To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions
In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For...
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| Format: | Article |
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MDPI AG
2022-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/17/3112 |
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| author | Chein-Shan Liu Chih-Wen Chang Yung-Wei Chen Jian-Hung Shen |
| author_facet | Chein-Shan Liu Chih-Wen Chang Yung-Wei Chen Jian-Hung Shen |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For this reason, it is challenging to solve such nonlinear problems by conventional numerical methods. First, we derive a nonlocal boundary shape function (NLBSF) from incorporating the Pascal triangle as free functions; hence, the new, two-parameter Pascal bases are created to automatically satisfy the specified conditions for the solution. To satisfy the wave equation in the domain by the collocation method, the solution of the forward nonlocal wave problem can be quickly obtained with high precision. For the backward nonlocal wave problem, we construct the corresponding NLBSF and Pascal bases, which exactly implement two final time conditions, a left-boundary condition and a nonlocal boundary condition; in addition, the numerical method for the backward nonlocal wave problem under two-side, nonlocal boundary conditions is also developed. Nine numerical examples, including forward and backward problems, are tested, demonstrating that this scheme is more effective and stable. Even for boundary conditions with a large noise at final time, the solution recovered in the entire domain for the backward nonlocal wave problem is accurate and stable. The accuracy and efficiency of the method are validated by comparing the estimation results with the existing literature. |
| first_indexed | 2024-03-10T01:32:56Z |
| format | Article |
| id | doaj.art-c2f28e44d9ac4ba693ebb113f976d1d4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:32:56Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c2f28e44d9ac4ba693ebb113f976d1d42023-11-23T13:38:36ZengMDPI AGMathematics2227-73902022-08-011017311210.3390/math10173112To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified ConditionsChein-Shan Liu0Chih-Wen Chang1Yung-Wei Chen2Jian-Hung Shen3Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Mechanical Engineering, National United University, Miaoli 360301, TaiwanDepartment of Marine Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Marine Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanIn this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For this reason, it is challenging to solve such nonlinear problems by conventional numerical methods. First, we derive a nonlocal boundary shape function (NLBSF) from incorporating the Pascal triangle as free functions; hence, the new, two-parameter Pascal bases are created to automatically satisfy the specified conditions for the solution. To satisfy the wave equation in the domain by the collocation method, the solution of the forward nonlocal wave problem can be quickly obtained with high precision. For the backward nonlocal wave problem, we construct the corresponding NLBSF and Pascal bases, which exactly implement two final time conditions, a left-boundary condition and a nonlocal boundary condition; in addition, the numerical method for the backward nonlocal wave problem under two-side, nonlocal boundary conditions is also developed. Nine numerical examples, including forward and backward problems, are tested, demonstrating that this scheme is more effective and stable. Even for boundary conditions with a large noise at final time, the solution recovered in the entire domain for the backward nonlocal wave problem is accurate and stable. The accuracy and efficiency of the method are validated by comparing the estimation results with the existing literature.https://www.mdpi.com/2227-7390/10/17/3112backward nonlocal wave equationPascal bases automatically satisfying specified conditionsintegral boundary conditionnonlocal boundary shape function |
| spellingShingle | Chein-Shan Liu Chih-Wen Chang Yung-Wei Chen Jian-Hung Shen To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions Mathematics backward nonlocal wave equation Pascal bases automatically satisfying specified conditions integral boundary condition nonlocal boundary shape function |
| title | To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions |
| title_full | To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions |
| title_fullStr | To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions |
| title_full_unstemmed | To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions |
| title_short | To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions |
| title_sort | to solve forward and backward nonlocal wave problems with pascal bases automatically satisfying the specified conditions |
| topic | backward nonlocal wave equation Pascal bases automatically satisfying specified conditions integral boundary condition nonlocal boundary shape function |
| url | https://www.mdpi.com/2227-7390/10/17/3112 |
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