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Solving Nonlinear Elliptic Inverse Source, Coefficient and Conductivity Problems by the Methods with Bases Satisfying the Boundary Conditions Automatically

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Abstract

In the paper, the inverse source, coefficient and conductivity problems of nonlinear elliptic partial differential equations in 2D and 3D regular domains are resolved. For this purpose, two types single-parameter bases that automatically meet the boundary conditions are deduced, namely the first and second bases satisfying boundary conditions method (BSBCM). We solve a linear algebraic equations system which satisfies the over-specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source and coefficient functions can be determined quickly. We also develop linear system methods to identify the coefficient and conductivity functions. It is remarkable that we do not need extra boundary data of the conductivity function to resolve the inverse conductivity problem. The present novel methods are verified to be accurate, effective, and robust, which are without solving nonlinear equations and executing iterations, and additional data used are quite economical.

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Authors and Affiliations

  1. Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 202301, Taiwan

    Chein-Shan Liu

  2. College of Mechanical and Electrical Engineering, National Engineering Research Center for Intelligent Electrical Vehicle Power System, Qingdao University, Qingdao, 266071, China

    Lin Qiu

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  1. Chein-Shan Liu
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Liu, CS., Qiu, L. Solving Nonlinear Elliptic Inverse Source, Coefficient and Conductivity Problems by the Methods with Bases Satisfying the Boundary Conditions Automatically. J Sci Comput 95, 42 (2023). https://doi.org/10.1007/s10915-023-02167-9

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  • DOI: https://doi.org/10.1007/s10915-023-02167-9

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