Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior
Various physical phenomena give rise to singularly perturbed differential equations with mixed shifts. Due to multiple parameters, singularly perturbed mixed delay boundary value problems are challenging to solve. This article considers a singularly perturbed differential-difference equation with de...
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Ram Arti Publishers
2025-10-01
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| Series: | International Journal of Mathematical, Engineering and Management Sciences |
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| Online Access: | https://www.ijmems.in/cms/storage/app/public/uploads/volumes/68-IJMEMS-24-0809-10-5-1447-1462-2025.pdf |
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| author | Shilpa Malge Ram Kishun Lodhi |
| author_facet | Shilpa Malge Ram Kishun Lodhi |
| author_sort | Shilpa Malge |
| collection | DOAJ |
| description | Various physical phenomena give rise to singularly perturbed differential equations with mixed shifts. Due to multiple parameters, singularly perturbed mixed delay boundary value problems are challenging to solve. This article considers a singularly perturbed differential-difference equation with delay and advance. To deal with the complexity of these equations, a non-polynomial spline numerical approach is adopted. For discretization, we have used a uniform mesh with equal spacing. Various theoretical results like stability and convergence are discussed. Numerical examples are solved to support the method and check the validity of the findings. The numerical order of convergence is determined and presented in the tables, along with the comparison of the results with the other existing methods. The comparison shows that the error is significantly less than the available solution in the literature. Also, the numerical order of convergence is determined to be equal to two. Graphs are drawn to observe the behavior of the solution for different values of parameters. |
| format | Article |
| id | doaj-art-b7404a9d3d244d0a9dc56b551a170f80 |
| institution | DOAJ |
| issn | 2455-7749 |
| language | English |
| publishDate | 2025-10-01 |
| publisher | Ram Arti Publishers |
| record_format | Article |
| series | International Journal of Mathematical, Engineering and Management Sciences |
| spelling | doaj-art-b7404a9d3d244d0a9dc56b551a170f802025-08-20T02:39:41ZengRam Arti PublishersInternational Journal of Mathematical, Engineering and Management Sciences2455-77492025-10-0110514471462https://doi.org/10.33889/IJMEMS.2025.10.5.068Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer BehaviorShilpa Malge 0Ram Kishun Lodhi1Department of Applied Sciences, Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Lavale, 412115, Pune, Maharashtra, India.Department of Applied Sciences, Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Lavale, 412115, Pune, Maharashtra, India.Various physical phenomena give rise to singularly perturbed differential equations with mixed shifts. Due to multiple parameters, singularly perturbed mixed delay boundary value problems are challenging to solve. This article considers a singularly perturbed differential-difference equation with delay and advance. To deal with the complexity of these equations, a non-polynomial spline numerical approach is adopted. For discretization, we have used a uniform mesh with equal spacing. Various theoretical results like stability and convergence are discussed. Numerical examples are solved to support the method and check the validity of the findings. The numerical order of convergence is determined and presented in the tables, along with the comparison of the results with the other existing methods. The comparison shows that the error is significantly less than the available solution in the literature. Also, the numerical order of convergence is determined to be equal to two. Graphs are drawn to observe the behavior of the solution for different values of parameters.https://www.ijmems.in/cms/storage/app/public/uploads/volumes/68-IJMEMS-24-0809-10-5-1447-1462-2025.pdfsingular perturbationdifferential-difference equationsboundary value problemsmixed delaynon-polynomial splineboundary layer |
| spellingShingle | Shilpa Malge Ram Kishun Lodhi Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior International Journal of Mathematical, Engineering and Management Sciences singular perturbation differential-difference equations boundary value problems mixed delay non-polynomial spline boundary layer |
| title | Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior |
| title_full | Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior |
| title_fullStr | Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior |
| title_full_unstemmed | Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior |
| title_short | Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior |
| title_sort | non polynomial spline for singularly perturbed differential difference equation with mixed shifts and layer behavior |
| topic | singular perturbation differential-difference equations boundary value problems mixed delay non-polynomial spline boundary layer |
| url | https://www.ijmems.in/cms/storage/app/public/uploads/volumes/68-IJMEMS-24-0809-10-5-1447-1462-2025.pdf |
| work_keys_str_mv | AT shilpamalge nonpolynomialsplineforsingularlyperturbeddifferentialdifferenceequationwithmixedshiftsandlayerbehavior AT ramkishunlodhi nonpolynomialsplineforsingularlyperturbeddifferentialdifferenceequationwithmixedshiftsandlayerbehavior |