Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings
The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference...
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MDPI AG
2025-02-01
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| Series: | AppliedMath |
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| author | Md Joni Alam Ahmed Ramady M. S. Abbas K. El-Rashidy Md Tauhedul Azam M. Mamun Miah |
| author_facet | Md Joni Alam Ahmed Ramady M. S. Abbas K. El-Rashidy Md Tauhedul Azam M. Mamun Miah |
| author_sort | Md Joni Alam |
| collection | DOAJ |
| description | The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula> norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme. |
| format | Article |
| id | doaj-art-906028cb08364f7bad752109e466cfd7 |
| institution | DOAJ |
| issn | 2673-9909 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
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| series | AppliedMath |
| spelling | doaj-art-906028cb08364f7bad752109e466cfd72025-08-20T02:42:41ZengMDPI AGAppliedMath2673-99092025-02-01511810.3390/appliedmath5010018Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating StringsMd Joni Alam0Ahmed Ramady1M. S. Abbas2K. El-Rashidy3Md Tauhedul Azam4M. Mamun Miah5Department of Mathematics, Comilla University, Cumilla 3506, BangladeshGRC Department, The Applied College, King Abdulaziz University, Jeddah 21589, Saudi ArabiaAdministrative and Financial Science Department, Ranyah University College, Taif University, Taif 21944, Saudi ArabiaDepartment of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 2722165, EgyptDepartment of Mathematics, Jahangirnagar University, Savar 1342, BangladeshDepartment of Mathematics, Khulna University of Engineering and Technology, Khulna 9203, BangladeshThe modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula> norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme.https://www.mdpi.com/2673-9909/5/1/18wave equationfinite difference methodstability analysisconvergence analysisnumerical analysis |
| spellingShingle | Md Joni Alam Ahmed Ramady M. S. Abbas K. El-Rashidy Md Tauhedul Azam M. Mamun Miah Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings AppliedMath wave equation finite difference method stability analysis convergence analysis numerical analysis |
| title | Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings |
| title_full | Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings |
| title_fullStr | Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings |
| title_full_unstemmed | Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings |
| title_short | Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings |
| title_sort | numerical investigation of the wave equation for the convergence and stability analysis of vibrating strings |
| topic | wave equation finite difference method stability analysis convergence analysis numerical analysis |
| url | https://www.mdpi.com/2673-9909/5/1/18 |
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