# Inverse Source Problem with Many Frequencies and Attenuation for One-Dimensional Domain

### Abstract

In this article, we consider the one-dimensional inverse source problem for Helmholtz equation with attenuation (damping) factor in a one layer medium. We establish a stability by using multiple frequencies at the two end points of the domain which contains the compact support of the source functions. The main result is an estimate which consists of two parts: the data discrepancy and the high frequency tail. We show that increasing stability possible using multi-frequency wave at the two endpoints.### Downloads

### References

Aralumallige S D, Isakov V 2010 Increasing stability of the continuation for the maxwell systemInverse Problems26074004, 14pp.

Ammari H, Bao G and Fleming J 2002 Inverse source problem for Maxwellâ€™s equation in magnetoencephalography SIAM J. Appl. Math.621369-82.

Balanis C 2005 Antenna Theory - Analysis and Design (Wiley, Hoboken, NJ).

Bao G, Lin J, Triki F 2010 A multi- frequency-inverse source problem J. Differential Equations2493443-3465.

Bao G, Lin J, Triki F 2011 An inverse source problem with multiple frequency data Comptes RendusMathematique349855-859.

Bao G, Lu S, Rundell W, and Xu B 2015 A recursive algorithm for multifrequency acoustic inverse source problemsSIAM Journal on Numerical Analysis53(3), 1608-1628.

Chen J, Fan D and Zhang C 2015 Estimate for damped fractional wave equations and applications, Electronic J. Differential Equations1621072-6691.

Cheng J, Isakov V and Lu S 2016 Increasing stability in the inverse source problem with many frequencies, J. Differential Equations2604786-4804.

Chen J, Fan D and Zhang C 2015 Estimate for damped fractional wave equations and applications, Electronic J. Differential Equations1621072-6691.

Entekhabi M N, 2018 Increasing stability in the two-dimensional inverse source scattering problem with attenuation and many frequencies Inverse Problems34115001.

Entekhabi M N, Isakov V 2017 On increasing stability in the two-dimensional inverse source scattering problem with many frequencies Inverse Problems34055005.

Entekhabi M N, Gunaratne A 2019 A logarithmic estimate for inverse source scattering problem with attenuation in a two-layered medium, arXiv: 1903.03475 [math. AP], To be appeared in ApplicableAnalysis.[13] Entekhabi M N, Isakov V 2018 Increasing stability in acoustic and elastic inverse source problems,arXiv: 1808.10528 [math. AP], To appear in SIAM J MATH ANAL.

Eller M and Valdivia N 2009 Acoustic source identification using multiple frequency information inverse Problems25115005.

Isakov V 2017 Inverse Problems for Partial Differential Equations ( Springer-Verlag, New York).

Isakov V, Kindermann S 2011 Regions of stability in the Cauchy problem for the Helmholtz equation methods Appl. Anal.181-30.

Isakov V, Lu S 2018 Increasing stability in the inverse source problem with attenuation and many frequencies To Be Appeared in SIAM J. Appl. Math181-18.

Isakov V, Lu S 2018 Inverse source problems without (pseudo)convexity assumptions Inverse Problems Imaging to be appeared.

John F 1982 Partial Differential Equations (Applied Mathematical Sciences, Springer-Verlag, NewYork, Berlin).[20] John F 1960 Continuous dependence on data for solutions of partial differential equations with a prescribed bound Comm. Pure Appl. Math.13551-587.

Kawashima S, Nakao M, Ono K 1995 On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term J. Math. Soc, Jap.47617-653.

Li P, Yuan G 2017 Increasing stability for the inverse source scattering problem with multi-frequencies Inverse Problems and Imaging11745-759.

Zhao Y, Li P 2017 Stability on the one-dimensional inverse source scattering problem in a two-layered medium, Applicable Analysis,98:4, 682-692, DOI: 10.1080/00036811.2017.1399365.

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