Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation
The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2022/4628634 |
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| author | Benedict Barnes Anthony Y. Aidoo Joseph Ackora-Prah |
| author_facet | Benedict Barnes Anthony Y. Aidoo Joseph Ackora-Prah |
| author_sort | Benedict Barnes |
| collection | DOAJ |
| description | The ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term 1+α2mem, where α>0 is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness. |
| format | Article |
| id | doaj-art-4dbbfbcd058442a899159316a624302e |
| institution | Kabale University |
| issn | 1687-0409 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-4dbbfbcd058442a899159316a624302e2025-02-03T05:58:56ZengWileyAbstract and Applied Analysis1687-04092022-01-01202210.1155/2022/4628634Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz EquationBenedict Barnes0Anthony Y. Aidoo1Joseph Ackora-Prah2Kwame Nkrumah University of Science and TechnologyEastern Connecticut State UniversityKwame Nkrumah University of Science and TechnologyThe ill-posed Helmholtz equation with inhomogeneous boundary deflection in a Hilbert space is regularized using the divergence regularization method (DRM). The DRM includes a positive integer scaler that homogenizes the inhomogeneous boundary deflection in the Helmholtz equation’s Cauchy issue. This guarantees the existence and uniqueness of the equation’s solution. To reestablish the stability of the regularized Helmholtz equation and regularized Cauchy boundary conditions, the DRM uses its regularization term 1+α2mem, where α>0 is the regularization parameter. As a result, DRM restores all three Hadamard requirements for well-posedness.http://dx.doi.org/10.1155/2022/4628634 |
| spellingShingle | Benedict Barnes Anthony Y. Aidoo Joseph Ackora-Prah Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation Abstract and Applied Analysis |
| title | Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation |
| title_full | Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation |
| title_fullStr | Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation |
| title_full_unstemmed | Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation |
| title_short | Using a Divergence Regularization Method to Solve an Ill-Posed Cauchy Problem for the Helmholtz Equation |
| title_sort | using a divergence regularization method to solve an ill posed cauchy problem for the helmholtz equation |
| url | http://dx.doi.org/10.1155/2022/4628634 |
| work_keys_str_mv | AT benedictbarnes usingadivergenceregularizationmethodtosolveanillposedcauchyproblemforthehelmholtzequation AT anthonyyaidoo usingadivergenceregularizationmethodtosolveanillposedcauchyproblemforthehelmholtzequation AT josephackoraprah usingadivergenceregularizationmethodtosolveanillposedcauchyproblemforthehelmholtzequation |