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25 pages, 2808 KB

Open AccessArticle

A Non-Standard Finite Difference Scheme for Time-Fractional Singularly Perturbed Convection–Diffusion Problems

by Pramod Chakravarthy Podila, Rahul Mishra and Higinio Ramos

Fractal Fract. 2025, 9(6), 333; https://doi.org/10.3390/fractalfract9060333 - 23 May 2025

Viewed by 505

This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be [...] Read more.

This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be easily extended to a Shishkin mesh in the spatial domain. We discuss error estimates to demonstrate the convergence of the numerical scheme. Additionally, various numerical examples are presented to illustrate the behavior of the solution for different values of the perturbation parameter

ϵ

and the order of the fractional derivative. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems, Second Edition)

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25 pages, 1055 KB

Open AccessArticle

A Layer-Adapted Numerical Method for Singularly Perturbed Partial Functional-Differential Equations

by Ahmed A. Al Ghafli, Fasika Wondimu Gelu and Hassan J. Al Salman

Axioms 2025, 14(5), 362; https://doi.org/10.3390/axioms14050362 - 12 May 2025

Viewed by 397

Abstract

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then [...] Read more.

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then discretized with the Crank–Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method’s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov–Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov–Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov–Shishkin mesh performs well for singularly perturbed problems without small negative shifts. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Related Topics)

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14 pages, 351 KB

Open AccessArticle

Efficient Numerical Methods for Reaction–Diffusion Problems Governed by Singularly Perturbed Fredholm Integro-Differential Equations

by Sekar Elango, Lolugu Govindarao, Muath Awadalla and Hajer Zaway

Mathematics 2025, 13(9), 1511; https://doi.org/10.3390/math13091511 - 4 May 2025

Cited by 2 | Viewed by 642

Abstract

A set of singularly perturbed systems comprising Fredholm integro-differential equations associated with reaction–diffusion problems is considered. To approximate solutions to these systems, a second-order scheme for the derivatives and the trapezoidal rule for the integral terms are utilized. The discretization is performed on [...] Read more.

A set of singularly perturbed systems comprising Fredholm integro-differential equations associated with reaction–diffusion problems is considered. To approximate solutions to these systems, a second-order scheme for the derivatives and the trapezoidal rule for the integral terms are utilized. The discretization is performed on non-standard grids known as Shishkin-type meshes. The numerical method demonstrates a second-order rate of convergence with respect to small parameters in the equations, and error estimates are derived in the discrete maximum norm. Numerical experiments are conducted to verify the theoretical results. Full article

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30 pages, 555 KB

Open AccessArticle

Efficient Layer-Resolving Fitted Mesh Finite Difference Approach for Solving a System of n Two-Parameter Singularly Perturbed Convection–Diffusion Delay Differential Equations

by Joseph Paramasivam Mathiyazhagan, Jenolin Arthur, George E. Chatzarakis and S. L. Panetsos

Axioms 2025, 14(4), 246; https://doi.org/10.3390/axioms14040246 - 24 Mar 2025

Viewed by 267

Abstract

This paper presents a robust fitted mesh finite difference method for solving a system of n singularly perturbed two parameter convection–reaction–diffusion delay differential equations defined on the interval

[0, 2]

. Leveraging a piecewise uniform Shishkin mesh, the method adeptly [...] Read more.

This paper presents a robust fitted mesh finite difference method for solving a system of n singularly perturbed two parameter convection–reaction–diffusion delay differential equations defined on the interval

[0, 2]

. Leveraging a piecewise uniform Shishkin mesh, the method adeptly captures the solution’s behavior arising from delay term and small perturbation parameters. The proposed numerical scheme is rigorously analyzed and proven to be parameter-robust, exhibiting nearly first-order convergence. A numerical illustration is included to validate the method’s efficiency and to confirm the theoretical predictions. Full article

(This article belongs to the Section Mathematical Analysis)

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25 pages, 489 KB

Open AccessArticle

Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System

by Jenolin Arthur, Joseph Paramasivam Mathiyazhagan, George E. Chatzarakis and S. L. Panetsos

Axioms 2025, 14(3), 171; https://doi.org/10.3390/axioms14030171 - 26 Feb 2025

Cited by 1 | Viewed by 521

Abstract

This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of n singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval

[0, 1]

, this system exhibits boundary layers [...] Read more.

This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of n singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval

[0, 1]

, this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solution’s behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers. Full article

(This article belongs to the Section Mathematical Analysis)

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27 pages, 1915 KB

Open AccessArticle

A Computational Study on Two-Parameter Singularly Perturbed Third-Order Delay Differential Equations

by Mahendran Rajendran, Senthilkumar Sethurathinam, Subburayan Veerasamy and Ravi P. Agarwal

Computation 2025, 13(2), 24; https://doi.org/10.3390/computation13020024 - 23 Jan 2025

Cited by 1 | Viewed by 906

Abstract

A class of third-order singularly perturbed two-parameter delay differential equations of boundary value problems is studied in this paper. Regular and singular components are used to estimate the solution’s a priori bounds and derivatives. A fitted finite-difference method is constructed to solve the [...] Read more.

A class of third-order singularly perturbed two-parameter delay differential equations of boundary value problems is studied in this paper. Regular and singular components are used to estimate the solution’s a priori bounds and derivatives. A fitted finite-difference method is constructed to solve the problem on a Shishkin mesh. The numerical solution converges uniformly to the exact solution; it is validated via numerical test problems. The order of convergence of the numerical method is almost first-order, which is independent of the parameters

ε

and

μ

. Full article

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12 pages, 324 KB

Open AccessArticle

A Quintic Spline-Based Computational Method for Solving Singularly Perturbed Periodic Boundary Value Problems

by Puvaneswari Arumugam, Valanarasu Thynesh, Chandru Muthusamy and Higinio Ramos

Axioms 2025, 14(1), 73; https://doi.org/10.3390/axioms14010073 - 20 Jan 2025

Cited by 1 | Viewed by 1043

Abstract

This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in [...] Read more.

This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in the maximum norm. Numerical examples are presented to validate the theoretical estimates. Additionally, the standard hybrid finite difference scheme, a cubic spline scheme, and the proposed method are compared to demonstrate the effectiveness of the present approach. Full article

(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)

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36 pages, 448 KB

Open AccessArticle

A Robust-Fitted-Mesh-Based Finite Difference Approach for Solving a System of Singularly Perturbed Convection–Diffusion Delay Differential Equations with Two Parameters

by Jenolin Arthur, George E. Chatzarakis, S. L. Panetsos and Joseph Paramasivam Mathiyazhagan

Symmetry 2025, 17(1), 68; https://doi.org/10.3390/sym17010068 - 3 Jan 2025

Cited by 2 | Viewed by 689

Abstract

This paper presents a robust fitted mesh finite difference method for solving a dynamical system of two parameter convection–reaction–diffusion delay differential equations defined on the interval

[0, 2]

. The method incorporates a piecewise uniform Shishkin mesh to accurately resolve [...] Read more.

This paper presents a robust fitted mesh finite difference method for solving a dynamical system of two parameter convection–reaction–diffusion delay differential equations defined on the interval

[0, 2]

. The method incorporates a piecewise uniform Shishkin mesh to accurately resolve the solution behavior caused by small perturbation parameters and delay terms. The proposed numerical scheme is proven to be parameter-robust and achieves almost first-order convergence. Numerical illustrations are provided to showcase the method’s effectiveness, highlighting its capability to address boundary and interior layers with improved accuracy. The results, supported by symmetrical considerations in the figures, enhance the precision and serve as validation for the theoretical results. Full article

(This article belongs to the Section Mathematics)

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17 pages, 3557 KB

Open AccessArticle

Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction–Diffusion System

by Manikandan Mariappan, Chandru Muthusamy and Higinio Ramos

Mathematics 2023, 11(12), 2685; https://doi.org/10.3390/math11122685 - 13 Jun 2023

Cited by 1 | Viewed by 1336

Abstract

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with

m < n

) contain a perturbation parameter while the rest do not contain [...] Read more.

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with

m < n

) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter

ε \overset{}{\to} 0

, proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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17 pages, 1630 KB

Open AccessArticle

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

by R. Soundararajan, V. Subburayan and Patricia J. Y. Wong

Mathematics 2023, 11(9), 2034; https://doi.org/10.3390/math11092034 - 25 Apr 2023

Cited by 3 | Viewed by 2168

Abstract

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) [...] Read more.

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations)

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14 pages, 333 KB

Open AccessArticle

Numerical Scheme for Singularly Perturbed Mixed Delay Differential Equation on Shishkin Type Meshes

by Sekar Elango and Bundit Unyong

Fractal Fract. 2023, 7(1), 43; https://doi.org/10.3390/fractalfract7010043 - 30 Dec 2022

Cited by 6 | Viewed by 1964

Abstract

Two non-uniform meshes used as part of the finite difference method to resolve singularly perturbed mixed-delay differential equations are studied in this article. The second-order derivative is multiplied by a small parameter which gives rise to boundary layers at

x = 0

and [...] Read more.

Two non-uniform meshes used as part of the finite difference method to resolve singularly perturbed mixed-delay differential equations are studied in this article. The second-order derivative is multiplied by a small parameter which gives rise to boundary layers at

x = 0

and

x = 3

and strong interior layers at

x = 1

and

x = 2

due to the delay terms. We prove that this method is almost first-order convergent on Shishkin mesh and is first-order convergent on Bakhvalov–Shishkin mesh. Error estimates are derived in the discrete maximum norm. Some examples are provided to validate the theoretical result. Full article

(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)

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26 pages, 460 KB

Open AccessArticle

Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems

by Yanjie Mei, Sulei Wang, Zhijie Xu, Chuanjing Song and Yao Cheng

Symmetry 2021, 13(12), 2291; https://doi.org/10.3390/sym13122291 - 1 Dec 2021

Viewed by 1683

Abstract

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive [...] Read more.

We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted

L^{2}

projection as well as an optimal convergence of order

k + 1

for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented. Full article

(This article belongs to the Section Mathematics)

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