior; in fluid mechanics to model flow transitions and turbulence; in elasticity to study material deformation under stress; and also in the fields of optimal control theory, chemical processes, biological modeling, population dynamics, and ecology [
1,
2,
3,
4,
5,
6,
7,
8,
9], demonstrating their wide-ranging relevance. Singularly perturbed delay differential equations (SPDDEs) have a wide range of applications in diverse fields [
10,
11,
12,
13,
14].
The presence of discontinuity in the source term further complicates solving SPPs as the solution tends to behave unevenly or non-smoothly at the discontinuity point. This leads to the formation of an interior layer at the region in the domain where the discontinuity occurs. To tackle such difficulties in estimating the solution of SPPs, classical numerical schemes, such as fitted mesh methods [
15], are modified by employing highly refined meshes (i.e., Shishkin mesh or Bakhvalov mesh) to provide numerical approximations by preserving the monotonicity of the original problem. Many specialized techniques have been introduced by researchers across the globe in order to overcome difficulties arising in solving SPPs.
Abagero et al. [
16] employed a fitted nonstandard numerical method to solve SPPs with Robin-type boundary conditions and discontinuous source terms. Sahoo & Gupta [
17] employed a first-order upwind scheme on a Shishkin mesh to solve convection–diffusion SPPs with discontinuous convective and source terms. They further employed the Richardson extrapolation scheme to enhance the order of convergence. Singh et al. [
18] devised spline-based numerical techniques, such as the Crank–Nicolson scheme and trigonometric B-spline basis function, to solve two-parameter SPPs with discontinuity in the convection coefficient and source term. The Richardson extrapolation scheme was implemented by the authors to enhance the accuracy in the spatial direction. Chawla et al. [
19] employed a backward difference scheme on Shishkin and Bakhvalov meshes to solve first-order singularly perturbed differential equations (SPDEs) with discontinuous source terms. Cen et al. [
20] developed a quadratic B-spline collocation method on a Shishkin-type mesh to solve a semilinear reaction–diffusion SPP with a discontinuous source term. Soundararajan et al. [
21] employed the backward Euler Method for time discretization and the streamline-diffusion Finite Element Method (FEM) on a Shishkin mesh to solve 1D parabolic SPPs with a discontinuous source term. Ajay Singh Rathore & Vembu Shanthi [
22] proposed an exponentially fitted mesh method to solve a singularly perturbed Fredholm integro-differential equation with a discontinuous source term.
In [
23], Ismail & Elmekkawy utilized the Restrictive Padé Approximation to solve a singularly perturbed first-order hyperbolic Partial Differential Equation, where a small perturbation is multiplied by the term containing the time derivative. The present paper differs from [
23] in the treatment of the numerical analysis as well as in the problem formulation. The problems considered in this article involve solving a system of two singularly perturbed time-dependent initial value problems. These problems contain distinct perturbation parameters that multiply the spatial term involving the highest-order derivative. Additionally, they contain a spatial delay, and a discontinuity occurs at
in the source terms. Initial and interior layers occur in the solution domain due to the presence of perturbation parameters, discontinuous source terms, and delay terms. This divergent problem structure and the complex behavior of the solution necessitate developing a classical layer-resolving finite difference scheme (FDS) [
24] using a highly refined Shishkin mesh. This method is able to precisely capture the layer behavior of solutions and achieves first-order convergence in both time and space. Numerical illustrations confirm the effectiveness of the proposed method and clearly exhibit its first-order convergence. The subsequent section outlines the organizing framework of the paper. The notation and terminology are discussed in
Section 2.
Section 3 outlines the problem statement. The solution components and layer functions are discussed in
Section 4. In
Section 5, the bounds on the solution and its components are derived. Discrete solution components and error estimates are discussed in
Section 6. A numerical scheme is constructed for example problems in
Section 7.
2. Notation and Terminology
This section defines the key notation and terminology used throughout the paper to ensure clarity and consistency.
singular perturbation parameter
independent variables
solution of the continuous problem (
1)
smooth component of the solution
singular component of the solution
solution of the discrete problem (
29)
discrete smooth component of the solution
discrete singular component of the solution
continuous barrier function
discrete barrier function
backward difference operators
forward difference operators
is the point at which discontinuity occurs in the domain
The jump at is denoted by a function where
For any
n-vector
, the norm
For any scalar-valued function
is a closed set in
,
For any vector-valued function
y,
For any function
, the vector discrete maximum norm defined on the Shishkin mesh
Domain defined for continuous case
Domain defined for discrete case
Temporal domain
: , :
Spatial domain
: , : .
The presence of discontinuity in domain necessitates the consideration of two different cases. In Case 1, the discontinuity occurs in the interval , and, in Case 2, the discontinuity occurs in the interval .
Case 1: Here,
,
, and
Case 2: Here,
and
,
3. The Problem Statement
In this section, we consider the following system of two singularly perturbed time-dependent initial value problems with spatial delay and discontinuity source terms.
with Robin initial conditions in space variable and Dirichlet initial condition in time variable
For all ,,, with , , and . The functions are assumed to be in , and can operate on functions in the domain .
Furthermore, for all
, the components
of
and
, respectively, satisfy the following conditions:
and, for any positive number
,
The function is discontinuous at due to a finite jump. So, the solution does not possess a continuous first-order derivative at .
Case 1:
Problem (
1) can be reformulated as
where
.
For problems (6) and (7), the solution components and exhibit an initial layer at and interior layers at , , and , each of width . Furthermore, the solution component exhibits additional layers of width at , , , and .
Case 2:
Problem (
1) can be reformulated as
For problems (8) and (9), the solution components and exhibit an initial layer at and interior layers at and , each of width . Furthermore, the solution component exhibits additional layers of width at , , and .
The existence of the solution of problem (
1) is discussed in the following theorem by adopting similar procedure to that in [
24].
Theorem 1. The system of Equations (1)–(3) has a solution , where Proof. The proof is by construction.
Case 1: Let
be the particular solutions of
Consider the function
where
, and
are solutions of
and
, where
are any particular vector constants.
can be derived in the following way so as to have
.
The product between vectors is the Schur product of vectors.
Case 2: Let
be the particular solutions of
Consider the function
where
, and
are solutions of
and
, where
are any particular vector constants.
can be determined as follows to ensure that
.
A similar approach confirms the existence of a solution for . When exists and is continuous at as is both well defined and continuous at this point. The proof of the theorem is complete. □
Uniqueness and Stability Analysis
Lemma 1. Let and satisfy (4) and (5). Consider as a function defined in the domain of such that it satisfies on . Then, on implies that on Proof. For
, let us consider
. If
, there is nothing to prove. Let us assume that
. Suppose
and then
which contradicts our assumption, and, for
which also contradicts our assumption. Therefore,
. Also,
and
.
Thus, for
, it follows that
which contradicts our assumption.
For
, it follows that
which contradicts our assumption. If
, then
and there exists a neighborhood
such that
for all
. If
for any point
, and then
. If
for all
, it can be noted that
is an increasing function in
; as a consequence,
cannot attain its minimum at
, which contradicts our assumption. This implies that
and for all
, and then
. The proof of the lemma is complete. □
Lemma 2. Let conditions (4) (5) hold for and . Consider a function in the domain of and then Proof. Consider the barrier functions
where
If
, then
and
Lemma 1 implies that
on
. Then,
The proof of the lemma is complete. □
4. Solution Components and Layer Functions
The continuous solution
of (
1) is decomposed into
.
represents the solution of the following equation.
Case 1:
Case 2:
represents the solution of the following equation.
Case 2:
with
,
Functions
, known as layer functions, are introduced and associated with the solution
as follows
where
5. Solution Bounds
Theorem 2. Let conditions (4) (5) hold for and . Let represent the solution of problems (1), (2), and (3). Then, for each , there exists a constant C such that, for all for Case 1 and for all Proof. The bounds on the solutions are derived by employing steps and techniques analogous to those of Lemma 3.1 in [
25]. □
The succeeding lemmas deal with finding bounds on and their derivatives.
Lemma 3. Let conditions (4) (5) hold for and . Then, and its derivatives satisfy the following bounds for all for Case 1 and for Case 2 Proof. The bounds on the
and its derivatives are derived by employing steps and techniques analogous to those of Lemma 5.1 in [
26]. □
Lemma 4. Let conditions (4) (5) hold for and . Then, and its derivatives satisfy the following bounds. Case 1:
For all and Case 2:
For all and Proof. The bounds on the
and its derivatives are derived by employing steps and techniques analogous to those of Lemma 5.2 in [
26]. □
5.1. Sharper Estimates
Definition 1. For each , for Case 1 and for Case 2, the unique point in is defined by Definition 2. For all such that , there exist points that are uniquely defined and satisfy the inequalities presented below for Case 1 and Case 2.
Lemma 5. Let conditions (4) (5) hold for and . Then, for each there exists a decomposition for which the following estimates hold for each .
Case 1:
For all and Case 2:
For all and Proof. The required estimates are derived by employing steps and techniques analogous to those of Lemma 10.3 in [
26]. □
5.2. Domain Discretization
Temporal domain is meticulously discretized into a uniform mesh composed of M mesh intervals on . Now, to capture the intricate layer behavior of the solutions, the spatial domain is discretized into a piecewise-uniform Shishkin mesh composed of N mesh intervals.
For Case 1, the interval
is partitioned into 12 sub-intervals, outlined as follows
The transition parameters
are defined as
The sub-intervals and are discretized using a uniform mesh consisting of mesh points. A uniform mesh consisting of mesh points is deployed on each of the sub-intervals , and .
For Case 2, the interval
is partitioned into 9 distinct sub-intervals, outlined as follows
The transition parameters
are defined as
The sub-intervals and are discretized using a uniform mesh consisting of mesh points. A uniform mesh consisting of mesh points is deployed on each of the sub-intervals , and
Also, , , .
For problems (
1)–(
3), a classical layer-resolving finite difference scheme is developed using the aforementioned discretization.
The problem represented in (
29) is reformulated for Case 1 and Case 2 as follows:
Case 2:
where
for
Lemma 6. Let conditions (4) (5) hold for and . Let us consider to be any mesh function such that on , on in Case 1 and on , on in Case 2. Then, it implies that on Proof. For , let and consider the case where the lemma does not hold. Then, From the stipulated hypotheses, it is simple to establish that , and
If
, then
which is a contradiction. For
, which leads to a contradiction.
Case 1: For
which contradicts our assumption. For
it follows that
which contradicts our assumption.
Case 2: For
which contradicts our assumption. For
it follows that
which contradicts our assumption.
If
for Case 1, then
Then, which contradicts our assumption.
If
for Case 2, then
Then, which contradicts our assumption. This implies that The proof of the lemma is complete. □
Lemma 7. Let conditions (4) (5) hold for and . Let us consider to be any mesh function, and then, for Case 1 for , and, for Case 2, for
Proof. It is evident that
on
and also
on
on
, and, for
, it follows that
Hence, from the result of Lemma 6, it follows that on for Case 1. By applying analogous procedure used in Case 1, the required result on for Case 2 is obtained. The proof of the lemma is complete. □
6. Discrete Solution Components and Error Estimates
The discrete solution
of (
29) is decomposed into
, where
and
are discrete smooth and discrete singular components, respectively.
The problems represented in (35) and (
36) are reformulated for Case 1 and Case 2 as follows:
The error at each point
is provided by
. For
in Case 1 and
in Case 2, the local truncation errors
can be expressed as follows:
Theorem 3. Let conditions (4) (5) hold for and . Let represent the solution of (10) and let represent the solution of (35). Then, for Case 1,and, for Case 2, Proof. From the expressions (
49) and (55), and using Lemma 3, for
it follows that
The proof of the theorem is complete. □
Theorem 4. Let conditions (4) (5) hold for and . Let represent the solution of (16) and let represent the solution of (36). Then, for Case 1,and, for Case 2, Proof. The procedure outlined in Theorem 10.5 of [
26] is employed in this theorem as it leads to a similar result. □
From the above results, it is concluded that, for Case 1,
or
or
, and, for Case 2,
or
,
Now, at the point
or
or
for Case 1 and
or
for Case 2, it follows that
Theorem 5. Let represent the solution of (1), (2), and (3) and represent the solution of (29) and (30). Thus, for sufficiently large N, Proof. For Case 1, consider the two mesh functions
and, for Case 2,
where
C is suitably chosen as sufficiently large constant. Hence, for Case 1,
or
or
, and, for Case 2,
or
and, for Case 1,
or
or
, and, for Case 2,
or
For Case 1,
or
or
, and, for Case 2,
or
Thus, for sufficiently large
N,
The proof of the theorem is complete. □
7. Numerical Illustrations
To elucidate the proposed numerical scheme, numerical examples are introduced in this section to address two distinct cases resulting from the presence of discontinuous source term in the domain.
The following are some of the notations present in this section:
-
: maximum point-wise two-mesh differences;
-
: —uniform maximum point-wise two-mesh differences;
-
: —uniform order of local convergence;
-
: —uniform order of convergence;
-
: —uniform error constant.
M and N are the numbers of mesh points in temporal and spatial domains, respectively. Two-mesh algorithm [
27] is applied to compute
along with
.
The
—uniform maximum point-wise two-mesh differences value is obtained as follows:
The
—uniform order of convergence is obtained as follows:
The
—uniform error constant is obtained as follows:
Example 1. Consider the following system of two singularly perturbed time-dependent delay initial value problems,where Case 1: presents in the interval Case 2: presents in the interval and, with Robin initial conditions in space and Dirichlet initial conditions in time The perturbation parameters
are expressed in terms of
as
and
, with
taking different values as outlined in
Table 1,
Table 2,
Table 3 and
Table 4. The values
and
are calculated using the procedure outlined above, and the results are presented in
Table 1 for Case 1 and
Table 3 for Case 2. The values
and
are obtained from
Table 1 for Case 1 and the values
and
are obtained from
Table 3 for Case 2. From both tables, it is evident that, as N increases, the values of
increase, whereas the values of
decrease. The computation times in seconds for obtaining the maximum point-wise errors for Case 1 are presented in
Table 2, while those for Case 2 are provided in
Table 4.
Figure 1 represents the solutions of Case 1 of Example 1 for
, and
. The solutions
and
exhibit rapid transitions in the initial and interior regions of the domain due to the influence of perturbation parameters, discontinuous source terms, and delay terms. The solution components
and
form initial layers at
near boundary region of the domain, while the interior layers emerge at
due to the delay terms, as well as at
and
as a result of discontinuities in the source terms.
Figure 2 represents the solutions of Case 2 of Example 1 for
, and
. The solution components
and
form initial layers at
near boundary region of the domain, while interior layers emerge at
due to the delay terms, and at
as a result of discontinuities in the source terms.
Example 2. Consider the following system of two singularly perturbed time-dependent delay initial value problems, where
Case 1: presents in the interval Case 2: presents in the interval and, with Robin initial conditions in space and Dirichlet initial conditions in time, The perturbation parameters
are expressed in terms of
as
and
, with
taking different values as outlined in
Table 5,
Table 6,
Table 7 and
Table 8. The values
and
are calculated using the procedure outlined above, and the results are presented in
Table 5 for Case 1 and
Table 7 for Case 2. The values
and
are obtained from
Table 5 for Case 1 and the values
and
are obtained from
Table 7 for Case 2. From both tables, it is evident that, as N increases, the values of
increase, whereas the values of
decrease. The computation times in seconds for obtaining the maximum point-wise errors for Case 1 are presented in
Table 6, while those for Case 2 are provided in
Table 8.
Figure 3 represents the solutions of Case 1 of Example 2 for
, and
. The solutions
and
exhibit rapid transitions in the initial and interior regions of the domain due to the influence of perturbation parameters, discontinuous source terms, and delay terms. The solution components
and
form initial layers at
near the boundary region of the domain, while the interior layers emerge at
due to the delay terms, as well as at
and
as a result of discontinuities in the source terms.
Figure 4 represents the solutions of Case 2 of Example 2 for
, and
. The solution components
and
form initial layers at
near the boundary region of the domain, while the interior layers emerge at
due to the delay terms, and at
as a result of discontinuities in the source terms.
Figure 5 and
Figure 6 present log-log plots of the maximum point-wise errors for Example 1 (Case 1 and Case 2) from
Table 1 and
Table 3, and for Example 2 (Case 1 and Case 2) from
Table 5 and
Table 7, respectively.
8. Conclusions
In this article, a classical layer-resolving numerical scheme was constructed for solving a system of two singularly perturbed time-dependent delay initial value problems with source terms that exhibit a jump at a specific point due to discontinuity and Robin initial conditions. This scheme incorporates a specially crafted piecewise uniform mesh to capture the layer behavior of the solutions in the initial and interior regions of the solution domain owing to the existence of delay and discontinuous source terms. The numerical experiments validated the computational outcomes, demonstrating strong concordance with the theoretical analysis. Additionally, the proposed method achieved first-order convergence. Moreover, the results clearly demonstrate that, as the number of mesh points increases, the order of convergence improves, while the error constant decreases. This demonstrates the robustness and efficiency of the proposed scheme. The results indicate that the proposed method remains stable and accurate across varying perturbation parameters, making it an efficient tool to solve complex equations. Future research will focus on extending the work to semilinear and nonlinear problems, with an emphasis on developing novel numerical methods to enhance computational efficiency.
Author Contributions
Conceptualization, J.P.M. and R.B.K.; methodology, R.B.K. and J.P.M.; software, J.P.M. and R.B.K.; validation, J.P.M., G.E.C., and S.L.P.; formal analysis, R.B.K. and J.P.M.; investigation, R.B.K., J.P.M., and G.E.C.; resources, J.P.M. and R.B.K.; data curation, R.B.K. and J.P.M.; writing—original draft preparation, R.B.K. and J.P.M.; writing—review and editing, R.B.K., J.P.M., G.E.C., and S.L.P.; visualization, R.B.K.; supervision, G.E.C. and J.P.M.; project administration, J.P.M. and G.E.C.; funding acquisition, G.E.C. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
SPPs | Singularly Perturbed Problems |
SPDEs | Singularly Perturbed Differential Equations |
SPDDEs | Singularly Perturbed Delay Differential Equations |
FEM | Finite Element Method |
FDS | Finite Difference Scheme |
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Figure 1.
Visualization of numerical solutions of Case 1 (Example 1), illustrating initial layers at , interior layers at due to the delay terms, as well as at and due to discontinuities.
Figure 1.
Visualization of numerical solutions of Case 1 (Example 1), illustrating initial layers at , interior layers at due to the delay terms, as well as at and due to discontinuities.
Figure 2.
Visualization of numerical solutions of Case 2 (Example 1), illustrating initial layers at , interior layers at due to the delay terms, and at due to discontinuities.
Figure 2.
Visualization of numerical solutions of Case 2 (Example 1), illustrating initial layers at , interior layers at due to the delay terms, and at due to discontinuities.
Figure 3.
Visualization of numerical solutions of Case 1 (Example 2), illustrating initial layers at , interior layers at due to the delay terms, as well as at and due to discontinuities.
Figure 3.
Visualization of numerical solutions of Case 1 (Example 2), illustrating initial layers at , interior layers at due to the delay terms, as well as at and due to discontinuities.
Figure 4.
Visualization of numerical solutions of Case 2 (Example 2), illustrating initial layers at , interior layers at due to the delay terms, and at due to discontinuities.
Figure 4.
Visualization of numerical solutions of Case 2 (Example 2), illustrating initial layers at , interior layers at due to the delay terms, and at due to discontinuities.
Figure 5.
Log−log plot depicting the maximum point-wise errors corresponding to
Table 1 and
Table 3 for Example 1 (Case 1 and Case 2).
Figure 5.
Log−log plot depicting the maximum point-wise errors corresponding to
Table 1 and
Table 3 for Example 1 (Case 1 and Case 2).
Figure 6.
Log−log plot depicting the maximum point-wise errors corresponding to
Table 5 and
Table 7 for Example 2 (Case 1 and Case 2).
Figure 6.
Log−log plot depicting the maximum point-wise errors corresponding to
Table 5 and
Table 7 for Example 2 (Case 1 and Case 2).
Table 1.
For Case 1 (Example 1), values of and generated for .
Table 1.
For Case 1 (Example 1), values of and generated for .
| : Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
1 × | 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
1 × | 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
1 × | 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
1 × | 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
1 × | 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
| 1.0596 × | 6.7301 × | 4.0582 × | 2.3597 × |
| 6.5482 × | 7.2980 × | 7.8222 × | |
| 5.7685 | 5.7685 | 5.4764 | 5.0135 |
|
|
Table 2.
Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 1).
Table 2.
Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 1).
| N: Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
| 0.17 | 2.16 | 4.2 | 17.88 |
| 0.14 | 4.39 | 3.7 | 12.8 |
| 0.14 | 1.03 | 2.65 | 13 |
| 0.11 | 3 | 2.69 | 12.49 |
| 0.12 | 1.05 | 2.48 | 13.21 |
Table 3.
For Case 2 (Example 1), values of and generated for .
Table 3.
For Case 2 (Example 1), values of and generated for .
| : Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
1 | 8.2939 | 5.1802 | 3.0901 | 1.7851 |
1 | 8.2939 | 5.1802 | 3.0901 | 1.7851 |
1 | 8.2939 | 5.1802 | 3.0901 | 1.7851 |
1 | 8.2939 | 5.1802 | 3.0901 | 1.7851 |
1 | 8.2939 | 5.1802 | 3.0901 | 1.7851 |
| 8.2939 | 5.1802 | 3.0901 | 1.7851 |
| 6.7904 | 7.4538 | 7.9166 | |
| 4.9011 | 4.9011 | 4.6808 | 4.3293 |
|
|
Table 4.
Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 2).
Table 4.
Computation time in seconds for obtaining maximum point-wise errors of Example 1 (Case 2).
| N: Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
| 0.18 | 1.26 | 3.25 | 15.31 |
| 0.17 | 1.19 | 2.98 | 16.03 |
| 0.23 | 1.14 | 3.14 | 15.86 |
| 0.23 | 1.18 | 3.21 | 15.64 |
| 0.14 | 1.25 | 3.72 | 15.43 |
Table 5.
For Case 1 (Example 2), values of and generated for .
Table 5.
For Case 1 (Example 2), values of and generated for .
| : Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
1 | 5.8093 | 3.6669 | 2.2022 | 1.2773 |
1 | 5.8093 | 3.6669 | 2.2022 | 1.2773 |
1 | 5.8093 | 3.6669 | 2.2022 | 1.2773 |
1 | 5.8093 | 3.6669 | 2.2022 | 1.2773 |
1 | 5.8093 | 3.6669 | 2.2022 | 1.2773 |
| 5.8093 | 3.6669 | 2.2022 | 1.2773 |
| 6.6375 | 7.3564 | 7.8578 | |
| 3.259 | 3.259 | 3.1006 | 2.8491 |
|
|
Table 6.
Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 1).
Table 6.
Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 1).
| N: Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
| 0.17 | 1.67 | 3.91 | 28.18 |
| 0.14 | 5.15 | 4.24 | 16.62 |
| 0.18 | 1.41 | 3.89 | 17.75 |
| 0.12 | 4.79 | 4.03 | 18.58 |
| 0.27 | 1.19 | 8.79 | 33 |
Table 7.
For Case 2 (Example 2), values of and generated for .
Table 7.
For Case 2 (Example 2), values of and generated for .
| : Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
1 | 4.4403 | 2.7565 | 1.638 | 9.4407 |
1 | 4.4403 | 2.7565 | 1.638 | 9.4407 |
1 | 4.4403 | 2.7565 | 1.638 | 9.4407 |
1 | 4.4403 | 2.7565 | 1.638 | 9.4407 |
1 | 4.4403 | 2.7565 | 1.638 | 9.4407 |
| 4.4403 | 2.7565 | 1.638 | 9.4407 |
| 6.8773 | 7.5091 | 7.9498 | |
| 2.7028 | 2.7028 | 2.5870 | 2.4017 |
|
|
Table 8.
Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 2).
Table 8.
Computation time in seconds for obtaining maximum point-wise errors of Example 2 (Case 2).
| N: Number of Mesh Points |
---|
| 96 | 192 | 384 | 768 |
---|
| 0.19 | 1.28 | 3.91 | 16.98 |
| 0.25 | 1.25 | 3.81 | 17.11 |
| 0.31 | 2.46 | 6.97 | 32.2 |
| 0.31 | 2.21 | 6.93 | 28.63 |
| 0.15 | 1.34 | 3.62 | 15.55 |
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