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This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.
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Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction principle
Abstract This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in the time domain with a polynomial–exponential basis. This truncation effectively removes the time variable, transforming the problem into a system of quasi-linear elliptic equations. We refer to this technique as the ‘time dimensional reduction method.’ To numerically solve this system comprehensively without the need for an accurate initial estimate, we used the newly developed Carleman contraction principle. We show the efficiency of our method through various numerical examples. The time dimensional reduction method stands out not only for its precise solutions but also for its remarkable speed in computation.
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- Award ID(s):
- 2208159
- PAR ID:
- 10557187
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 40
- Issue:
- 12
- ISSN:
- 0266-5611
- Format(s):
- Medium: X Size: Article No. 125021
- Size(s):
- Article No. 125021
- Sponsoring Org:
- National Science Foundation
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