Abstract
The establishment of relevant model and solving an inverse random source problem are one of the main tools for analyzing mechanical properties of elastic materials. In this paper, we study an inverse random source problem for biharmonic equation in two dimension. Under some regularity assumptions on the structure of random source, the well-posedness of the forward problem is established. Moreover, based on the explicit solution of the forward problem, we can solve the corresponding inverse random source problem via two transformed integral equations. Numerical examples are presented to illustrate the validity and effectiveness of the proposed inversion method.
Highlights
Motivated by significant scientific and industrial applications, the field of inverse problems has undergone huge growth in the past few decades
Mlinar [15] reviewed several inverse methods on quantifying these mechanical properties which could be utilized in designing semiconductors. These models and methods made a great success for one dimensional materials. to our best knowledge, little is known on for higher dimensional nanomaterials. We will investigate both the direct and inverse random source problems for biharmonic equation which describes the elastic deformation of twodimensional materials
For the forward problem, constructing a sequence of regular processes to approach the colored noise, we show the existence of a unique mild solution to the two dimensional biharmonic equation
Summary
Motivated by significant scientific and industrial applications, the field of inverse problems has undergone huge growth in the past few decades. We will focus on an inverse random source problems on quantifying mechanical properties for two dimensional materials which can be viewed as an extension of Bao et al [5, 6]. Mlinar [15] reviewed several inverse methods on quantifying these mechanical properties which could be utilized in designing semiconductors These models and methods made a great success for one dimensional materials. We will investigate both the direct and inverse random source problems for biharmonic equation which describes the elastic deformation of twodimensional materials. We introduce the biharmonic equation in two dimensions and discuss the solutions of the problem under the circumstances of deterministic and stochastic direct source.
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