^{1}

^{1}

^{*}

^{1}

^{2}

^{1}

^{2}

Edited by: Lijian Jiang, Tongji University, China

Reviewed by: Majid Darehmiraki, Behbahan Khatam Alanbia University of Technology, Iran; Khalid Hattaf, Centre Régional des Métiers de l'Education et de la Formation (CRMEF), Morocco

This article was submitted to Numerical Analysis and Scientific Computation, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at

Differential equations that involve a small parameter in their higher order derivative term are said to be singularly perturbed problems (SPPs) or singularly perturbed differential equations (SPDEs). Many mathematical models, starting from fluid dynamics to mathematical biology, are modeled using (SPPs). For example, high Reynold's number flow in fluid dynamics, heat transport problems with large Péclet numbers, elastic vibration, etc. [

In recent times, a class of SPPs involving nonlocal boundary conditions have been obtained great attention from scholars. To mention few of them, the authors in Bahuguna and Dabas [

In general, the classical numerical methods used for solving SPDEs are not well-posed and fail to provide an exact solution when a perturbation parameter (ε) goes to zero. Therefore, it is essential to develop a numerical method that offers suitable results for small values of the perturbation parameter. As far as the researchers' knowledge, singularly perturbed parabolic partial differential equations with nonlocal boundary conditions are first being considered and have not yet been treated numerically. In this study, we investigate a uniformly convergent numerical method for solving the problem under consideration. We used the nonstandard finite difference method for space direction and the implicit Euler method for time direction.

The contents of the paper are arranged in the following manner: A brief introduction of the given problem is discussed in Section 1. In Section 2, the properties of continuous solutions are given. In Section 3, a numerical method is formulated by using the method of lines for the given problem. Stability and convergence analysis for developed numerical methods are also studied. Numerical results and discussions are given in Section 4. In Section 5, the conclusion of the paper is given.

In this paper, we consider a class of singularly perturbed 1D parabolic partial differential equations of the reaction-diffusion type with non-local boundary conditions,

where _{l}, ϕ_{r}, ϕ_{b} are sufficiently smooth functions and _{l} and ϕ_{r} are only functions of _{b} is a function of _{b}(0, 0) = ϕ_{l}(0, 0), ϕ_{b}(1, 0) = ϕ_{r}(1, 0), and

Note that ϕ_{l}, ϕ_{r}, and ϕ_{b} are assumed to be sufficiently smooth for Equation (1) to make sense, namely

Next, we analyze some properties of the continuous solution (Equation 1) which guarantee the existence and uniqueness of the analytical solution. A replication of this belonging in semi-discrete form can be used to present the approximate solution, which we provide in the following section.

^{*}, ^{*}) be defined as ^{*}, ^{*}) ≤ 0. It is known that (^{*}, ^{*}) ∉ ∂^{*}, ^{*}) = 0,

where ||

For

For

For 0 <

where ε > 0,

The sufficient conditions for the existence of a unique solution is given in Lemma 3 and Theorem 1.

To estimate the error for the fitted numerical technique below, the idea that the solution of Equation (1) is more regular than the one guaranteed by using the result in Theorem 1. To attain this greater regularity, stronger compatibility conditions are imposed at the corners.

where

∀ Ñ_{δ} in _{δ}, δ > 0 is a neighborhood with diameter δ in

Hence, the proof is complete by using the bound on

The bounds of the derivatives of the solution given in Theorem 1 were derived from classical results. They are not adequate for the proof of the ε -uniform error estimate. Stronger bounds on these derivatives are now obtained by a method originally devised in Shishkin [

where

The fundamental idea of non-standard discrete modeling techniques is the development of the exact finite difference technique. Micken presented methods and rules for developing nonstandard FDMs for various types of problems [

To construct a genuine finite difference scheme for the problem of the form in Equation (1), we use the methods described in Woldaregay and Duressa [

By solving Equation (3), we obtain two independent solutions

The spatial domain [0, 1] is discretized on uniform mesh length Δ_{i}) will be denoted by _{i}. Here, the main aim is to compute difference equations that have similar results with the problem (Equation 1) at the mesh point _{i} which is given by

After simplification, Equation (4) becomes

which is an exact difference scheme for Equation (3). By performing some arithmetic manipulation and making rearrangement on Equation (5) for the variable coefficient problem, we obtain

The denominator function

where λ^{2} is a function of ε, β_{i},

For more information about nonstandard finite difference methods for reaction diffusion problems, an interested reader can refer to the study of Munyakazi and Patidar [

By using Equation (7), and applying the nonstandard finite difference method to a semi-discrete problem, we have

with boundary conditions

Here, for

Substituting Equation (10) in to Equation (9), we obtain

Equation (11) can be rewritten as

Assume that the approximation of _{i}, _{i}(

Equation (12) is the system of IVPs and its compact form is written as

where B is (_{i}(_{i}(

and

Here, we present the maximum principle and uniform stability estimate of the semi-discrete operator

_{0}(_{i}(

_{q}(_{q+1} − _{q} > 0 and _{q} − _{q−1} < 0. Here, we have

By using the above assumption, we get that _{i}(_{i}(

This Lemma 5 is used to obtain the bounds of the discrete solution given in Lemma 6. In general, the discrete maximum principle is widely used to show the boundedness and positivity of a discrete solution.

_{i}(

For the points on the boundary, we have

For 1 ≤

From Lemma 5, we get,

Next, we present the convergence analysis of spatial discretization. We denoted _{i}(_{i},

respectively, and the second order central finite difference operator as

where

_{i}(

Note that we have used Taylor expansions of _{i−1}(_{i+1}(

Using Equation (15) in Equation (14), we obtain

We use Lemma (7), to obtain the boundedness of Equation (16). Using Lemma (7) and Theorem (1), we obtain

The truncation error at _{N}, become

A mesh with length Δ_{j+1} − _{j},_{t} = [0, _{i}(_{j}) as

with the initial condition _{0}(_{l}(_{j}), and by rearranging Equation (19), we obtain

Using Taylor expansion, we obtain _{j−1}) as

However, _{t}(_{j}) = _{j}) − _{j})_{j}). Thus,

Now, the local truncation error _{j} becomes

Since the matrix ^{2} > (Δ^{3} for small Δ_{j}) ≤

□

_{j+1}|| ≤

^{th} time step is obtained by using the local error estimate up to ^{th} time step as follows.

Hence,

where

□

We summarizes the results of this work by considering the error estimate obtained in Equations (18) and (22) and we conclude by the following theorem.

where

Then, by combining the bound given in Theorem 3 and Lemma 9, the theorem gets proved.

Here, we developed an algorithm for the proposed method for the problem and perform experiments to validate the theoretical justifications and results. Since the exact solutions of the given examples are not known, we use double mesh techniques to obtain the maximum pointwise error of the developed scheme. Now, let ^{N,Δt} be a conducted solution of a problem using mesh points

We calculate the maximum absolute error as

The solutions of the above two examples exhibit strong boundary layers near

3-D graph of numerical solution for Example (1) which displays the existing layer. ^{−2}. ^{−12}.

3-D graph of numerical solution for Example (2) that displays the existing layer. ^{−2}. ^{−12}.

Maximum absolute error and rate of convergence of the scheme for Example (1).

^{2} |
^{3} |
^{4} |
|||
---|---|---|---|---|---|

10^{−6} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

10^{−8} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

10^{−10} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

10^{−12} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

10^{−14} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

10^{−20} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

1.8951 | 1.9645 | 1.9859 | 1.9938 | - | |

^{N,Δt} |
1.2294e-02 | 3.3054e-03 | 8.4694e-04 | 2.1381e-04 | 5.3681e-05 |

^{N,Δt} |
1.8951 | 1.9645 | 1.9859 | 1.9938 | - |

Maximum absolute error and rate of convergence of the scheme for Example (2).

^{2} |
^{3} |
^{4} |
|||
---|---|---|---|---|---|

10^{−6} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

1.5354 | 1.8919 | 1.9736 | 1.9935 | - | |

10^{−8} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

1.5354 | 1.8919 | 1.9736 | 1.9935 | - | |

10^{−10} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

1.5354 | 1.8919 | 1.9736 | 1.9935 | - | |

10^{−12} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

1.5354 | 1.8919 | 1.9736 | 1.9935 | - | |

10^{−14} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

10^{−20} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

1.5354 | 1.8919 | 1.9736 | 1.9935 | - | |

^{N,Δt} |
1.5809e-02 | 5.4540e-03 | 1.4696e-03 | 3.7419e-04 | 9.3970e-05 |

^{N,Δt} |
1.5354 | 1.8919 | 1.9736 | 1.9935 | - |

The Log-Log plot of the maximum absolute error for different values of ε for Examples 1 and 2, respectively.

This paper investigates a numerical treatment for a class of singularly perturbed parabolic partial differential equations of the reaction-diffusion type with nonlocal boundary conditions. To solve the problem at hand, we employed the method of lines. A nonstandard finite difference scheme is used to semi-discretize the spatial direction, and the implicit Euler method is used to discretize the results of initial value problems. To deal with the integral boundary condition, we used a composite Simpson's

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Authors are grateful to their reviewers for their contributions.