Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique
This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal deri...
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/13/15/2391 |
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| author | Khawaja Shams Ul Haq Muhammad Asif Muhammad Faheem Ioan-Lucian Popa |
| author_facet | Khawaja Shams Ul Haq Muhammad Asif Muhammad Faheem Ioan-Lucian Popa |
| author_sort | Khawaja Shams Ul Haq |
| collection | DOAJ |
| description | This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision. |
| format | Article |
| id | doaj-art-4f2d36d6776a40268fb4457e731bd0a6 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-4f2d36d6776a40268fb4457e731bd0a62025-08-20T03:36:31ZengMDPI AGMathematics2227-73902025-07-011315239110.3390/math13152391Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution TechniqueKhawaja Shams Ul Haq0Muhammad Asif1Muhammad Faheem2Ioan-Lucian Popa3Department of Mathematics, University of Peshawar, Peshawar 25120, PakistanDepartment of Mathematics, University of Peshawar, Peshawar 25120, PakistanHigher Education Department, Govt. Degree College Badaber, Peshawar 25000, PakistanDepartment of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, RomaniaThis study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision.https://www.mdpi.com/2227-7390/13/15/2391Haar wavelettelegraph interface modelpartial differential equationshyperbolic equations |
| spellingShingle | Khawaja Shams Ul Haq Muhammad Asif Muhammad Faheem Ioan-Lucian Popa Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique Mathematics Haar wavelet telegraph interface model partial differential equations hyperbolic equations |
| title | Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique |
| title_full | Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique |
| title_fullStr | Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique |
| title_full_unstemmed | Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique |
| title_short | Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique |
| title_sort | capturing discontinuities with precision a numerical exploration of 3d telegraph interface models via multi resolution technique |
| topic | Haar wavelet telegraph interface model partial differential equations hyperbolic equations |
| url | https://www.mdpi.com/2227-7390/13/15/2391 |
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