Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation

The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In...

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Published in:Mathematics (Basel) 2022-01, Vol.10 (2), p.241
Main Authors: Yang, Judy P., Li, Hsiang-Ming
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Collocation methods
fourth-order PDE
gradient approximation
heat source
Inverse problems
Kernels
Mathematical analysis
Mathematics
Numerical analysis
Partial differential equations
Poisson equation
Regularization methods
reproducing kernel approximation
Shape functions
weighted collocation method
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description The weighted gradient reproducing kernel collocation method is introduced to recover the heat source described by Poisson’s equation. As it is commonly known that there is no unique solution to the inverse heat source problem, the weak solution based on a priori assumptions is considered herein. In view of the fourth-order partial differential equation (PDE) in the mathematical model, the high-order gradient reproducing kernel approximation is introduced to efficiently untangle the problem without calculating the high-order derivatives of reproducing kernel shape functions. The weights of the weighted collocation method for high-order inverse analysis are first determined. In the benchmark analysis, the unclear illustration in the literature is clarified, and the correct interpretation of numerical results is given particularly. Two mathematical formulations with four examples are provided to demonstrate the viability of the method, including the extreme cases of the limited accessible boundary.
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subjects Approximation
Boundary conditions
Collocation methods
fourth-order PDE
gradient approximation
heat source
Inverse problems
Kernels
Mathematical analysis
Mathematics
Numerical analysis
Partial differential equations
Poisson equation
Regularization methods
reproducing kernel approximation
Shape functions
weighted collocation method
title Recovering Heat Source from Fourth-Order Inverse Problems by Weighted Gradient Collocation
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